PRECALCULUS Mathematics for Calculus FIFTH EDITION
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PRECALCULUS Mathematics for Calculus FIFTH EDITION
James Stewart McMaster University
Lothar Redlin The Pennsylvania State University
Saleem Watson California State University, Long Beach
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Precalculus: Mathematics for Calculus, Fifth Edition, Enhanced WebAssign Edition James Stewart, Lothar Redlin, Saleem Watson Acquisitions Editor: Gary Whalen Assistant Editor: Natasha Coats
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To our students, from whom we have learned so much.
About the Cover The art on the cover was created by Bill Ralph, a mathematician who uses modern mathematics to produce visual representations of “dynamical systems.” Examples of dynamical systems in nature include the weather, blood pressure, the motions of the planets, and other phenomena that involve continual change. Such systems, which tend to be unpredictable and even chaotic at times, are modeled mathematically using the concepts of composition and iteration of functions (see Section 2.7 and the Discovery Project on pages 223–224). The basic idea is to start with a particular function and evaluate it at some point in its domain, yielding a new number. The function is then evaluated at the new number. Repeating this process produces a sequence of numbers called iterates of the function. The original domain is “painted” by assigning a color to each starting point; the color is determined by certain properties of its sequence of iterates and the mathematical concept of “dimension.” The result is a picture that reveals the complex patterns of the dynamical system. In a sense, these pictures allow us to look, through the lens of mathematics, at exotic little universes that have never been seen before. Professor Ralph teaches at Brock University in Canada. He can be contacted by email at [emailprotected].
About the Authors James Stewart was educated at the University of Toronto and Stanford University, did research at the University of London, and now teaches at McMaster University. His research ﬁeld is harmonic analysis.
He is the author of a bestselling calculus textbook series published by Brooks/Cole, including Calculus, 5th Ed., Calculus: Early Transcendentals, 5th Ed., and Calculus: Concepts and Contexts, 3rd Ed., as well as a series of highschool mathematics textbooks.
Lothar Redlin grew up on Vancouver Island, received a Bachelor of Science degree from the University of Victoria, and a Ph.D. from McMaster University in 1978. He subsequently did research and taught at the University of Washington, the University of Waterloo, and California State University, Long Beach. He is currently Professor of Mathematics at The Pennsylvania State University, Abington College. His research ﬁeld is topology.
Saleem Watson received his Bachelor of Science degree from Andrews University in Michigan. He did graduate studies at Dalhousie University and McMaster University, where he received his Ph.D. in 1978. He subsequently did research at the Mathematics Institute of the University of Warsaw in Poland. He also taught at The Pennsylvania State University. He is currently Professor of Mathematics at California State University, Long Beach. His research ﬁeld is functional analysis.
The authors have also published College Algebra, Fourth Edition (Brooks/Cole, 2004), Algebra and Trigonometry, Second Edition (Brooks/Cole, 2007), and Trigonometry (Brooks/Cole, 2003).
Contents
Preface xiii To the Student xxi Calculators and Calculations
1
Fundamentals ■
1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11
2
2.1 2.2
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Chapter Overview 1 Real Numbers 2 Exponents and Radicals 12 Algebraic Expressions 24 ● DISCOVERY PROJECT Visualizing a Formula 34 Rational Expressions 35 Equations 44 Modeling with Equations 58 ● DISCOVERY PROJECT Equations through the Ages 75 Inequalities 76 Coordinate Geometry 87 Graphing Calculators; Solving Equations and Inequalities Graphically 101 Lines 111 Modeling Variation 123 Chapter 1 Review 130 Chapter 1 Test 135 ■ FOCUS ON PROBLEM SOLVING General Principles 138
Functions ■
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Chapter Overview 147 What is a Function? 148 Graphs of Functions 158 ● DISCOVERY PROJECT Relations and Functions
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2.3 2.4 2.5 2.6 2.7 2.8
3
Polynomial and Rational Functions ■
3.1 3.2 3.3 3.4 3.5 3.6
4
Increasing and Decreasing Functions; Average Rate of Change 173 Transformations of Functions 182 Quadratic Functions; Maxima and Minima 193 Modeling with Functions 203 Combining Functions 214 ● DISCOVERY PROJECT Iteration and Chaos 223 OnetoOne Functions and Their Inverses 225 Chapter 2 Review 233 Chapter 2 Test 237 ■ FOCUS ON MODELING Fitting Lines to Data 239
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Chapter Overview 249 Polynomial Functions and Their Graphs 250 Dividing Polynomials 265 Real Zeros of Polynomials 272 ● DISCOVERY PROJECT Zeroing in on a Zero 283 Complex Numbers 285 Complex Zeros and the Fundamental Theorem of Algebra 291 Rational Functions 299 Chapter 3 Review 316 Chapter 3 Test 319 ■ FOCUS ON MODELING Fitting Polynomial Curves to Data 320
Exponential and Logarithmic 326 Functions ■
4.1 4.2 4.3 4.4 4.5
Chapter Overview 327 Exponential Functions 328 ● DISCOVERY PROJECT Exponential Explosion 341 Logarithmic Functions 342 Laws of Logarithms 352 Exponential and Logarithmic Equations 358 Modeling with Exponential and Logarithmic Functions 369 Chapter 4 Review 382 Chapter 4 Test 385 ■ FOCUS ON MODELING Fitting Exponential and Power Curves to Data 386
Contents
5
Trigonometric Functions of Real 398 Numbers ■
5.1 5.2 5.3 5.4 5.5
6
Trigonometric Functions of Angles ■
6.1 6.2 6.3 6.4 6.5
7
Chapter Overview 399 The Unit Circle 400 Trigonometric Functions of Real Numbers 408 Trigonometric Graphs 418 ● DISCOVERY PROJECT Predator/Prey Models 432 More Trigonometric Graphs 434 Modeling Harmonic Motion 442 Chapter 5 Review 454 Chapter 5 Test 458 ■ FOCUS ON MODELING Fitting Sinusoidal Curves to Data 459
Chapter Overview 467 Angle Measure 468 Trigonometry of Right Triangles 478 Trigonometric Functions of Angles 488 ● DISCOVERY PROJECT Similarity 499 The Law of Sines 501 The Law of Cosines 508 Chapter 6 Review 516 Chapter 6 Test 520 ■ FOCUS ON MODELING Surveying 522
Analytic Trigonometry ■
7.1 7.2 7.3 7.4 7.5
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526
Chapter Overview 527 Trigonometric Identities 528 Addition and Subtraction Formulas 535 DoubleAngle, HalfAngle, and SumProduct Formulas 541 Inverse Trigonometric Functions 550 ● DISCOVERY PROJECT Where to Sit at the Movies Trigonometric Equations 561 Chapter 7 Review 571 Chapter 7 Test 574 ■ FOCUS ON MODELING Traveling and Standing Waves 575
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8
Polar Coordinates and Vectors ■
8.1 8.2 8.3
8.4 8.5
9
580
Chapter Overview 581 Polar Coordinates 582 Graphs of Polar Equations 587 Polar Form of Complex Numbers; DeMoivre’s Theorem 596 ● DISCOVERY PROJECT Fractals 605 Vectors 607 The Dot Product 617 ● DISCOVERY PROJECT Sailing Against the Wind 626 Chapter 8 Review 627 Chapter 8 Test 629 ■ FOCUS ON MODELING Mapping the World 630
Systems of Equations and 634 Inequalities ■
9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9
Chapter Overview 635 Systems of Equations 636 Systems of Linear Equations in Two Variables 644 Systems of Linear Equations in Several Variables 651 ● DISCOVERY PROJECT Best Fit versus Exact Fit 660 Systems of Linear Equations: Matrices 662 The Algebra of Matrices 675 ● DISCOVERY PROJECT Will the Species Survive? 688 Inverses of Matrices and Matrix Equations 689 ● DISCOVERY PROJECT Computer Graphics I 700 Determinants and Cramer’s Rule 704 Partial Fractions 715 Systems of Inequalities 721 Chapter 9 Review 728 Chapter 9 Test 733 ■ FOCUS ON MODELING Linear Programming 735
10 Analytic Geometry ■
10.1 10.2 10.3
Chapter Overview Parabolas 744 Ellipses 753 Hyperbolas 762
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● DISCOVERY PROJECT
10.4 10.5 10.6 10.7
Conics in Architecture 771 Shifted Conics 775 Rotation of Axes 783 ● DISCOVERY PROJECT Computer Graphics II 792 Polar Equations of Conics 795 Plane Curves and Parametric Equations 801 Chapter 10 Review 810 Chapter 10 Test 814 ■ FOCUS ON MODELING The Path of a Projectile 816
11 Sequences and Series ■
11.1 11.2 11.3 11.4 11.5 11.6
820
Chapter Overview 821 Sequences and Summation Notation 822 Arithmetic Sequences 833 Geometric Sequences 838 ● DISCOVERY PROJECT Finding Patterns 847 Mathematics of Finance 848 Mathematical Induction 854 The Binomial Theorem 860 Chapter 11 Review 870 Chapter 11 Test 873 ■ FOCUS ON MODELING Modeling with Recursive Sequences 874
12 Limits: A Preview of Calculus ■
12.1 12.2 12.3 12.4 12.5
880
Chapter Overview 881 Finding Limits Numerically and Graphically 882 Finding Limits Algebraically 890 Tangent Lines and Derivatives 898 ● DISCOVERY PROJECT Designing a Roller Coaster 908 Limits at Inﬁnity: Limits of Sequences 908 Areas 916 Chapter 12 Review 925 Chapter 12 Test 928 ■ FOCUS ON MODELING Interpretations of Area 929
Cumulative Review Answers A1 Index I1 Photo Credits C1
CR1
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Preface
The art of teaching is the art of assisting discovery. MARK VAN DOREN
What do students really need to know to be prepared for calculus? What tools do instructors really need to assist their students in preparing for calculus? These two questions have motivated the writing of this book. To be prepared for calculus a student needs not only technical skill but also a clear understanding of concepts. Indeed, conceptual understanding and technical skill go hand in hand, each reinforcing the other. A student also needs to gain an appreciation for the power and utility of mathematics in modeling the realworld. Every feature of this textbook is devoted to fostering these goals. We are keenly aware that good teaching comes in many different forms, and that each instructor brings unique strengths and imagination to the classroom. Some instructors use technology to help students become active learners; others use the rule of four, “topics should be presented geometrically, numerically, algebraically, and verbally,” to promote conceptual reasoning; some use an expanded emphasis on applications to promote an appreciation for mathematics in everyday life; still others use group learning, extended projects, or writing exercises as a way of encouraging students to explore their own understanding of a given concept; and all present mathematics as a problemsolving endeavor. In this book we have included all these methods of teaching precalculus as enhancements to a central core of fundamental skills. These methods are tools to be utilized by instructors and their students to navigate their own course of action in preparing for calculus. In writing this ﬁfth edition our purpose was to further enhance the utility of the book as an instructional tool. The main change in this edition is an expanded emphasis on modeling and applications: In each section the applications exercises have been expanded and are grouped together under the heading Applications, and each chapter (except Chapter 1) now ends with a Focus on Modeling section. We have also made some organizational changes, including dividing the chapter on analytic trigonometry into two chapters, each of more manageable size. There are numerous other smaller changes—as we worked through the book we sometimes realized that an additional example was needed, or an explanation could be clariﬁed, or a section could beneﬁt from different types of exercises. Throughout these changes, however, we have retained the overall structure and the main features that have contributed to the success of this book.
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Many of the changes in this edition have been drawn from our own experience in teaching, but, more importantly, we have listened carefully to the users of the current edition, including many of our closest colleagues. We are also grateful to the many letters and emails we have received from users of this book, instructors as well as students, recommending changes and suggesting additions. Many of these have helped tremendously in making this edition even more userfriendly.
Special Features The most important way to foster conceptual understanding and hone technical skill is through the problems that the instructor assigns. To that end we have provided a wide selection of exercises.
EXERCISE SETS
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Exercises Each exercise set is carefully graded, progressing from basic conceptual exercises and skilldevelopment problems to more challenging problems requiring synthesis of previously learned material with new concepts. Applications Exercises We have included substantial applied problems that we believe will capture the interest of students. These are integrated throughout the text in both examples and exercises. In the exercise sets, applied problems are grouped together under the heading, Applications. (See, for example, pages 127, 156, 314, and 451.) Discovery, Writing, and Group Learning Each exercise set ends with a block of exercises called Discovery•Discussion. These exercises are designed to encourage students to experiment, preferably in groups, with the concepts developed in the section, and then to write out what they have learned, rather than simply look for “the answer.” (See, for example, pages 232 and 369.)
A COMPLETE REVIEW CHAPTER We have included an extensive review chapter primarily as a handy reference for the student to revisit basic concepts in algebra and analytic geometry. ■
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Chapter 1 This is the review chapter; it contains the fundamental concepts a student needs to begin a precalculus course. As much or as little of this chapter can be covered in class as needed, depending on the background of the students. Chapter 1 Test The test at the end of Chapter 1 is intended as a diagnostic instrument for determining what parts of this review chapter need to be taught. It also serves to help students gauge exactly what topics they need to review.
The trigonometry chapters of this text have been written so that either the right triangle approach or the unit circle approach may be taught ﬁrst. Putting these two approaches in different chapters, each with its relevant applications, helps clarify the purpose of each approach. The chapters introducing trigonometry are as follows:
FLEXIBLE APPROACH TO TRIGONOMETRY
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Chapter 5: Trigonometric Functions of Real Numbers This chapter introduces trigonometry through the unit circle approach. This approach emphasizes that the trigonometric functions are functions of real numbers, just like the polynomial and exponential functions with which students are already familiar. Chapter 6: Trigonometric Functions of Angles This chapter introduces trigonometry through the right triangle approach. This approach builds on the foundation of a conventional highschool course in trigonometry.
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Another way to teach trigonometry is to intertwine the two approaches. Some instructors teach this material in the following order: Sections 5.1, 5.2, 6.1, 6.2, 6.3, 5.3, 5.4, 6.4, 6.5. Our organization makes it easy to do this without obscuring the fact that the two approaches involve distinct representations of the same functions. GRAPHING CALCULATORS AND COMPUTERS Calculator and computer technology extends in a powerful way our ability to calculate and visualize mathematics. The availability of graphing calculators makes it not less important, but far more important to understand the concepts that underlie what the calculator produces. Accordingly, all our calculatororiented subsections are preceded by sections in which students must graph or calculate by hand, so that they can understand precisely what the calculator is doing when they later use it to simplify the routine, mechanical part of their work. The graphing calculator sections, subsections, examples, and exercises, all marked with the special symbol , are optional and may be omitted without loss of continuity. We use the following capabilities of the calculator: ■
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Graphing Calculators The use of the graphing calculator is integrated throughout the text to graph and analyze functions, families of functions, and sequences, to calculate and graph regression curves, to perform matrix algebra, to graph linear inequalities, and other powerful uses. Simple Programs We exploit the programming capabilities of a graphing calculator to simulate reallife situations, to sum series, or to compute the terms of a recursive sequence. (See, for instance, pages 702, 825, and 829.)
FOCUS ON MODELING The “modeling” theme has been used throughout to unify and clarify the many applications of precalculus. We have made a special effort, in these modeling sections and subsections, to clarify the essential process of translating problems from English into the language of mathematics. (See pages 204 or 647.) ■
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Constructing Models There are numerous applied problems throughout the book where students are given a model to analyze (see, for instance, page 200). But the material on modeling, where students are required to construct mathematical models for themselves, has been organized into clearly deﬁned sections and subsections (see, for example, pages 203, 369, 442, and 848). Focus on Modeling Each chapter concludes with a Focus on Modeling section. The ﬁrst such section, after Chapter 2, introduces the basic idea of modeling a reallife situation by ﬁtting lines to data (linear regression). Other sections present ways in which polynomial, exponential, logarithmic, and trigonometric functions, and systems of inequalities can all be used to model familiar phenomena from the sciences and from everyday life (see, for example, pages 320, 386, or 459). Chapter 1 concludes with a section entitled Focus on Problem Solving.
One way to engage students and make them active learners is to have them work (perhaps in groups) on extended projects that give a feeling of substantial accomplishment when completed. Each chapter contains one or more Discovery Projects (see the table of contents); these provide a challenging but accessible set of activities that enable students to explore in greater depth an interesting aspect of the topic they have just learned. (See, for instance, pages 223, 432, or 700.) DISCOVERY PROJECTS
MATHEMATICAL VIGNETTES Throughout the book we make use of the margins to provide historical notes, key insights, or applications of mathematics in the mod
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ern world. These serve to enliven the material and show that mathematics is an important, vital activity, and that even at this elementary level it is fundamental to everyday life. ■
Mathematical Vignettes These vignettes include biographies of interesting mathematicians and often include a key insight that the mathematician discovered and which is relevant to precalculus. (See, for instance, the vignettes on Viète, page 49; coordinates as addresses, page 88; and radiocarbon dating, page 360.)
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Mathematics in the Modern World This is a series of vignettes that emphasizes the central role of mathematics in current advances in technology and the sciences. (See pages 256, 656, and 746, for example).
The Check Your Answer feature is used wherever possible to emphasize the importance of looking back to check whether an answer is reasonable. (See, for instance, page 363.)
CHECK YOUR ANSWER
REVIEW MATERIAL The review material in this edition covers individual chapters as well as groups of chapters. This material is an important tool for helping students see the unity of the different precalculus topics. The questions and exercises in each review section combine the topics from an entire chapter or from groups of chapters. The review material is organized as follows. ■
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Concept Check The endofchapter material begins with a Concept Check designed to get the students to think about and explain in their own words the ideas presented in the chapter. These can be used as writing exercises, in a classroom discussion setting, or for personal study. Review Exercises The Concept Checks are followed by review exercises designed to provide additional practice for working with the chapter material. Answers to oddnumbered review exercises are given in the back of the book. Chapter Test Each chapter ends with a Chapter Test designed to help the students assess their ability to work with the chapter material as a whole. Answers to both even and odd test questions are given in the back of the book. Cumulative Review The Cumulative Reviews at the end of the text cover the material of several related chapters, very much like midterm exams. Each such review begins with a checklist of the topics the students should have mastered after completing the respective chapters. This is followed by a Cumulative Review Test. As with the Chapter Tests, answers to all cumulative test questions are given in the back of the book.
Major Changes for the Fifth Edition ■
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More than 20 percent of the exercises are new. New exercises have been chosen to provide more practice with basic concepts, as well as to explore ideas that we do not have space to cover in the discussion and examples in the text itself. Many new applied exercises have been added. Each chapter now begins with a Chapter Overview that introduces the main themes of the chapter and explains why the material is important. Six new Focus on Modeling sections have been added, with topics ranging from Mapping the World (Chapter 8) to Traveling and Standing Waves (Chapter 7).
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Five new Discovery Projects have been added, with topics ranging from the uses of vectors in sailing (see page 626) to the uses of conics in architecture (see page 771). A few more mathematical vignettes have been added (see for example the vignette on splines, page 252, and the one on Maria Agnesi, page 802.) We have moved the section on variation from Chapter 2 to Chapter 1, thus focusing Chapter 2 more clearly on the essential concept of a function. In Chapter 5, Trigonometric Functions of Real Numbers, we have incorporated the material on harmonic motion as a new section. The Focus on Modeling section is now about ﬁtting sinusoidal curves to data. In Chapter 7, Analytic Trigonometry, we now include only the material on trigonometric identities and equations. This change was done at the request of users. Chapter 8, Polar Coordinates and Vectors, is a new chapter, incorporating material that was previously in other chapters. The topics in this chapter, which also include the polar representation of complex numbers, are united by the theme of using the trigonometric functions to locate the coordinates of a point or describe the components of a vector. In Chapter 9, Systems of Equations and Inequalities, we have put the section on graphing of inequalities as the last section, so it now immediately precedes the material on linear programming in the Focus on Modeling section. Chapter 10, Analytic Geometry, now includes only the conic sections and parametric equations. The material on polar coordinates is in the new Chapter 8. In Chapter 11, Sequence and Series, we have expanded the material on recursive sequences by adding a Focus on Modeling section on the use of such sequences in modeling realworld phenomena.
Acknowledgments We thank the following reviewers for their thoughtful and constructive comments. Michelle Benedict, Augusta State University; Linda Crawford, Augusta State University; Vivian G. Kostyk, Inver Hills Community College; and Heather C. McGilvray, Seattle University.
REVIEWERS FOR THE FOURTH EDITION
REVIEWERS FOR THE FIFTH EDITION Kenneth Berg, University of Maryland; Elizabeth Bowman, University of Alabama at Huntsville; William Cherry, University of North Texas; Barbara Cortzen, DePaul University; Gerry Fitch, Louisiana State University; Lana Grishchenko, Cal Poly State University, San Luis Obispo; Bryce Jenkins, Cal Poly State University, San Luis Obispo; Margaret Mary Jones, Rutgers University; Victoria Kauffman, University of New Mexico; Sharon Keener, Georgia Perimeter College; YongHee KimPark, California State University Long Beach; Mangala Kothari, Rutgers University; Andre Mathurin, Bellarmine College Prep; Donald Robertson, Olympic College; Jude Socrates, Pasadena City College; Eneﬁok Umana, Georgia Perimeter College; Michele Wallace, Washington State University; and Linda Waymire, Daytona Beach Community College.
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We have beneﬁted greatly from the suggestions and comments of our colleagues who have used our books in previous editions. We extend special thanks in this regard to Linda Byun, Bruce Chaderjian, David Gau, Daniel Hernandez, YongHee KimPark, Daniel Martinez, David McKay, Robert Mena, Kent Merryﬁeld, Florence Newberger, Viet Ngo, Marilyn Oba, Alan Safer, Angelo Segalla, Robert Valentini, and Derming Wang, from California State University, Long Beach; to Karen Gold, Betsy Gensamer, Cecilia McVoy, Mike McVoy, Samir Ouzomgi, and Ralph Rush, of the Pennsylvania State University, Abington College; to Gloria Dion, of Educational Testing Service, Princeton, New Jersey; to Mark Ashbaugh and Nakhlé Asmar of the University of Missouri, Columbia; to Fred Saﬁer, of the City College of San Francisco; and Steve Edwards, of Southern Polytechnic State University in Marietta, Georgia. We have also received much invaluable advice from our students, especially Devaki Shah and Ellen Newman. We especially thank Martha Emry, manager of our production service, for her excellent work and her tireless attention to quality and detail. Her energy, devotion, experience, and intelligence were essential components in the creation of this book. We are grateful to Luana Richards, our copy editor, who over the years has expertly shaped the language and style of all of our books. At Matrix Art Services we thank Jade Myers for his elegant graphics. We thank the team at G&S Book Services for the high quality and consistency brought to the page composition. Our special thanks to Phyllis PanmanWatson for her dedication and care in creating the answer section. At Brooks/Cole our thanks go to the book team: Assistant Editor Stacy Green; Editorial Assistant Katherine Cook; Senior Marketing Manager Karin Sandberg; Marketing Assistant Jennifer Velasquez; Marketing Communications Project Manager Bryan Vann; Senior Project Production Manager Janet Hill; Senior Art Director Vernon Boes; and Technology Project Manager Earl Perry. We are particularly grateful to our publisher Bob Pirtle for guiding this book through every stage of writing and production. His support and editorial insight when crucial decisions had to be made were invaluable.
ANCILLARIES FOR PRECALCULUS, MATHEMATICS FOR CALCULUS, FIFTH EDITION For the Instructor LECTURE PREPARATION Instructor’s Guide 0534493009 This helpful teaching companion is written by Doug Shaw, author of the Instructor Guides for the Stewart calculus texts. It contains points to stress, suggested time to allot, text discussion topics, core materials for lecture, workshop/ discussion suggestions, group work exercises in a form suitable for handout, solutions to group work exercises, and suggested homework problems. Complete Solutions Manual 0534493165 This manual provides workedout solutions to all of the problems in the text. Solution Builder 0495108200 This is an electronic version of the complete solutions manual. It provides instructors with an efﬁcient method for creating solution sets to homework or exams that can then be printed or posted.
PRESENTATION TOOLS Text Speciﬁc DVDs 0534493092 This set of DVDs is available free upon adoption of the text. Each chapter of the text is broken down into 10 to 20minute problemsolving lessons that cover each section of the chapter. JoinIn™ on TurningPoint ® 0495107549 With this bookspeciﬁc JoinIn™ content for electronic response systems, and Microsoft® PowerPoint® slides of your own lecture, you can transform your classroom and assess your students’ progress with instant inclass quizzes and polls.
TESTING TOOLS iLrn™ Assessment iLrn Assessment is a powerful and fully integrated teaching and course management system that can be used for homework, quizzing, or testing purposes. Easy to use, it offers you complete control when creating assessments; you can draw from the wealth of exercises provided or create your own questions. A real timesaver, iLrn Assessment offers automatic grading of textspeciﬁc homework, quizzes, and tests with results ﬂowing directly into the gradebook. The autoenrollment feature also saves time with course set up as students selfenroll into the course gradebook. A wide range of problem types provides greater variety and more diverse challenges in your tests. iLrn Assessment provides seamless integration with Blackboard® and WebCT®.
Test Bank 0534492991 The Test Bank consists of two parts. Part 1 includes six tests per chapter including 3 ﬁnal exams. Part 2 of the Test Bank contains test questions broken down by section. Question types are free response and multiple choice.
For the Student HOMEWORK TOOLS Student Solutions Manual 0534492908 The student solutions manual provides worked out solutions to the oddnumbered problems in the text. Enhanced WebAssign Enhanced WebAssign is designed for you to do your homework online. This proven and reliable system uses pedagogy and content found in Stewart, Redlin, and Watson’s text, and then enhances it to help you learn college algebra more effectively. Automatically graded homework allows you to focus on your learning and get interactive study assistance outside of class. Enhanced WebAssign is compatible with recent operating systems, works with most web browsers, and requires no proprietary plugins. And, it’s fully supported by a comprehensive service and training program.
LEARNING TOOLS Interactive Video Skillbuilder CD ROM 0534492878 The Interactive Video Skillbuilder CDROM contains hours of video instruction. To help students evaluate their progress, each section contains a 10question web quiz and each chapter contains a chapter test, with answers to each problem on each test. Also includes MathCue Tutorial, dualplatform software that presents and scores problems and tutors students by displaying annotated, stepbystep solutions. Problem sets may be customized as desired. Study Guide 0534492894 Contains detailed explanations, workedout examples, practice problems, and key ideas to master. Each section of the main text has a corresponding section in the Study Guide.
ADDITIONAL RESOURCES Book Companion Website http://mathematics.brookscole.com This outstanding site features chapterbychapter online tutorial quizzes, a sample ﬁnal exam, chapter outlines, chapter review, chapterbychapter web links, ﬂashcards, and more! Plus, the Brooks/Cole Mathematics Resource Center features historical notes, math news, and career information.
ExamView® Computerized Testing 0495019941 Create, deliver, and customize tests (both print and online) in minutes with this easytouse assessment system. ExamView is a registered trademark of FSCreations, Inc. Used herein under license.
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To the Student
This textbook was written for you to use as a guide to mastering precalculus mathematics. Here are some suggestions to help you get the most out of your course. First of all, you should read the appropriate section of text before you attempt your homework problems. Reading a mathematics text is quite different from reading a novel, a newspaper, or even another textbook. You may ﬁnd that you have to reread a passage several times before you understand it. Pay special attention to the examples, and work them out yourself with pencil and paper as you read. With this kind of preparation you will be able to do your homework much more quickly and with more understanding. Don’t make the mistake of trying to memorize every single rule or fact you may come across. Mathematics doesn’t consist simply of memorization. Mathematics is a problemsolving art, not just a collection of facts. To master the subject you must solve problems—lots of problems. Do as many of the exercises as you can. Be sure to write your solutions in a logical, stepbystep fashion. Don’t give up on a problem if you can’t solve it right away. Try to understand the problem more clearly—reread it thoughtfully and relate it to what you have learned from your teacher and from the examples in the text. Struggle with it until you solve it. Once you have done this a few times you will begin to understand what mathematics is really all about. Answers to the oddnumbered exercises, as well as all the answers to each chapter test, appear at the back of the book. If your answer differs from the one given, don’t immediately assume that you are wrong. There may be a calculation that connects the two answers and makes both correct. For example, if you get 1/( 12 1) but the answer given is 1 12, your answer is correct, because you can multiply both numerator and denominator of your answer by 12 1 to change it to the given answer. The symbol is used to warn against committing an error. We have placed this symbol in the margin to point out situations where we have found that many of our students make the same mistake.
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Calculators and Calculations
Calculators are essential in most mathematics and science subjects. They free us from performing routine tasks, so we can focus more clearly on the concepts we are studying. Calculators are powerful tools but their results need to be interpreted with care. In what follows, we describe the features that a calculator suitable for a precalculus course should have, and we give guidelines for interpreting the results of its calculations.
Scientiﬁc and Graphing Calculators For this course you will need a scientiﬁc calculator—one that has, as a minimum, the usual arithmetic operations (, , , ) as well as exponential, logarithmic, and trigonometric functions (e x, 10 x, ln, log, sin, cos, tan). In addition, a memory and at least some degree of programmability will be useful. Your instructor may recommend or require that you purchase a graphing calculator. This book has optional subsections and exercises that require the use of a graphing calculator or a computer with graphing software. These special subsections and exercises are indicated by the symbol . Besides graphing functions, graphing calculators can also be used to ﬁnd functions that model reallife data, solve equations, perform matrix calculations (which are studied in Chapter 9), and help you perform other mathematical operations. All these uses are discussed in this book. It is important to realize that, because of limited resolution, a graphing calculator gives only an approximation to the graph of a function. It plots only a ﬁnite number of points and then connects them to form a representation of the graph. In Section 1.9, we give guidelines for using a graphing calculator and interpreting the graphs that it produces.
Calculations and Signiﬁcant Figures Most of the applied examples and exercises in this book involve approximate values. For example, one exercise states that the moon has a radius of 1074 miles. This does not mean that the moon’s radius is exactly 1074 miles but simply that this is the radius rounded to the nearest mile. One simple method for specifying the accuracy of a number is to state how many signiﬁcant digits it has. The signiﬁcant digits in a number are the ones from the ﬁrst nonzero digit to the last nonzero digit (reading from left to right). Thus, 1074 has four signiﬁcant digits, 1070 has three, 1100 has two, and 1000 has one signiﬁcant digit. This rule may sometimes lead to ambiguities. For example, if a distance is 200 km to xxii
Calculators and Calculations
xxiii
the nearest kilometer, then the number 200 really has three signiﬁcant digits, not just one. This ambiguity is avoided if we use scientiﬁc notation—that is, if we express the number as a multiple of a power of 10: 2.00 10 2 When working with approximate values, students often make the mistake of giving a ﬁnal answer with more signiﬁcant digits than the original data. This is incorrect because you cannot “create” precision by using a calculator. The ﬁnal result can be no more accurate than the measurements given in the problem. For example, suppose we are told that the two shorter sides of a right triangle are measured to be 1.25 and 2.33 inches long. By the Pythagorean Theorem, we ﬁnd, using a calculator, that the hypotenuse has length 21.252 2.332 ⬇ 2.644125564 in. But since the given lengths were expressed to three signiﬁcant digits, the answer cannot be any more accurate. We can therefore say only that the hypotenuse is 2.64 in. long, rounding to the nearest hundredth. In general, the ﬁnal answer should be expressed with the same accuracy as the leastaccurate measurement given in the statement of the problem. The following rules make this principle more precise.
Rules for Working with Approximate Data 1. When multiplying or dividing, round off the ﬁnal result so that it has as many signiﬁcant digits as the given value with the fewest number of signiﬁcant digits. 2. When adding or subtracting, round off the ﬁnal result so that it has its last signiﬁcant digit in the decimal place in which the leastaccurate given value has its last signiﬁcant digit. 3. When taking powers or roots, round off the ﬁnal result so that it has the same number of signiﬁcant digits as the given value.
As an example, suppose that a rectangular table top is measured to be 122.64 in. by 37.3 in. We express its area and perimeter as follows: Area length width 122.64 37.3 ⬇ 4570 in2
Three signiﬁcant digits
Perimeter 2Ólength widthÔ 2Ó122.64 37.3Ô ⬇ 319.9 in.
Tenths digit
Note that in the formula for the perimeter, the value 2 is an exact value, not an approximate measurement. It therefore does not affect the accuracy of the ﬁnal result. In general, if a problem involves only exact values, we may express the ﬁnal answer with as many signiﬁcant digits as we wish. Note also that to make the ﬁnal result as accurate as possible, you should wait until the last step to round off your answer. If necessary, use the memory feature of your calculator to retain the results of intermediate calculations.
Abbreviations
cm dB F ft g gal h H Hz in. J kcal kg km kPa L lb lm M m
xxiv
centimeter decibel farad foot gram gallon hour henry Hertz inch Joule kilocalorie kilogram kilometer kilopascal liter pound lumen mole of solute per liter of solution meter
mg MHz mi min mL mm N qt oz s ⍀ V W yd yr °C °F K ⇒ ⇔
milligram megahertz mile minute milliliter millimeter Newton quart ounce second ohm volt watt yard year degree Celsius degree Fahrenheit Kelvin implies is equivalent to
Mathematical Vignettes
No Smallest or Largest Number in an Open Interval 8 Diophantus 20 François Viète 49 Pythagoras 54 Coordinates as Addresses 88 Alan Turing 103 Rene Descartes 112 George Polya 138 Einstein’s Letter 141 Bhaskara 144 Donald Knuth 165 Sonya Kovalevsky 188 Evariste Galois 273 Leonhard Euler 288 Carl Friedrich Gauss 294 Gerolamo Cardano 296 The Gateway Arch 331 John Napier 346 Radiocarbon Dating 360 Standing Room Only 372 HalfLives of Radioactive Elements 373 Radioactive Waste 374 pH for Some Common Substances 377 Largest Earthquakes 378 Intensity Levels of Sounds 379 The Value of p 414 Periodic Functions 427 AM and FM Radio 428 RootMean Square 448 Hipparchus 479 Aristarchus of Samos 480 Thales of Miletus 482 Surveying 504 Euclid 532 Jean Baptiste Joseph Fourier 536 Pierre de Fermat 652 Olga TausskyTodd 672 Julia Robinson 678 Arthur Cayley 692 David Hilbert 708 Emmy Noether 710
The Rhind Papyrus 716 Linear Programming 737 Archimedes 748 Eccentricities of the Orbits of the Planets 758 Paths of Comets 766 Johannes Kepler 780 Maria Gaetana Agnesi 802 Galileo Galilei 817 Large Prime Numbers 824 Eratosthenes 825 Fibonacci 826 Golden Ratio 829 Srinavasa Ramanujan 840 Blaise Pascal 858 Pascal’s Triangle 862 Isaac Newton 894 Newton and Limits 902
MATHEMATICS IN THE MODERN WORLD Mathematics in the Modern World 16 Changing Words, Sounds, and Pictures into Numbers 30 Error Correcting Codes 38 Computers 178 Model Airplanes 245 Splines 252 Automotive Design 256 Unbreakable Codes 308 Law Enforcement 344 Evaluating Functions on a Calculator 436 Weather Prediction 562 Fractals 600 Global Positioning System 656 Fair Voting Methods 682 Mathematical Ecology 696 Looking Inside Your Head 746 Fair Division of Assets 834 Mathematical Economics 850 xxv
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PRECALCULUS Mathematics for Calculus FIFTH EDITION
1
Fundamentals
1.1
Real Numbers
1.7
Inequalities
1.2
Exponents and Radicals
1.8
Coordinate Geometry
1.3
Algebraic Expressions
1.9
1.4
Rational Expressions
Graphing Calculators; Solving Equations and Inequalities Graphically
1.5
Equations
1.10
Lines
1.6
Modeling with Equations
1.11
Modeling Variation
Chapter Overview In this ﬁrst chapter we review the real numbers, equations, and the coordinate plane. You are probably already familiar with these concepts, but it is helpful to get a fresh look at how these ideas work together to solve problems and model (or describe) realworld situations. Let’s see how all these ideas are used in the following reallife situation: Suppose you get paid $8 an hour at your parttime job. We are interested in how much money you make. To describe your pay we use real numbers. In fact, we use real numbers every day—to describe how tall we are, how much money we have, how cold (or warm) it is, and so on. In algebra, we express properties of the real numbers by using letters to stand for numbers. An important property is the distributive property: A1B C2 AB AC To see that this property makes sense, let’s consider your pay if you work 6 hours one day and 5 hours the next. Your pay for those two days can be calculated in two different ways: $816 5 2 or $8 # 6 $8 # 5, and both methods give the same answer. This and other properties of the real numbers constitute the rules for working with numbers, or the rules of algebra. We can also model your pay for any number of hours by a formula. If you work x hours then your pay is y dollars, where y is given by the algebraic formula y 8x
So if you work 10 hours, your pay is y 8 # 10 80 dollars. An equation is a sentence written in the language of algebra that expresses a fact about an unknown quantity x. For example, how many hours would you need to work to get paid 60 dollars? To answer this question we need to solve the equation
y y = 8x Pay ($)
60 8x y = 60
20
Bob Krist /Corbis
7.5 Hours worked
x
We use the rules of algebra to ﬁnd x. In this case we divide both sides of the equation by 8, so x 608 7.5 hours. The coordinate plane allows us to sketch a graph of an equation in two variables. For example, by graphing the equation y 8x we can “see” how pay increases with hours worked. We can also solve the equation 60 8x graphically by ﬁnding the value of x at which the graphs of y 8x and y 60 intersect (see the ﬁgure). In this chapter we will see many examples of how the real numbers, equations, and the coordinate plane all work together to help us solve reallife problems. 1
2
CHAPTER 1
Fundamentals
1.1
Real Numbers Let’s review the types of numbers that make up the real number system. We start with the natural numbers: 1, 2, 3, 4, . . .
The different types of real numbers were invented to meet speciﬁc needs. For example, natural numbers are needed for counting, negative numbers for describing debt or belowzero temperatures, rational numbers for concepts like “half a gallon of milk,” and irrational numbers for measuring certain distances, like the diagonal of a square.
The integers consist of the natural numbers together with their negatives and 0: . . . , 3, 2, 1, 0, 1, 2, 3, 4, . . . We construct the rational numbers by taking ratios of integers. Thus, any rational number r can be expressed as m r n where m and n are integers and n 0. Examples are: 37
1 2
46 461
17 0.17 100
(Recall that division by 0 is always ruled out, so expressions like 03 and 00 are undeﬁned.) There are also real numbers, such as 12, that cannot be expressed as a ratio of integers and are therefore called irrational numbers. It can be shown, with varying degrees of difﬁculty, that these numbers are also irrational: 13
15
3 1 2
p
3 p2
The set of all real numbers is usually denoted by the symbol ⺢. When we use the word number without qualiﬁcation, we will mean “real number.” Figure 1 is a diagram of the types of real numbers that we work with in this book. Rational numbers
Irrational numbers
–21 , –37 ,
3 œ3 , œ5 , œ2 , π , — 2
Integers
3
46, 0.17, 0.6, 0.317
π
Natural numbers
. . . , −3, −2, −1, 0, 1, 2, 3, . . .
A repeating decimal such as x 3.5474747. . . is a rational number. To convert it to a ratio of two integers, we write 1000x 3547.47474747. . . 10x 35.47474747. . . 990x 3512.0 Thus, x 3512 990 . (The idea is to multiply x by appropriate powers of 10, and then subtract to eliminate the repeating part.)
Figure 1 The real number system
Every real number has a decimal representation. If the number is rational, then its corresponding decimal is repeating. For example, 1 2 157 495
0.5000. . . 0.50
2 3
0.66666. . . 0.6
0.3171717. . . 0.317
9 7
1.285714285714. . . 1.285714
(The bar indicates that the sequence of digits repeats forever.) If the number is irrational, the decimal representation is nonrepeating: 12 1.414213562373095. . .
p 3.141592653589793. . .
SECTION 1.1
Real Numbers
3
If we stop the decimal expansion of any number at a certain place, we get an approximation to the number. For instance, we can write p ⬇ 3.14159265 where the symbol ⬇ is read “is approximately equal to.” The more decimal places we retain, the better our approximation.
Properties of Real Numbers We all know that 2 3 3 2 and 5 7 7 5 and 513 87 87 513, and so on. In algebra, we express all these (inﬁnitely many) facts by writing abba where a and b stand for any two numbers. In other words, “a b b a” is a concise way of saying that “when we add two numbers, the order of addition doesn’t matter.” This fact is called the Commutative Property for addition. From our experience with numbers we know that the properties in the following box are also valid.
Properties of Real Numbers Property
Example
Description
Commutative Properties abba ab ba
7337 3#55#3
When we add two numbers, order doesn’t matter. When we multiply two numbers, order doesn’t matter.
Associative Properties 1a b2 c a 1b c 2
12 42 7 2 14 72
1ab 2c a1bc2
13 # 72 # 5 3 # 17 # 52
Distributive Property a1b c2 ab ac 1b c 2a ab ac
2 # 13 52 2 # 3 2 # 5 13 52 # 2 2 # 3 2 # 5
When we add three numbers, it doesn’t matter which two we add ﬁrst. When we multiply three numbers, it doesn’t matter which two we multiply ﬁrst. When we multiply a number by a sum of two numbers, we get the same result as multiplying the number by each of the terms and then adding the results.
The Distributive Property applies whenever we multiply a number by a sum. Figure 2 explains why this property works for the case in which all the numbers are positive integers, but the property is true for any real numbers a, b, and c. The Distributive Property is crucial because it describes the way addition and multiplication interact with each other.
2(3+5)
Figure 2 The Distributive Property
2#3
2#5
4
CHAPTER 1
Fundamentals
Example 1
Using the Distributive Property
(a) 21x 32 2 # x 2 # 3
Distributive Property
2x 6
Simplify
c
(b) 1a b2 1x y2 1a b2x 1a b 2y
1ax bx 2 1ay by 2
ax bx ay by
Distributive Property Distributive Property Associative Property of Addition
In the last step we removed the parentheses because, according to the Associative Property, the order of addition doesn’t matter. Don’t assume that a is a negative number. Whether a is negative or positive depends on the value of a. For example, if a 5, then a 5, a negative number, but if a 5, then a 152 5 (Property 2), a positive number.
■
The number 0 is special for addition; it is called the additive identity because a 0 a for any real number a. Every real number a has a negative, a, that satisﬁes a 1a2 0. Subtraction is the operation that undoes addition; to subtract a number from another, we simply add the negative of that number. By deﬁnition a b a 1b2 To combine real numbers involving negatives, we use the following properties.
Properties of Negatives Property
Example
1. 112a a
1125 5
2. 1a 2 a
152 5
4. 1a 2 1b2 ab
142 132 4 # 3
3. 1a 2b a1b2 1ab2
1527 5172 15 # 72
5. 1a b2 a b
13 52 3 5
6. 1a b2 b a
15 82 8 5
Property 6 states the intuitive fact that a b and b a are negatives of each other. Property 5 is often used with more than two terms: 1a b c2 a b c
Example 2
Using Properties of Negatives
Let x, y, and z be real numbers. (a) 1x 22 x 2
(b) 1x y z2 x y 1z2 x y z
Property 5: (a b) a b Property 5: (a b) a b Property 2: (a) a
■
SECTION 1.1
Real Numbers
5
The number 1 is special for multiplication; it is called the multiplicative identity because a 1 a for any real number a. Every nonzero real number a has an inverse, 1/a, that satisﬁes a # 11/a2 1. Division is the operation that undoes multiplication; to divide by a number, we multiply by the inverse of that number. If b 0, then, by deﬁnition, aba#
1 b
We write a # 11/b2 as simply a/b. We refer to a/b as the quotient of a and b or as the fraction a over b; a is the numerator and b is the denominator (or divisor). To combine real numbers using the operation of division, we use the following properties.
Properties of Fractions Property
Example
Description
1.
a#c ac b d bd
2 5 # 2 ## 5 10 3 7 3 7 21
2.
a c a d # b d b c
2 5 2 7 14 # 3 7 3 5 15
When dividing fractions, invert the divisor and multiply.
3.
a b ab c c c
2 7 27 9 5 5 5 5
When adding fractions with the same denominator, add the numerators.
4.
a c ad bc b d bd
3 2#73#5 29 2 5 7 35 35
5.
ac a bc b
2 2#5 # 3 5 3
Cancel numbers that are common factors in numerator and denominator.
2 6 , so 2 # 9 3 # 6 3 9
Cross multiply.
6. If
a c , then ad bc b d
When multiplying fractions, multiply numerators and denominators.
When adding fractions with different denominators, ﬁnd a common denominator. Then add the numerators.
When adding fractions with different denominators, we don’t usually use Property 4. Instead we rewrite the fractions so that they have the smallest possible common denominator (often smaller than the product of the denominators), and then we use Property 3. This denominator is the Least Common Denominator (LCD) described in the next example.
Example 3 Evaluate:
Using the LCD to Add Fractions
5 7 36 120
Solution Factoring each denominator into prime factors gives 36 22 # 32
and
120 23 # 3 # 5
We ﬁnd the least common denominator (LCD) by forming the product of all the factors that occur in these factorizations, using the highest power of each factor.
6
CHAPTER 1
Fundamentals
Thus, the LCD is 23 # 32 # 5 360. So 5 7 5 # 10 7#3 36 120 36 # 10 120 # 3
Use common denominator
21 71 50 360 360 360
Property 3: Adding fractions with the same denominator
■
The Real Line The real numbers can be represented by points on a line, as shown in Figure 3. The positive direction (toward the right) is indicated by an arrow. We choose an arbitrary reference point O, called the origin, which corresponds to the real number 0. Given any convenient unit of measurement, each positive number x is represented by the point on the line a distance of x units to the right of the origin, and each negative number x is represented by the point x units to the left of the origin. The number associated with the point P is called the coordinate of P, and the line is then called a coordinate line, or a real number line, or simply a real line. Often we identify the point with its coordinate and think of a number as being a point on the real line. 1 _ 16
_3.1725 _2.63
_4.9 _4.7 _5 _4 _4.85
_3
_ œ∑2 _2
_1
1 1 8 4 1 2
œ∑2 1
œ∑3 œ∑5 2
4.2 4.4 4.9999 π 3
4 5 4.3 4.5
0.3 ∑
Figure 3 The real line
The real numbers are ordered. We say that a is less than b and write a b if b a is a positive number. Geometrically, this means that a lies to the left of b on the number line. Equivalently, we can say that b is greater than a and write b a. The symbol a b 1or b a2 means that either a b or a b and is read “a is less than or equal to b.” For instance, the following are true inequalities (see Figure 4): 7 7.4 7.5
p 3
_π _4
_3
12 2
2 2 7.4 7.5
œ∑2 _2
_1
1
2
3
4
5
6
7
8
Figure 4
Sets and Intervals A set is a collection of objects, and these objects are called the elements of the set. If S is a set, the notation a 僆 S means that a is an element of S, and b 僆 S means that b is not an element of S. For example, if Z represents the set of integers, then 3 僆 Z but p 僆 Z. Some sets can be described by listing their elements within braces. For instance, the set A that consists of all positive integers less than 7 can be written as A 51, 2, 3, 4, 5, 66
SECTION 1.1
Real Numbers
7
We could also write A in setbuilder notation as A 5x 0 x is an integer and 0 x 76 which is read “A is the set of all x such that x is an integer and 0 x 7.” If S and T are sets, then their union S 傼 T is the set that consists of all elements that are in S or T (or in both). The intersection of S and T is the set S 傽 T consisting of all elements that are in both S and T. In other words, S 傽 T is the common part of S and T. The empty set, denoted by , is the set that contains no element.
Example 4
Union and Intersection of Sets
If S {1, 2, 3, 4, 5}, T {4, 5, 6, 7}, and V {6, 7, 8}, ﬁnd the sets S 傼 T, S 傽 T, and S 傽 V. Solution T 64748 1, 2, 3, 4, 5, 6, 7, 8
14243 123
S
a
V
b
Figure 5 The open interval 1a, b 2
S 傼 T 51, 2, 3, 4, 5, 6, 76
S 艚 T 54, 56 S艚V
All elements in S or T Elements common to both S and T S and V have no element in common
Certain sets of real numbers, called intervals, occur frequently in calculus and correspond geometrically to line segments. If a b, then the open interval from a to b consists of all numbers between a and b and is denoted 1a, b2 . The closed interval from a to b includes the endpoints and is denoted 3a, b4 . Using setbuilder notation, we can write 1a, b2 5x 0 a x b6
a
b
Figure 6 The closed interval 3 a, b4
3a, b4 5x 0 a x b6
Note that parentheses 1 2 in the interval notation and open circles on the graph in Figure 5 indicate that endpoints are excluded from the interval, whereas square brackets 3 4 and solid circles in Figure 6 indicate that the endpoints are included. Intervals may also include one endpoint but not the other, or they may extend inﬁnitely far in one direction or both. The following table lists the possible types of intervals.
Notation
The symbol q (“inﬁnity”) does not stand for a number. The notation 1a, q 2 , for instance, simply indicates that the interval has no endpoint on the right but extends inﬁnitely far in the positive direction.
■
Set description
1a, b 2
5x 0 a x b6
3a, b 4
5x 0 a x b6
3 a, b 2
5x 0 a x b6
1a, b 4
5x 0 a x b6
1a, q 2
5x 0 a x6
3a, q 2
5x 0 a x6
1q, b 2
5x 0 x b6
1q, b 4
5x 0 x b6
1q, q 2
⺢ (set of all real numbers)
Graph
a
b
a
b
a
b
a
b
a a b b
8
CHAPTER 1
Fundamentals
No Smallest or Largest Number in an Open Interval Any interval contains inﬁnitely many numbers—every point on the graph of an interval corresponds to a real number. In the closed interval 3 0, 14 , the smallest number is 0 and the largest is 1, but the open interval 10, 12 contains no smallest or largest number. To see this, note that 0.01 is close to zero, but 0.001 is closer, 0.0001 closer yet, and so on. So we can always ﬁnd a number in the interval 10, 1 2 closer to zero than any given number. Since 0 itself is not in the interval, the interval contains no smallest number. Similarly, 0.99 is close to 1, but 0.999 is closer, 0.9999 closer yet, and so on. Since 1 itself is not in the interval, the interval has no largest number.
Example 5
Graphing Intervals
Express each interval in terms of inequalities, and then graph the interval. (a) 31, 2 2 5x 0 1 x 26 _1
(b) 31.5, 44 5x 0 1.5 x 46
(c) 13, q 2 5x 0 3 x6
1.5
4 ■
_3
Example 6
2
Finding Unions and Intersections of Intervals
Graph each set. (a) 11, 3 2 艚 32, 74
(b) 11, 3 2 傼 32, 74
Solution (a) The intersection of two intervals consists of the numbers that are in both intervals. Therefore 11, 32 艚 32, 74 5x 0 1 x 3 and 2 x 76 5x 0 2 x 36 32, 3 2
This set is illustrated in Figure 7. (b) The union of two intervals consists of the numbers that are in either one interval or the other (or both). Therefore 11, 32 傼 32, 74 5x 0 1 x 3 or 2 x 76
0.01
5x 0 1 x 76 11, 7 4
0.1
This set is illustrated in Figure 8. 0
0.001
0.01
(1,3)
(1,3) 0 0.0001
0.001
1
3
1
3 [2,7]
[2,7] 0
2
7
2 (1,7]
[2,3) 0
2
3
Figure 7 11, 3 2 艚 3 2, 74 32, 32
7
1
Figure 8 11, 32 傼 3 2, 74 11, 74
7
■
Absolute Value and Distance _3=3 _3 Figure 9
5=5 0
5
The absolute value of a number a, denoted by 0 a 0 , is the distance from a to 0 on the real number line (see Figure 9). Distance is always positive or zero, so we have 0 a 0 0 for every number a. Remembering that a is positive when a is negative, we have the following deﬁnition.
SECTION 1.1
Real Numbers
9
Deﬁnition of Absolute Value If a is a real number, then the absolute value of a is 0a0 e
Example 7 (a) (b) (c) (d)
if a 0 if a 0
a a
Evaluating Absolute Values of Numbers
030 3
0 3 0 132 3 000 0 0 3 p 0 13 p2 p 3
1since 3 p 1 3 p 02
■
When working with absolute values, we use the following properties.
Properties of Absolute Value Property
Example
1. 0 a 0 0
0 3 0 3 0
2. 0 a 0 0 a 0
0 5 0 0 5 0
3. 0 ab 0 0 a 0 0 b 0
0 2 # 5 0 0 2 0 0 5 0
4.
13 _2
11
Figure 10
`
0a0 a ` b 0b0
`
0 12 0 12 ` 3 0 3 0
Description The absolute value of a number is always positive or zero. A number and its negative have the same absolute value. The absolute value of a product is the product of the absolute values. The absolute value of a quotient is the quotient of the absolute values.
What is the distance on the real line between the numbers 2 and 11? From Figure 10 we see that the distance is 13. We arrive at this by ﬁnding either 0 11 122 0 13 or 0 122 11 0 13. From this observation we make the following deﬁnition (see Figure 11).
Distance between Points on the Real Line ba a
b
Figure 11 Length of a line segment 0 b a 0
If a and b are real numbers, then the distance between the points a and b on the real line is d1a, b 2 0 b a 0
10
CHAPTER 1
Fundamentals
From Property 6 of negatives it follows that 0 b a 0 0 a b 0 . This conﬁrms that, as we would expect, the distance from a to b is the same as the distance from b to a.
Example 8
The distance between the numbers 8 and 2 is
10 _8
d1a, b2 0 8 2 0 0 10 0 10
2
We can check this calculation geometrically, as shown in Figure 12.
Figure 12
1.1
Distance between Points on the Real Line
Exercises
1–2 ■ List the elements of the given set that are (a) natural numbers (b) integers (c) rational numbers (d) irrational numbers 3 1. 50, 10, 50, 227, 0.538, 17, 1.23, 13, 1 26
15 2. 51.001, 0.333. . . , p, 11, 11, 13 15 , 116, 3.14, 3 6
3–10
■
■
21–26 21. (a) 22. (a)
2 3
25. (a)
27–28
7. 15x 12 3 15x 3
28. (a)
2 2 3
■
6. 21A B 2 2A 2B
8. 1x a 2 1x b2 1x a 2 x 1x a 2 b
29–32
11. Commutative Property of addition,
x3
12. Associative Property of multiplication, 713x2 13. Distributive Property,
41A B 2
14. Distributive Property,
5x 5y
■
Use properties of real numbers to write the expression 15–20 without parentheses. 15. 31x y 2 17. 412m2 19. 52 12x 4y2
16. 1a b 2 8 18. 43 16y2
20. 13a 2 1b c 2d 2
2 3
■
15
(b) 0.25A 89 12 B
(b) A 12 13 B A 12 13 B
2 3
(b)
2
(b)
1 12 1 8 2 5 1 10
19 12 153
Place the correct symbol (, , or ) in the space. 7 2
(b) 3
0.67
(b)
2 3
72 0.67
(c) 3.5 (c) 0 0.67 0
10 12 11 13
(b) 12 1.41 1 (b) 1 2
31. (a) p 3
(b) 8 9
32. (a) 1.1 1.1
(b) 8 8
33–34
■
7 2
State whether each inequality is true or false.
29. (a) 6 10 30. (a)
11–14 ■ Rewrite the expression using the given property of real numbers.
1 4
(b) 1 58 16
35
2 34 1 1 2 3
27. (a) 3
10. 71a b c 2 71a b 2 7c
(b)
24. (a) A3 14 B A1 45 B
5. 1x 2y 2 3z x 12y 3z2
9. 2x13 y 2 13 y2 2x
154
23. (a) 23 A6 32 B
26. (a)
4. 213 5 2 13 5 2 2
Perform the indicated operations.
3 10
State the property of real numbers being used.
3. 7 10 10 7
■
Write each statement in terms of inequalities.
33. (a) x is positive (b) t is less than 4 (c) a is greater than or equal to p (d) x is less than 13 and is greater than 5 (e) The distance from p to 3 is at most 5 34. (a) y is negative (b) z is greater than 1 (c) b is at most 8
0 0.67 0
SECTION 1.1
Real Numbers
1 0 1 0
(d) „ is positive and is less than or equal to 17
63. (a) @ 0 6 0 0 4 0 @
(b)
(e) y is at least 2 units from p
64. (a) @ 2 0 12 0 @
(b) 1 @ 1 0 1 0 @
35–38
■
Find the indicated set if A {1, 2, 3, 4, 5, 6, 7}
B {2, 4, 6, 8}
C {7, 8, 9, 10}
65. (a) 0 12 2 # 6 0
35. (a) A 傼 B
(b) A 傽 B
36. (a) B 傼 C
(b) B 傽 C
67–70
37. (a) A 傼 C
(b) A 傽 C
67.
38. (a) A 傼 B 傼 C
(b) A 傽 B 傽 C
39–40
■
68.
Find the indicated set if B 5x 0 x 46
A 5x 0 x 26
C 5x 0 1 x 56
■
−3 −2 −1
1
2
3
−3 −2 −1
1
2
3
(b) 3 and 21 (c)
11 8
and 103
40. (a) A 傽 C
(b) A 傽 B
70. (a)
7 15
and 211
42. 12, 8 4
45. 3 2, q 2
46. 1q, 12
44. 36, 12 4
43. 3 2, 8 2
47–52 ■ Express the inequality in interval notation, and then graph the corresponding interval.
7 12 ` 12 7
69. (a) 2 and 17
(b) B 傽 C
41. 13, 0 2
(b) `
Find the distance between the given numbers.
39. (a) B 傼 C
41–46 ■ Express the interval in terms of inequalities, and then graph the interval.
(b) 0 A 13 B 1152 0
6 ` 24
66. (a) `
11
(b) 38 and 57 (c) 2.6 and 1.8 71–72 ■ Express each repeating decimal as a fraction. (See the margin note on page 2.) 71. (a) 0.7
(b) 0.28
(c) 0.57
72. (a) 5.23
(b) 1.37
(c) 2.135
47. x 1
48. 1 x 2
49. 2 x 1
50. x 5
Applications
51. x 1
52. 5 x 2
73. Area of a Garden Mary’s backyard vegetable garden measures 20 ft by 30 ft, so its area is 20 30 600 ft 2. She decides to make it longer, as shown in the ﬁgure, so that the area increases to A 20130 x 2 . Which property of real numbers tells us that the new area can also be written A 600 20x?
53–54
■
53. (a) (b)
Express each set in interval notation. −3
5
−3
5
54. (a) 0 (b) 55–60
−2 ■
20 ft
55. 12, 0 2 傼 11, 1 2
56. 12, 0 2 艚 11, 1 2
59. 1q, 4 2 傼 14, q 2
60. 1q, 64 艚 12, 102
61–66
■
58. 34, 62 傼 3 0, 82
Evaluate each expression.
61. (a) 0 100 0
62. (a) 0 15 5 0
x
Graph the set.
57. 34, 64 艚 3 0, 82
30 ft
2
(b) 0 73 0
(b) 0 10 p 0
74. Temperature Variation The bar graph shows the daily high temperatures for Omak, Washington, and Geneseo, New York, during a certain week in June. Let TO represent the temperature in Omak and TG the temperature in Geneseo. Calculate TO TG and 0 TO TG 0 for each day shown.
12
CHAPTER 1
Fundamentals
product of two rational numbers are rational numbers. Is the product of two irrational numbers necessarily irrational? What about the sum?
Daily high temperature (*F)
Which of these two values gives more information? Omak, WA Geneseo, NY
80 75 70 65
Sun
Mon
Tue
Wed Day
Thu
Fri
Sat
78. Combining Rational Numbers with Irrational Numbers Is 12 12 rational or irrational? Is 12 # 12 rational or irrational? In general, what can you say about the sum of a rational and an irrational number? What about the product? 79. Limiting Behavior of Reciprocals Complete the tables. What happens to the size of the fraction 1/x as x gets large? As x gets small?
75. Mailing a Package The post ofﬁce will only accept packages for which the length plus the “girth” (distance around) is no more than 108 inches. Thus, for the package in the ﬁgure, we must have
x
L
5 ft=60 in. x
6 in. y
1/x
1.0 0.5 0.1 0.01 0.001
1 2 10 100 1000
L 21x y 2 108 (a) Will the post ofﬁce accept a package that is 6 in. wide, 8 in. deep, and 5 ft long? What about a package that measures 2 ft by 2 ft by 4 ft? (b) What is the greatest acceptable length for a package that has a square base measuring 9 in. by 9 in?
x
1/x
80. Irrational Numbers and Geometry Using the following ﬁgure, explain how to locate the point 12 on a number line. Can you locate 15 by a similar method? What about 16? List some other irrational numbers that can be located this way.
8 in. œ∑2
1
Discovery • Discussion 76. Signs of Numbers Let a, b, and c be real numbers such that a 0, b 0, and c 0. Find the sign of each expression. (a) a (b) b (c) bc (d) a b (e) c a (f ) a bc (g) ab ac (h) abc (i) ab2 77. Sums and Products of Rational and Irrational Numbers Explain why the sum, the difference, and the
1.2
_1
1
2
81. Commutative and Noncommutative Operations We have seen that addition and multiplication are both commutative operations. (a) Is subtraction commutative? (b) Is division of nonzero real numbers commutative?
Exponents and Radicals In this section we give meaning to expressions such as a m/n in which the exponent m/n is a rational number. To do this, we need to recall some facts about integer exponents, radicals, and nth roots.
Integer Exponents A product of identical numbers is usually written in exponential notation. For example, 5 # 5 # 5 is written as 53. In general, we have the following deﬁnition.
SECTION 1.2
Exponents and Radicals
13
Exponential Notation If a is any real number and n is a positive integer, then the nth power of a is an a # a # . . . # a 1442443
n factors
The number a is called the base and n is called the exponent.
Example 1
Exponential Notation
(a) A 12 B 5 A 12 BA 12 BA 12 BA 12 BA 12 B 321 Note the distinction between 132 4 and 34. In 13 2 4 the exponent applies to 3, but in 34 the exponent applies only to 3.
(b) 132 4 132 # 132 # 132 # 132 81 (c) 34 13 # 3 # 3 # 32 81
■
We can state several useful rules for working with exponential notation. To discover the rule for multiplication, we multiply 54 by 52: 54 # 52 15 # 5 # 5 # 5215 # 52 5 # 5 # 5 # 5 # 5 # 5 56 542 144 4244 43 123
1444442444443
4 factors 2 factors
6 factors
It appears that to multiply two powers of the same base, we add their exponents. In general, for any real number a and any positive integers m and n, we have aman 1a # a # . . . # a2 1a # a # . . . # a2 a # a # a # . . . # a amn 144 4244 43 1442443
m factors
n factors
144424443 m n factors
Thus aman amn. We would like this rule to be true even when m and n are 0 or negative integers. For instance, we must have 20 # 23 203 23
But this can happen only if 20 1. Likewise, we want to have 54 # 54 54 142 544 50 1
and this will be true if 54 1/54. These observations lead to the following deﬁnition.
Zero and Negative Exponents If a 0 is any real number and n is a positive integer, then a0 1
Example 2
a n
1 an
Zero and Negative Exponents
A 47 B 0
1 1 1 (b) x 1 1 x x 1 1 1 (c) 122 3 8 8 122 3 (a)
and
■
14
CHAPTER 1
Fundamentals
Familiarity with the following rules is essential for our work with exponents and bases. In the table the bases a and b are real numbers, and the exponents m and n are integers.
Laws of Exponents Law
Example
1. a a a m n
2.
mn
am a mn an
3. 1a m 2 n a mn
4. 1ab2 a b n
n n
a n an 5. a b n b b
Description
3 #3 3 2
5
25
3
7
To multiply two powers of the same number, add the exponents.
35 352 33 32 # 132 2 5 32 5 310
To divide two powers of the same number, subtract the exponents. To raise a power to a new power, multiply the exponents.
13 # 42 3 # 4 2
2
2
To raise a product to a power, raise each factor to the power.
3 2 32 a b 2 4 4
To raise a quotient to a power, raise both numerator and denominator to the power.
■
Proof of Law 3 If m and n are positive integers, we have 1a m2 n 1a # a # . . . # a2 n 1444442444443
m factors
1a # a # . . . # a2 1a # a # . . . # a2 . . . 1a # a # . . . # a2 1444442444443 1444442444443
1444442444443
m factors m factors m factors 144444444444424444444444443 n groups of factors
a # a # . . . # a amn
1444 442444 443 mn factors
The cases for which m 0 or n 0 can be proved using the deﬁnition of negative exponents. ■ ■
Proof of Law 4 If n is a positive integer, we have
1ab2 n 1ab2 1ab2 . . . 1ab2 1a # a # . . . # a2 # 1b # b # . . . # b2 a n b n 144424443 n factors
1442443 n factors
1442443 n factors
Here we have used the Commutative and Associative Properties repeatedly. If n 0, Law 4 can be proved using the deﬁnition of negative exponents. ■ You are asked to prove Laws 2 and 5 in Exercise 88.
Example 3 (a) x x x 4 7
47
Using Laws of Exponents x11
(b) y 4y 7 y 47 y 3 (c)
c9 c 95 c 4 c5
Law 1: aman amn
1 y3
Law 1: aman amn Law 2: am/an amn
SECTION 1.2
# (d) 1b 4 2 5 b 4 5 b 20
Exponents and Radicals
15
Law 3: (am)n amn
(e) 13x2 3 33x 3 27x 3
Law 4: (ab)n anbn
x 5 x5 x5 (f) a b 5 2 32 2
Law 5: (a/b)n an/bn
■
Example 4 Simplifying Expressions with Exponents Simplify: x 3 y 2x 4 b (b) a b a z y
(a) 12a 3b 2 2 13ab 4 2 3
Solution (a) 12a 3b 2 2 13ab 4 2 3 12a 3b 2 2 333a 3 1b 4 2 3 4 12a 3b 2 2 127a 3b 12 2
122 1272a 3a 3b 2b 12 54a 6b 14
x 3 y 2x 4 x 3 1y 2 2 4x 4 b 3 (b) a b a z y y z4
x 3 y 8x 4 y 3 z4
1x 3x 4 2 a
Law 4: (ab)n anbn Law 3: (am)n amn Group factors with the same base Law 1: aman amn Laws 5 and 4 Law 3
y8 1 b y 3 z4
x 7y 5 z4
Group factors with the same base Laws 1 and 2
■
When simplifying an expression, you will ﬁnd that many different methods will lead to the same result; you should feel free to use any of the rules of exponents to arrive at your own method. We now give two additional laws that are useful in simplifying expressions with negative exponents.
Laws of Exponents Law a 6. a b b 7.
Example n
b a b a
a n bm b m an
n
3 a b 4
2
Description
4 a b 3
32 45 32 45
2
To raise a fraction to a negative power, invert the fraction and change the sign of the exponent. To move a number raised to a power from numerator to denominator or from denominator to numerator, change the sign of the exponent.
■
Proof of Law 7 Using the deﬁnition of negative exponents and then Property 2 of fractions (page 5), we have 1/a n a n 1 bm bm n# n m m b a 1 a 1/b You are asked to prove Law 6 in Exercise 88.
■
16
CHAPTER 1
Fundamentals
Mathematics in the Modern World Although we are often unaware of its presence, mathematics permeates nearly every aspect of life in the modern world. With the advent of modern technology, mathematics plays an ever greater role in our lives. Today you were probably awakened by a digital alarm clock, made a phone call that used digital transmission, sent an email message over the Internet, drove a car with digitally controlled fuel injection, listened to music on a CD player, then fell asleep in a room whose temperature is controlled by a digital thermostat. In each of these activities mathematics is crucially involved. In general, a property such as the intensity or frequency of sound, the oxygen level in the exhaust emission from a car, the colors in an image, or the temperature in your bedroom is transformed into sequences of numbers by sophisticated mathematical algorithms. These numerical data, which usually consist of many millions of bits (the digits 0 and 1), are then transmitted and reinterpreted. Dealing with such huge amounts of data was not feasible until the invention of computers, machines whose logical processes were invented by mathematicians. The contributions of mathematics in the modern world are not limited to technological advances. The logical processes of mathematics are now used to analyze complex problems in the social, political, and life sciences in new and surprising ways. Advances in mathematics continue to be made, some of the most exciting of these just within the past decade. In other Mathematics in the Modern World, we will describe in more detail how mathematics affects all of us in our everyday activities.
Example 5
Simplifying Expressions with Negative Exponents
Eliminate negative exponents and simplify each expression. y 2 6st 4 (a) (b) a 3 b 2 2 2s t 3z Solution (a) We use Law 7, which allows us to move a number raised to a power from the numerator to the denominator (or vice versa) by changing the sign of the exponent. t4 moves to denominator and becomes t4.
6st 4 6ss 2 2 2 2s t 2t 2t 4 s2 moves to numerator and becomes s2.
3s 3 t6
Law 7
Law 1
(b) We use Law 6, which allows us to change the sign of the exponent of a fraction by inverting the fraction. a
y 2 3z 3 2 b b a y 3z 3
9z 6 y2
Law 6
Laws 5 and 4
■
Scientiﬁc Notation Exponential notation is used by scientists as a compact way of writing very large numbers and very small numbers. For example, the nearest star beyond the sun, Proxima Centauri, is approximately 40,000,000,000,000 km away. The mass of a hydrogen atom is about 0.00000000000000000000000166 g. Such numbers are difﬁcult to read and to write, so scientists usually express them in scientiﬁc notation.
Scientiﬁc Notation A positive number x is said to be written in scientiﬁc notation if it is expressed as follows: x a 10n
where 1 a 10 and n is an integer
For instance, when we state that the distance to the star Proxima Centauri is 4 1013 km, the positive exponent 13 indicates that the decimal point should be moved 13 places to the right: 4 1013 40,000,000,000,000 Move decimal point 13 places to the right.
SECTION 1.2
Exponents and Radicals
17
When we state that the mass of a hydrogen atom is 1.66 1024 g, the exponent 24 indicates that the decimal point should be moved 24 places to the left: 1.66 1024 0.00000000000000000000000166 Move decimal point 24 places to the left.
Example 6
Writing Numbers in Scientiﬁc Notation (b) 0.000627 6.27 104 14243
(a) 327,900 3.279 105 14243 5 places
To use scientiﬁc notation on a calculator, press the key labeled EE or EXP or EEX to enter the exponent. For example, to enter the number 3.629 1015 on a TI83 calculator, we enter 3.629 2ND
EE
■
4 places
Scientiﬁc notation is often used on a calculator to display a very large or very small number. For instance, if we use a calculator to square the number 1,111,111, the display panel may show (depending on the calculator model) the approximation 1.234568 12
1.23468
or
E12
Here the ﬁnal digits indicate the power of 10, and we interpret the result as
15
1.234568 1012
and the display reads 3.629E15
Example 7 Calculating with Scientiﬁc Notation If a ⬇ 0.00046, b ⬇ 1.697 1022, and c ⬇ 2.91 1018, use a calculator to approximate the quotient ab/c. Solution We could enter the data using scientiﬁc notation, or we could use laws of exponents as follows: 14.6 104 2 11.697 1022 2 ab ⬇ c 2.91 1018
14.62 11.6972 1042218 2.91
⬇ 2.7 1036 We state the answer correct to two signiﬁcant ﬁgures because the least accurate of the given numbers is stated to two signiﬁcant ﬁgures.
■
Radicals We know what 2n means whenever n is an integer. To give meaning to a power, such as 24/5, whose exponent is a rational number, we need to discuss radicals. The symbol 1 means “the positive square root of.” Thus It is true that the number 9 has two square roots, 3 and 3, but the notation 19 is reserved for the positive square root of 9 (sometimes called the principal square root of 9). If we want the negative root, we must write 19, which is 3.
1a b
means
b2 a
and
b0
Since a b2 0, the symbol 1a makes sense only when a 0. For instance, 19 3
because
32 9
and
30
18
CHAPTER 1
Fundamentals
Square roots are special cases of nth roots. The nth root of x is the number that, when raised to the nth power, gives x.
Deﬁnition of nth Root If n is any positive integer, then the principal nth root of a is deﬁned as follows: n 1 ab
bn a
means
If n is even, we must have a 0 and b 0.
Thus 4 1 81 3
because
3 1 8 2
because
34 81
and
122 3 8
30
4 6 But 18, 1 8, and 1 8 are not deﬁned. (For instance, 18 is not deﬁned because the square of every real number is nonnegative.) Notice that
242 116 4
2142 2 116 4 0 4 0
but
So, the equation 2a 2 a is not always true; it is true only when a 0. However, we can always write 2a 2 0 a 0 . This last equation is true not only for square roots, but for any even root. This and other rules used in working with nth roots are listed in the following box. In each property we assume that all the given roots exist.
Properties of nth Roots Property
Example
n
n
18 # 27 18127 122 132 6
n
3
1. 2ab 2a2b n
2.
4
_ mn m n 3. 3 1 a 3a n
3 6 31729 1729 3
if n is odd
5. 2a n 0 a 0 n
if n is even
3 2 152 3 5,
5 5 2 2 2
4 2 132 4 0 3 0 3
Simplifying Expressions Involving nth Roots
Example 8
3 3 (a) 2x 4 2x 3x 3
3
4 16 1 16 2 4 3 B 81 181
a 2a n Bb 2b n
4. 2a n a
3
3 3
Factor out the largest cube
2x 2x
3 3 3 Property 1: 1 ab 1a1b
3 x2 x
3 3 Property 4: 2 a a
SECTION 1.2
4 4 4 8 4 4 (b) 2 81x 8y 4 2 812 x 2y
Exponents and Radicals
19
4 4 4 4 Property 1: 2 abc 2 a2 b2 c
4 32 1x 2 2 4 0 y 0
4 Property 5: 2a4 0 a 0
3x 2 0 y 0
4 Property 5: 2a4 0 a 0 , 0 x2 0 x2
■
It is frequently useful to combine like radicals in an expression such as 213 513. This can be done by using the Distributive Property. Thus 213 513 12 52 13 713
The next example further illustrates this process.
Example 9
Avoid making the following error:
Combining Radicals
(a) 132 1200 116 # 2 1100 # 2
1a b 1a 1b For instance, if we let a 9 and b 16, then we see the error: 19 16 ⱨ 19 116 125 ⱨ 3 4 5ⱨ7
Wrong!
Factor out the largest squares
11612 110012
Property 1: 1ab 1a1b
412 1012 1412
Distributive Property
(b) If b 0, then 225b 2b 3 2252b 2b 2 2b 52b b2b
15 b2 2b
Property 1: 1ab 1a1b Property 5, b 0 Distributive Property
■
Rational Exponents To deﬁne what is meant by a rational exponent or, equivalently, a fractional exponent such as a1/3, we need to use radicals. In order to give meaning to the symbol a1/n in a way that is consistent with the Laws of Exponents, we would have to have 1a 1/n 2 n a 11/n2n a 1 a So, by the deﬁnition of nth root, n a 1/n 1 a
In general, we deﬁne rational exponents as follows.
Deﬁnition of Rational Exponents For any rational exponent m/n in lowest terms, where m and n are integers and n 0, we deﬁne n a m/n 1 1 a2 m
or equivalently
n
a m/n 2a m
If n is even, then we require that a 0.
With this deﬁnition it can be proved that the Laws of Exponents also hold for rational exponents.
CHAPTER 1
Fundamentals
Diophantus lived in Alexandria about 250 A.D. His book Arithmetica is considered the ﬁrst book on algebra. In it he gives methods for ﬁnding integer solutions of algebraic equations. Arithmetica was read and studied for more than a thousand years. Fermat (see page 652) made some of his most important discoveries while studying this book. Diophantus’ major contribution is the use of symbols to stand for the unknowns in a problem. Although his symbolism is not as simple as what we use today, it was a major advance over writing everything in words. In Diophantus’ notation the equation x5 7x2 8x 5 24 is written
K©å h ©zM° ´iskd Our modern algebraic notation did not come into common use until the 17th century.
Example 10
Using the Deﬁnition of Rational Exponents
(a) 41/2 14 2
3 (b) 82/3 1 1 82 2 22 4
(c) 1251/3 (d)
1 3
2x
4
1 1/3
125
1 x
1
3
1125
3 2 3 Alternative solution: 82/3 2 8 2 64 4
1 5
x 4/3
4/3
Example 11
■
Using the Laws of Exponents with Rational Exponents
(a) a 1/3a 7/3 a 8/3 (b)
a 2/5a 7/5 a 3/5
Law 1: aman amn
a 2/57/53/5 a 6/5
Law 1, Law 2:
(c) 12a 3b 4 2 3/2 23/2 1a 3 2 3/2 1b 4 2 3/2
am amn an
Law 4: 1abc 2 n anbncn
1 122 3a 313/22b 413/22
c
20
Law 3: 1am 2 n amn
212a 9/2b 6 (d) a
2x 3/4 y 1/3
b a
y4
3
x
b 1/2
23 1x 3/4 2 3 1y 1/3 2 3
# 1y 4x 1/2 2
8x 9/4 # 4 1/2 y x y
Law 3
8x 11/4y 3
Example 12
Laws 1 and 2
■
Simplifying by Writing Radicals as Rational Exponents
3 (a) 121x2 131 x2 12x 1/2 2 13x 1/3 2
6x 1/21/3 6x 5/6
(b) 3x2x 1xx
2
1/2 1/2
1x 3/2 2 1/2 x
Laws 5, 4, and 7
3/4
Deﬁnition of rational exponents Law 1 Deﬁnition of rational exponents Law 1 Law 3
■
Rationalizing the Denominator It is often useful to eliminate the radical in a denominator by multiplying both numerator and denominator by an appropriate expression. This procedure is called rationalizing the denominator. If the denominator is of the form 1a, we multiply numerator and denominator by 1a. In doing this we multiply the given quantity by 1, so we do not change its value. For instance, 1 1 # 1 # 1a 1a 1 a 1a 1a 1a 1a
SECTION 1.2
Exponents and Radicals
21
Note that the denominator in the last fraction contains no radical. In general, if the n denominator is of the form 2a m with m n, then multiplying the numerator and den nm nominator by 2a will rationalize the denominator, because (for a 0) n
n
n
n
2a m 2a nm 2a mnm 2a n a
Example 13 (a) (b)
(c)
1.2
2 213 2 # 13 3 13 13 13 1 3 2 2 x
1
1.
1 15
2.
3 2 2 7
Exponential expression
6.
21.5
7.
a2/5
23–26
10. (a) 52 # A 15 B 3
(b)
43 11. (a) 8 2
32 (b) 9
12. (a)
(b)
8 B 27 3
(c) 322/5
(b) A 278 B 2/3
3/2 (c) A 25 64 B
7
10 104
A 32 B 2
(c)
(c)
A 12 B 4
# A 52 B 2
(c) 11/ 16
3
5
(b) 164 1 B 64 3
27. a9a5 29. 112x 2y 4 2 A 12 x 5yB 31. 33.
(c)
5 13 5
196
24. 175 148 4 4 26. 148 13
x 12x 2 x
37. 39.
30. 16y 2 3 32.
3
b4 A 13b2 B 112b8 2 16y 3 2 4 2y
28. 13y 2 2 14y 5 2
4
35. 1rs 2 3 12s 2 2 14r 2 4
4
(c) 132
22. 1xy 2 2z
27–44 ■ Simplify the expression and eliminate any negative exponent(s).
9
4
(b) 116
(b)
3 32
(c) A 14 B 2
# 169
5
25. 196 13
(c) 13 2 0
4 3 20. 2 x 14y 2z
Simplify the expression.
5
Evaluate each expression. (b) 13 2 2
■
23. 132 118
2x 5
15. (a)
(b) 1322 2/5
21. 19x 2 2/3 12y 2 2/3 z 2/3
1
14. (a) 164
4 4 (c) 1 241 54
17. (a) A 49 B 1/2
19. 2x 2 y 2
5 3 2 5
9. (a) 32
148 13
(b)
19–22 ■ Evaluate the expression using x 3, y 4, and z 1. 113/2
13. (a) 116
■
16. (a) 17128
18. (a) 10240.1
4.
A 23 B 3
3 3 2 x
3 1 x x
7 5 7 5 7 5 1 1 1 2 a 2 a 2 a 2 7 7 7 7 2 2 5 7 a Ba 2a 2a 2a 2a
42/3
■
3 1 x
7
3.
9–18
3
Exercises
Radical expression
8.
3 1 x
3 2 2 x 1x
1–8 ■ Write each radical expression using exponents, and each exponential expression using radicals.
5.
Rationalizing Denominators
34. 12s3t1 2 A 14s6 B 116t4 2 36. 12u2√3 2 3 13u3√2 2 38.
5
1x 2y 3 2 4 1xy 4 2 3 2
x y
a 3b 4 a 5b 5
12x 3 2 2 13x 4 2
40. a
1x 3 2 4
d2 3 c 4d 3 ba 3b 2 cd c
22
41.
CHAPTER 1
1xy 2z 3 2 4
1x y z2 3 2
43. a
42. a
3
q 1rs 2 5
r sq
Fundamentals
8
b
1
xy 2z 3 2 3 4
x y z
44. 13ab 2c 2 a
b
3
2a 2b 2 b c3
■
Simplify the expression. Assume the letters denote any 45–52 real numbers. 4 4 45. 2 x
5 10 46. 2 x
4 47. 2 16x 8
3 3 6 48. 2 x y
49. 2a 2b 6
3 2 3 4 50. 2 a b2a b
3 51. 3 264x 6
4 52. 2x 4y 2z 2
53–70 ■ Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. 53. x 2/3x 1/5 55. 13a 1/4 2 19a2 3/2 57. 14b 2 1/2 18b 2/5 2 59. 1c 2d 3 2 1/3 61. 1 y
2
63. 12x y 65. a 67. a 69.
x 6y y4
2 18y 2
b
b 4b 1/3 19st2 3/2 3a
2 2/3
71–72
■
(c) 0.000028536 72. (a) 129,540,000 (c) 0.0000000014 73–74
■
(c) 2.670 108 74. (a) 7.1 1014 (c) 8.55 103
78. 11.062 1024 2 18.61 1019 2
62. 1a
2
64. 1x y z 2
5 3 10 3/5
68.
2x 1/3
b 1/6
81. 4
y 1/2z 1y 10z 5 2 1/5 1y 2z 3 2 1/3
70. a
a 2b 3 3 x 2b 1 b a 3/2 1/3 b x 1y 2 a y
1.295643 109 13.610 1017 2 12.511 106 2 173.12 11.6341 1028 2 0.0000000019
10.00001622 10.015822
1594,621,000 2 10.0058 2
83–86
(d) 0.0001213 (b) 7,259,000,000 (d) 0.0007029
(b) 2.721 108 (d) 9.999 109 (b) 6 1012 (d) 6.257 1010
■
82.
13.542 106 2 9 15.05 104 2 12
Rationalize the denominator.
83. (a)
1 110
(b)
2 Bx
(c)
x B3
84. (a)
5 B 12
(b)
x B6
(c)
y B 2z
85. (a)
(b) 7,200,000,000,000
Write each number in decimal notation.
73. (a) 3.19 105
77. 17.2 109 2 11.806 1012 2
80.
2/5 3/4
Write each number in scientiﬁc notation.
71. (a) 69,300,000
77–82 ■ Use scientiﬁc notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer correct to the number of signiﬁcant digits indicated by the given data.
79.
1
127s 3t 4 2 2/3
76. (a) The distance from the earth to the sun is about 93 million miles. (b) The mass of an oxygen molecule is about 0.000000000000000000000053 g. (c) The mass of the earth is about 5,970,000,000,000,000,000,000,000 kg.
58. 18x 6 2 2/3
66. a
5/2
2
56. 12a 3/4 2 15a 3/2 2 60. 14x 6y 8 2 3/2
3/4 2/3 4 4/5 3
54. 12x 3/2 2 14x2 1/2
(b) The diameter of an electron is about 0.0000000000004 cm. (c) A drop of water contains more than 33 billion billion molecules.
86. (a)
2 3
1x 1 4
1a
(b) (b)
1 4
2y
3
a 3
2b
2
(c) (c)
x y 2/5 1 c 3/7
87. Let a, b, and c be real numbers with a 0, b 0, and c 0. Determine the sign of each expression. (a) b5
(b) b10
(c) ab2c3
(d) 1b a 2 3
(e) 1b a 2 4
(f)
a 3c 3 b 6c 6
88. Prove the given Laws of Exponents for the case in which m and n are positive integers and m n. (a) Law 2
(b) Law 5
(c) Law 6
75–76 ■ Write the number indicated in each statement in scientiﬁc notation.
Applications
75. (a) A lightyear, the distance that light travels in one year, is about 5,900,000,000,000 mi.
89. Distance to the Nearest Star Proxima Centauri, the star nearest to our solar system, is 4.3 lightyears away. Use the
SECTION 1.2
information in Exercise 75(a) to express this distance in miles. 90. Speed of Light The speed of light is about 186,000 mi/s. Use the information in Exercise 76(a) to ﬁnd how long it takes for a light ray from the sun to reach the earth. 91. Volume of the Oceans The average ocean depth is 3.7 103 m, and the area of the oceans is 3.6 1014 m2. What is the total volume of the ocean in liters? (One cubic meter contains 1000 liters.)
Exponents and Radicals
23
95. Speed of a Skidding Car Police use the formula s 230fd to estimate the speed s (in mi/h) at which a car is traveling if it skids d feet after the brakes are applied suddenly. The number f is the coefﬁcient of friction of the road, which is a measure of the “slipperiness” of the road. The table gives some typical estimates for f.
Dry Wet
Tar
Concrete
Gravel
1.0 0.5
0.8 0.4
0.2 0.1
(a) If a car skids 65 ft on wet concrete, how fast was it moving when the brakes were applied? (b) If a car is traveling at 50 mi/h, how far will it skid on wet tar?
92. National Debt As of November 2004, the population of the United States was 2.949 108, and the national debt was 7.529 1012 dollars. How much was each person’s share of the debt? 93. Number of Molecules A sealed room in a hospital, measuring 5 m wide, 10 m long, and 3 m high, is ﬁlled with pure oxygen. One cubic meter contains 1000 L, and 22.4 L of any gas contains 6.02 1023 molecules (Avogadro’s number). How many molecules of oxygen are there in the room? 94. How Far Can You See? Due to the curvature of the earth, the maximum distance D that you can see from the top of a tall building of height h is estimated by the formula D 22rh h 2 where r 3960 mi is the radius of the earth and D and h are also measured in miles. How far can you see from the observation deck of the Toronto CN Tower, 1135 ft above the ground?
96. Distance from the Earth to the Sun It follows from Kepler’s Third Law of planetary motion that the average distance from a planet to the sun (in meters) is d a
GM 1/3 2/3 b T 4p2
where M 1.99 1030 kg is the mass of the sun, G 6.67 1011 N # m2/kg2 is the gravitational constant, and T is the period of the planet’s orbit (in seconds). Use the fact that the period of the earth’s orbit is about 365.25 days to ﬁnd the distance from the earth to the sun. 97. Flow Speed in a Channel The speed of water ﬂowing in a channel, such as a canal or river bed, is governed by the Manning Equation V 1.486
CN Tower r
A2/3S 1/2 p 2/3n
Here V is the velocity of the ﬂow in ft/s; A is the crosssectional area of the channel in square feet; S is the downward slope of the channel; p is the wetted perimeter in feet (the distance from the top of one bank, down the side of the channel, across the bottom, and up to the top of the other bank); and n is the roughness coefﬁcient (a measure of the roughness of the channel bottom). This equation is used to predict the capacity of ﬂood channels to handle runoff from
24
CHAPTER 1
Fundamentals
heavy rainfalls. For the canal shown in the ﬁgure, A 75 ft 2, S 0.050, p 24.1 ft, and n 0.040. (a) Find the speed with which water ﬂows through this canal. (b) How many cubic feet of water can the canal discharge per second? [Hint: Multiply V by A to get the volume of the ﬂow per second.]
99. Easy Powers That Look Hard Calculate these expressions in your head. Use the Laws of Exponents to help you. 185 (a) 5 (b) 206 # 10.5 2 6 9 100. Limiting Behavior of Powers Complete the following tables. What happens to the nth root of 2 as n gets large? What about the nth root of 12 ?
21/n
n 20 ft
1 2 5 10 100
5 ft 10 ft
A 12 B 1/n
1 2 5 10 100
Construct a similar table for n1/n. What happens to the nth root of n as n gets large?
Discovery • Discussion 98. How Big Is a Billion? If you have a million (106) dollars in a suitcase, and you spend a thousand (10 3) dollars each day, how many years would it take you to use all the money? Spending at the same rate, how many years would it take you to empty a suitcase ﬁlled with a billion (10 9 ) dollars?
1.3
n
101. Comparing Roots Without using a calculator, determine which number is larger in each pair. (a) 21/2 or 21/3 (b) A 12 B 1/2 or A 12 B 1/3 (c) 71/4 or 41/3
3 (d) 1 5 or 13
Algebraic Expressions A variable is a letter that can represent any number from a given set of numbers. If we start with variables such as x, y, and z and some real numbers, and combine them using addition, subtraction, multiplication, division, powers, and roots, we obtain an algebraic expression. Here are some examples: 2x 2 3x 4
1x 10
y 2z y2 4
A monomial is an expression of the form ax k, where a is a real number and k is a nonnegative integer. A binomial is a sum of two monomials and a trinomial is a sum of three monomials. In general, a sum of monomials is called a polynomial. For example, the ﬁrst expression listed above is a polynomial, but the other two are not.
Polynomials A polynomial in the variable x is an expression of the form a n x n a n1x n1 . . . a 1x a 0 where a0, a1, . . . , an are real numbers, and n is a nonnegative integer. If an 0, then the polynomial has degree n. The monomials a k x k that make up the polynomial are called the terms of the polynomial.
SECTION 1.3
Algebraic Expressions
25
Note that the degree of a polynomial is the highest power of the variable that appears in the polynomial. Polynomial
Type
2x 2 3x 4
trinomial
x 5x 8
3xx 2
1 3 2x
5x 1 9x
5
6
Terms
2x 2, 3x, 4 8
binomial
x , 5x
four terms
12 x 3,
binomial
5x, 1
monomial
9x
monomial
6
Degree
2 8
x , x, 3 2
5
3 1 5 0
Combining Algebraic Expressions
Distributive Property
We add and subtract polynomials using the properties of real numbers that were discussed in Section 1.1. The idea is to combine like terms (that is, terms with the same variables raised to the same powers) using the Distributive Property. For instance,
ac bc 1a b 2 c
5x 7 3x 7 15 32x 7 8x 7 In subtracting polynomials we have to remember that if a minus sign precedes an expression in parentheses, then the sign of every term within the parentheses is changed when we remove the parentheses: 1b c 2 b c [This is simply a case of the Distributive Property, a1b c2 ab ac, with a 1.]
Example 1
Adding and Subtracting Polynomials
(a) Find the sum 1x 3 6x 2 2x 42 1x 3 5x 2 7x2 . (b) Find the difference 1x 3 6x 2 2x 42 1x 3 5x 2 7x2 . Solution (a) 1x 3 6x 2 2x 42 1x 3 5x 2 7x2
1x 3 x 3 2 16x 2 5x 2 2 12x 7x2 4
Group like terms
2x 3 x 2 5x 4
Combine like terms
(b) 1x 6x 2x 42 1x 5x 7x2 3
2
3
2
x 3 6x 2 2x 4 x 3 5x 2 7x
1x x 2 16x 5x 2 12x 7x2 4 3
3
11x 2 9x 4
2
2
Distributive Property Group like terms Combine like terms
■
To ﬁnd the product of polynomials or other algebraic expressions, we need to use the Distributive Property repeatedly. In particular, using it three times on the product of two binomials, we get 1a b2 1c d2 a1c d2 b1c d2 ac ad bc bd
26
CHAPTER 1
Fundamentals
This says that we multiply the two factors by multiplying each term in one factor by each term in the other factor and adding these products. Schematically we have The acronym FOIL helps us remember that the product of two binomials is the sum of the products of the First terms, the Outer terms, the Inner terms, and the Last terms.
1a b2 1c d2 ac ad bc bd 앖 F
앖 O
앖 I
앖 L
In general, we can multiply two algebraic expressions by using the Distributive Property and the Laws of Exponents.
Example 2
Multiplying Algebraic Expressions
(a) 12x 12 13x 52 6x 2 10x 3x 5 앖 F
앖 O
앖 I
Distributive Property
앖 L
6x 2 7x 5
Combine like terms
(b) 1x 2 32 1x 3 2x 12 x 2 1x 3 2x 12 31x 3 2x 12
Distributive Property
x 5 2x 3 x 2 3x 3 6x 3
Distributive Property
x 5 x 3 x 2 6x 3
Combine like terms
(c) 11 1x2 12 31x2 2 31x 21x 31 1x2 2 2 1x 3x
Distributive Property Combine like terms
■
Certain types of products occur so frequently that you should memorize them. You can verify the following formulas by performing the multiplications. See the Discovery Project on page 34 for a geometric interpretation of some of these formulas.
Special Product Formulas If A and B are any real numbers or algebraic expressions, then 1. 1A B2 1A B2 A2 B 2 Sum and product of same terms 2. 1A B2 2 A2 2AB B 2 3. 1A B2 A 2AB B 2
2
Square of a sum
2
4. 1A B2 A 3A B 3AB B 3
3
2
2
5. 1A B2 A 3A B 3AB B 3
3
2
2
Square of a difference 3
Cube of a sum
3
Cube of a difference
The key idea in using these formulas (or any other formula in algebra) is the Principle of Substitution: We may substitute any algebraic expression for any letter in a formula. For example, to ﬁnd 1x 2 y 3 2 2 we use Product Formula 2, substituting x 2 for A and y 3 for B, to get 1x 2 y 3 2 2 1x 2 2 2 21x 2 2 1 y 3 2 1y 3 2 2 (A B)2 A2
2AB
B2
SECTION 1.3
Example 3
Algebraic Expressions
27
Using the Special Product Formulas
Use the Special Product Formulas to ﬁnd each product. (a) 13x 52 2 (b) 1x 2 22 3 (c) 12x 1y2 12x 1y2 Solution (a) Substituting A 3x and B 5 in Product Formula 2, we get
13x 52 2 13x2 2 213x2 152 52 9x 2 30x 25
(b) Substituting A x 2 and B 2 in Product Formula 5, we get
1x 2 22 3 1x 2 2 3 31x 2 2 2 122 31x 2 2 12 2 2 23 x 6 6x 4 12x 2 8
(c) Substituting A 2x and B 1y in Product Formula 1, we get 12x 1y2 12x 1y2 12x2 2 1 1y2 2 4x 2 y
■
Factoring We use the Distributive Property to expand algebraic expressions. We sometimes need to reverse this process (again using the Distributive Property) by factoring an expression as a product of simpler ones. For example, we can write x 2 4 1x 22 1x 22
We say that x 2 and x 2 are factors of x 2 4. The easiest type of factoring occurs when the terms have a common factor.
Example 4
Factoring Out Common Factors
Factor each expression. (a) 3x 2 6x (c) 12x 42 1x 32 51x 3 2
Check Your Answer
Solution (a) The greatest common factor of the terms 3x 2 and 6x is 3x, so we have 3x 2 6x 3x 1x 22
Multiplying gives 3x1x 2 2 3x 2 6x
(b) 8x 4y 2 6x 3y 3 2xy 4
(b) We note that 8, 6, and 2 have the greatest common factor 2 x 4, x 3, and x have the greatest common factor x
Check Your Answer Multiplying gives 2xy 14x 3x y y 2 2
3
2
2
8x 4y 2 6x 3y 3 2xy 4
y 2, y 3, and y 4 have the greatest common factor y 2 So the greatest common factor of the three terms in the polynomial is 2xy 2, and we have 8x 4y 2 6x 3y 3 2xy 4 12xy 2 2 14x 3 2 12xy 2 2 13x 2y2 12xy 2 2 1y 2 2 2xy 2 14x 3 3x 2y y 2 2
28
CHAPTER 1
Fundamentals
(c) The two terms have the common factor x 3.
12x 42 1x 32 51x 3 2 3 12x 42 54 1x 32 12x 12 1x 32
Distributive Property Simplify
■
To factor a trinomial of the form x 2 bx c, we note that 1x r2 1x s2 x 2 1r s2x rs
so we need to choose numbers r and s so that r s b and rs c.
Example 5
Factoring x 2 ⴙ bx ⴙ c by Trial and Error
Factor: x 2 7x 12 Solution We need to ﬁnd two integers whose product is 12 and whose sum is 7. By trial and error we ﬁnd that the two integers are 3 and 4. Thus, the factorization is
Check Your Answer
x 2 7x 12 1x 32 1x 42
Multiplying gives 1x 32 1x 42 x 7x 12 2
factors of 12
factors of a 앗 앗
To factor a trinomial of the form ax2 bx c with a 1, we look for factors of the form px r and qx s: ax 2 bx c 1 px r2 1qx s2 pqx 2 1 ps qr2x rs
ax 2 bx c Ó px rÔÓqx sÔ 앖 앖 factors of c
Therefore, we try to ﬁnd numbers p, q, r, and s such that pq a, rs c, ps qr b. If these numbers are all integers, then we will have a limited number of possibilities to try for p, q, r, and s.
Example 6 Factor: Check Your Answer
■
Factoring ax 2 ⴙ bx ⴙ c by Trial and Error
6x 2 7x 5
Solution We can factor 6 as 6 # 1 or 3 # 2, and 5 as 25 # 1 or 5 # 112 . By trying these possibilities, we arrive at the factorization
Multiplying gives
factors of 6
13x 5 2 12x 1 2 6x 2 7x 5
6x 2 7x 5 13x 52 12x 12 factors of 5
Example 7
Recognizing the Form of an Expression
Factor each expression. (a) x 2 2x 3 (b) 15a 12 2 215a 1 2 3 Solution (a) x 2 2x 3 1x 32 1x 12 (b) This expression is of the form
Trial and error 2
2
3
■
SECTION 1.3
Algebraic Expressions
29
where represents 5a 1. This is the same form as the expression in part (a), so it will factor as 1 321 12. 1 5a 1 2 2 21 5a 1 2 3 31 5a 1 2 34 31 5a 1 2 14 15a 22 15a 22
■
Some special algebraic expressions can be factored using the following formulas. The ﬁrst three are simply Special Product Formulas written backward.
Special Factoring Formulas Formula
Name
1. A2 B 2 1A B2 1A B2 2. A 2AB B 1A B2 2
2
Difference of squares
2
Perfect square
3. A2 2AB B 2 1A B2 2
Perfect square
4. A B 1A B2 1A AB B 2 3
3
2
2
5. A3 B 3 1A B2 1A2 AB B 2 2
Example 8
Difference of cubes Sum of cubes
Factoring Differences of Squares
Factor each polynomial. (a) 4x 2 25 (b) 1x y2 2 z 2 Solution (a) Using the Difference of Squares Formula with A 2x and B 5, we have 4x 2 25 12x2 2 52 12x 5 2 12x 52 A2 B2 (A B)(A B)
(b) We use the Difference of Squares Formula with A x y and B z. 1x y2 2 z 2 1x y z2 1x y z2
Example 9
Factoring Differences and Sums of Cubes
Factor each polynomial. (a) 27x 3 1 (b) x 6 8 Solution (a) Using the Difference of Cubes Formula with A 3x and B 1, we get 27x 3 1 13x2 3 13 13x 12 3 13x2 2 13x2 112 12 4 13x 12 19x 2 3x 12
■
30
CHAPTER 1
Fundamentals
Mathematics in the Modern World Changing Words, Sound, and Pictures into Numbers Pictures, sound, and text are routinely transmitted from one place to another via the Internet, fax machines, or modems. How can such things be transmitted through telephone wires? The key to doing this is to change them into numbers or bits (the digits 0 or 1). It’s easy to see how to change text to numbers. For example, we could use the correspondence A 00000001, B 00000010, C 00000011, D 00000100, E 00000101, and so on. The word “BED” then becomes 000000100000010100000100. By reading the digits in groups of eight, it is possible to translate this number back to the word “BED.” Changing sound to bits is more complicated. A sound wave can be graphed on an oscilloscope or a computer. The graph is then broken down mathematically into simpler components corresponding to the different frequencies of the original sound. (A branch of mathematics called Fourier analysis is used here.) The intensity of each component is a number, and the original sound can be reconstructed from these numbers. For example, music is stored on a CD as a sequence of bits; it may look like 101010001010010100101010 1000001011110101000101011. . . . (One second of music requires 1.5 million bits!) The CD player reconstructs the music from the numbers on the CD. Changing pictures into numbers involves expressing the color and brightness of each dot (or pixel) into a number. This is done very efﬁciently using a branch of mathematics called wavelet theory. The FBI uses wavelets as a compact way to store the millions of ﬁngerprints they need on ﬁle.
(b) Using the Sum of Cubes Formula with A x 2 and B 2, we have x 6 8 1x 2 2 3 23 1x 2 22 1x 4 2x 2 42
■
A trinomial is a perfect square if it is of the form A2 2AB B 2
or
A2 2AB B 2
So, we recognize a perfect square if the middle term (2AB or 2AB) is plus or minus twice the product of the square roots of the outer two terms.
Example 10
Recognizing Perfect Squares
Factor each trinomial. (a) x 2 6x 9
(b) 4x 2 4xy y 2
Solution (a) Here A x and B 3, so 2AB 2 # x # 3 6x. Since the middle term is 6x, the trinomial is a perfect square. By the Perfect Square Formula, we have x 2 6x 9 1x 32 2
(b) Here A 2x and B y, so 2AB 2 # 2x # y 4xy. Since the middle term is 4xy, the trinomial is a perfect square. By the Perfect Square Formula, we have 4x 2 4xy y 2 12x y2 2
■
When we factor an expression, the result can sometimes be factored further. In general, we ﬁrst factor out common factors, then inspect the result to see if it can be factored by any of the other methods of this section. We repeat this process until we have factored the expression completely.
Example 11
Factoring an Expression Completely
Factor each expression completely. (a) 2x 4 8x 2 (b) x 5y 2 xy 6 Solution (a) We ﬁrst factor out the power of x with the smallest exponent. 2x 4 8x 2 2x 2 1x 2 42 Common factor is 2x 2 2x 2 1x 22 1x 22
Factor x 2 4 as a difference of squares
(b) We ﬁrst factor out the powers of x and y with the smallest exponents. x 5y 2 xy 6 xy 2 1x 4 y 4 2 Common factor is xy 2 xy 2 1x 2 y 2 2 1x 2 y 2 2
xy 2 1x 2 y 2 2 1x y2 1x y2
Factor x 4 y 4 as a difference of squares Factor x 2 y 2 as a difference of squares
■
In the next example we factor out variables with fractional exponents. This type of factoring occurs in calculus.
SECTION 1.3
Example 12 To factor out x1/2 from x 3/2, we subtract exponents:
x 3/2 x 1/2 1x 3/2 11/22 2 x
1/2
x
1/2
1x
3/21/2
1x 2
2
2
Factor each expression. (a) 3x 3/2 9x 1/2 6x 1/2
(a) 3x
1x 3x 22
3x
2
3/2
9x
1/2
6x
(b) 12 x2 2/3x 12 x2 1/3
Solution (a) Factor out the power of x with the smallest exponent, that is, x1/2. 3x 3/2 9x 1/2 6x 1/2 3x 1/2 1x 2 3x 22
12 x 2 2/3x 12 x 2 1/3
Factor the quadratic x 2 3x 2
(b) Factor out the power of 2 x with the smallest exponent, that is, 12 x2 2/3. 12 x2 2/3x 12 x2 1/3 12 x2 2/3 3x 12 x 2 4 12 x2 2/3 12 2x2
1/2
(b) 12 x2 2/3 3 x 12 x2 4
212 x2
2/3
Factor out 12 x2 2/3 Simplify
11 x2
Factor out 2
Factoring by Grouping
Factor each polynomial. (a) x 3 x 2 4x 4
(b) x 3 2x 2 3x 6
Solution (a) x 3 x 2 4x 4 1x 3 x 2 2 14x 42
Group terms
x 1x 12 41x 12 2
Factor out common factors
1x 2 42 1x 12
(b) x 2x 3x 6 1x 2x 2 13x 62 3
2
3
2
x 1x 22 31x 22 2
1x 32 1x 22
Factor out x 1 from each term Group terms Factor out common factors Factor out x 2 from each term
2
Exercises
1–6 ■ Complete the following table by stating whether the polynomial is a monomial, binomial, or trinomial; then list its terms and state its degree.
7–42
■
Perform the indicated operations and simplify.
7. 112x 72 15x 122
1. x 2 3x 7 2. 2x 5 4x 2 3. 8 4. 12 x 7 5. x x 2 x 3 x 4 6. 12 x 13
Type
Terms
Degree
8. 15 3x2 12x 82
9. 13x x 1 2 12x 3x 5 2 2
Polynomial
■
Polynomials with at least four terms can sometimes be factored by grouping terms. The following example illustrates the idea.
Example 13
1.3
Factor out 3x1/2
3x 1/2 1x 12 1x 22
To see that you have factored correctly, multiply using the Laws of Exponents.
31
Factoring Expressions with Fractional Exponents
Check Your Answer
1/2
Algebraic Expressions
2
10. 13x 2 x 1 2 12x 2 3x 5 2
11. 1x 3 6x 2 4x 7 2 13x 2 2x 4 2 12. 31x 1 2 41x 2 2
13. 812x 5 2 71x 9 2
14. 41x 2 3x 5 2 31x 2 2x 12 15. 212 5t 2 t 2 1t 1 2 1t 4 12
16. 513t 4 2 1t 2 22 2t1t 3 2
■
32
CHAPTER 1
Fundamentals
17. 1x 1x 1x2
18. x 3/2 1 1x 1/ 1x2
21. 1x 2y 2 13x y2
22. 14x 3y2 12x 5y2
19. 13t 22 17t 5 2 23. 11 2y 2 2
20. 14x 1 2 13x 7 2 24. 13x 42 2
25. 12x 3y 2
27. 12x 5 2 1x 2 x 12
28. 11 2x 2 1x 2 3x 1 2
2 2
29. 1x 2 a 2 2 1x 2 a 2 2 31. a1a
30. 1x 1/2 y 1/2 2 1x 1/2 y 1/2 2
1 1 b a1a b b b
32. 1 2h 2 1 12 1 2h 2 1 1 2 33. 11 a 3 2 3
34. 11 2y2 3
36. 13x 3 x 2 2 2 1x 2 2x 12 37. 11 x 4/3 2 11 x 2/3 2
38. 11 b 2 2 11 b 2 2
41. 1x y z 2 1x y z 2
42. 1x 2 y z 2 1x 2 y z2
39. 13x 2y 7xy 2 2 1x 2y 3 2y 2 2 40. 1x 4y y 5 2 1x 2 xy y 2 2
■
Factor out the common factor.
43. 2x 3 16x
45. y1y 6 2 91y 6 2 47. 2x 2y 6xy 2 3xy 49–54
■
44. 2x 4 4x 3 14x 2
46. 1z 2 2 2 51z 2 2 48. 7x 4y 2 14xy 3 21xy 4
49. x 2x 3
50. x 6x 5
51. 8x 14x 15
52. 6y 11y 21
2
2
53. 13x 22 813x 2 2 12
55. 9a 16
56. 1x 3 2 4
57. 27x y
58. 8s 125t
59. x 12x 36
60. 16z 24z 9
2
61–66
■
2
3
6
2
Factor the expression by grouping terms.
61. x 4x x 4
62. 3x x 6x 2
63. 2x x 6x 3
64. 9x 3x 3x 1
65. x x x 1
66. x 5 x 4 x 1
3
2
3
3
2
2
71. 12x 18x
72. 5ab 8abc
73. x 2x 8
74. y 2 8y 15
75. 2x 2 5x 3
76. 9x 2 36x 45
77. 6x 2 5x 6
78. r 2 6rs 9s2
79. 25s 2 10st t 2
80. x 2 36
81. 4x 2 25
82. 49 4y2
2
84. a 1
1 2 1 2 b a1 b x x
85. x 2 1x 2 12 91x 2 12
86. 1a 2 12 b 2 41a 2 12
87. 8x 3 125
88. x 6 64
89. x6 8y 3
90. 27a3 b6
91. x 3 2 x 2 x
92. 3x 3 27x
93. y3 3y 2 4y 12
94. x 3 3x 2 x 3
95. 2x 3 4x 2 x 2
96. 3x 3 5x 2 6x 10
97. 1x 12 1x 22 2 1x 12 2 1x 22 98. y 4 1 y 22 3 y 5 1y 22 4
100. 1a 2 2a2 2 21a 2 2a2 3
101. 51x 2 42 4 12x 2 1x 22 4 1x 2 42 5 14 2 1x 22 3
55–60 ■ Use a Special Factoring Formula to factor the expression. 3
Factor the expression completely.
3
101–104 ■ Factor the expression completely. (This type of expression arises in calculus when using the “product rule.”)
54. 21a b2 2 51a b 2 3
3
■
2
2
2
71–100
99. 1a 2 12 2 71a 2 12 10
Factor the trinomial.
2
68. x3/2 2x1/2 x 1/2
69. 1x 2 12 1/2 21x 2 12 1/2
83. 1a b 2 2 1a b2 2
35. 1x 2 x 12 12x 2 x 2 2
43–48
67. x 5/2 x 1/2
70. 2x 1/3 1x 22 2/3 5x 4/3 1x 22 1/3
1 2 26. a c b c
2
67–70 ■ Factor the expression completely. Begin by factoring out the lowest power of each common factor.
3
2
3
102. 312x 12 2 12 2 1x 32 1/2 12x 12 3 A 12 B 1x 3 2 1/2 103. 1x 2 32 1/3 23 x 2 1x 2 32 4/3
104. 12 x1/2 13x 42 1/2 32 x1/2 13x 42 1/2
105. (a) Show that ab 12 3 1a b2 2 1a 2 b 2 2 4 . (b) Show that 1a 2 b 2 2 2 1a 2 b 2 2 2 4a 2b 2. (c) Show that 1a 2 b 2 2 1c 2 d 2 2 1ac bd 2 2 1ad bc2 2
(d) Factor completely: 4a 2c 2 1a 2 b 2 c 2 2 2.
2
106. Verify Special Factoring Formulas 4 and 5 by expanding their righthand sides.
SECTION 1.3
Applications 107. Volume of Concrete A culvert is constructed out of large cylindrical shells cast in concrete, as shown in the ﬁgure. Using the formula for the volume of a cylinder given on the inside back cover of this book, explain why the volume of the cylindrical shell is V pR 2h pr 2h V 2p # average radius # height # thickness
110. The Power of Algebraic Formulas Use the Difference of Squares Formula to factor 172 162. Notice that it is easy to calculate the factored form in your head, but not so easy to calculate the original form in this way. Evaluate each expression in your head: (a) 5282 5272 (b) 1222 1202 (c) 10202 10102 Now use the Special Product Formula
to evaluate these products in your head: (d) 79 # 51 (e) 998 # 1002
Use the “unrolled” diagram to explain why this makes sense geometrically. R r h
h
108. Mowing a Field A square ﬁeld in a certain state park is mowed around the edges every week. The rest of the ﬁeld is kept unmowed to serve as a habitat for birds and small animals (see the ﬁgure). The ﬁeld measures b feet by b feet, and the mowed strip is x feet wide. (a) Explain why the area of the mowed portion is b 2 1b 2x 2 2. (b) Factor the expression in (a) to show that the area of the mowed portion is also 4x1b x 2 . b x
x
33
1A B2 1A B2 A2 B 2
Factor to show that
b
Algebraic Expressions
x
111. Differences of Even Powers (a) Factor the expressions completely: A4 B 4 and A6 B 6. (b) Verify that 18,335 124 74 and that 2,868,335 126 76. (c) Use the results of parts (a) and (b) to factor the integers 18,335 and 2,868,335. Then show that in both of these factorizations, all the factors are prime numbers. 112. Factoring An ⴚ 1 Verify these formulas by expanding and simplifying the righthand side. A2 1 1A 1 2 1A 12
A3 1 1A 1 2 1A2 A 12
A4 1 1A 1 2 1A3 A2 A 12 Based on the pattern displayed in this list, how do you think A5 1 would factor? Verify your conjecture. Now generalize the pattern you have observed to obtain a factoring formula for An 1, where n is a positive integer. 113. Factoring x 4 ⴙ ax 2 ⴙ b A trinomial of the form x 4 ax 2 b can sometimes be factored easily. For example, x 4 3x 2 4 1x 2 42 1x 2 12 . But x 4 3x 2 4 cannot be factored in this way. Instead, we can use the following method. x 4 3x 2 4 1x 4 4x 2 42 x 2
x
1x 2 22 2 x 2
Discovery ● Discussion
3 1x 2 22 x 4 3 1x 2 22 x 4
109. Degrees of Sums and Products of Polynomials Make up several pairs of polynomials, then calculate the sum and product of each pair. Based on your experiments and observations, answer the following questions. (a) How is the degree of the product related to the degrees of the original polynomials? (b) How is the degree of the sum related to the degrees of the original polynomials?
1x 2 x 2 2 1x 2 x 22
Add and subtract x2 Factor perfect square Difference of squares
Factor the following using whichever method is appropriate. (a) x 4 x 2 2 (b) x 4 2x 2 9 (c) x 4 4x 2 16 (d) x 4 2x 2 1
34
CHAPTER 1
Fundamentals
Visualizing a Formula DISCOVERY PROJECT
Many of the Special Product Formulas that we learned in this section can be “seen” as geometrical facts about length, area, and volume. For example, the ﬁgure shows how the formula for the square of a binomial can be interpreted as a fact about areas of squares and rectangles. b
b
ab
b™
a
a™
ab
(a+b)™ a
a b a (a+b)™=a™+2ab+b™
b
In the ﬁgure, a and b represent lengths, a2, b2, ab, and 1a b2 2 represent areas. The ancient Greeks always interpreted algebraic formulas in terms of geometric ﬁgures as we have done here. 1. Explain how the ﬁgure veriﬁes the formula a 2 b 2 1a b 2 1a b2 . a
a b b
2. Find a ﬁgure that veriﬁes the formula 1a b2 2 a 2 2ab b 2. 3. Explain how the ﬁgure veriﬁes the formula 1a b2 3 a 3 3a 2b 3ab 2 b 3.
a
b a b
a
b
4. Is it possible to draw a geometric ﬁgure that veriﬁes the formula for 1a b2 4? Explain. 5. (a) Expand 1a b c 2 2.
(b) Make a geometric ﬁgure that veriﬁes the formula you found in part (a).
SECTION 1.4
1.4
Rational Expressions
35
Rational Expressions A quotient of two algebraic expressions is called a fractional expression. Here are some examples: y2 y2 4
1x 3 x1
2x x1
A rational expression is a fractional expression where both the numerator and denominator are polynomials. For example, the following are rational expressions: 2x x1
x3 x x 5x 6
x x 1 2
2
In this section we learn how to perform algebraic operations on rational expressions.
The Domain of an Algebraic Expression Expression
Domain
1 x
5x 0 x 06
1x
5x 0 x 06
1 1x
5x 0 x 06
In general, an algebraic expression may not be deﬁned for all values of the variable. The domain of an algebraic expression is the set of real numbers that the variable is permitted to have. The table in the margin gives some basic expressions and their domains.
Example 1
Finding the Domain of an Expression
Find the domains of the following expressions. x (a) 2x 2 3x 1 (b) 2 x 5x 6
(c)
1x x5
Solution (a) This polynomial is deﬁned for every x. Thus, the domain is the set ⺢ of real numbers. (b) We ﬁrst factor the denominator. x x 1x 22 1x 32 x 2 5x 6 Denominator would be 0 if x 2 or x 3.
Since the denominator is zero when x 2 or 3, the expression is not deﬁned for these numbers. The domain is 5x 0 x 2 and x 36 . (c) For the numerator to be deﬁned, we must have x 0. Also, we cannot divide by zero, so x 5. Must have x 0 to take square root.
1x x5
Thus, the domain is 5x 0 x 0 and x 56.
Denominator would be 0 if x 5. ■
36
CHAPTER 1
Fundamentals
Simplifying Rational Expressions To simplify rational expressions, we factor both numerator and denominator and use the following property of fractions: AC A BC B This allows us to cancel common factors from the numerator and denominator.
Example 2
Simplifying Rational Expressions by Cancellation
x 1 x x2 2
Simplify: Solution We can’t cancel the x 2’s in x 1 because x 2 is not a factor. x2 x 2 2
2
1x 12 1x 12 x2 1 2 1x 12 1x 22 x x2
x1 x2
Factor
Cancel common factors
■
Multiplying and Dividing Rational Expressions To multiply rational expressions, we use the following property of fractions: A#C AC B D BD This says that to multiply two fractions we multiply their numerators and multiply their denominators.
Example 3
Multiplying Rational Expressions
Perform the indicated multiplication and simplify:
x 2 2x 3 # 3x 12 x 2 8x 16 x 1
Solution We ﬁrst factor.
1x 12 1x 32 31x 42 x 2 2x 3 # 3x 12 # 2 x1 x 8x 16 x 1 1x 42 2
31x 12 1x 32 1x 42 1x 12 1x 42 2
31x 32 x4
Factor
Property of fractions Cancel common factors
To divide rational expressions, we use the following property of fractions: A C A D # B D B C
■
SECTION 1.4
Rational Expressions
37
This says that to divide a fraction by another fraction we invert the divisor and multiply.
Example 4
Dividing Rational Expressions
Perform the indicated division and simplify:
x4 x 2 3x 4 2 2 x 4 x 5x 6
Solution x4 x 2 3x 4 x 4 # x 2 5x 6 x2 4 x 2 5x 6 x 2 4 x 2 3x 4
1x 42 1x 22 1x 32 1x 22 1x 22 1x 42 1x 12
x3 1x 22 1x 12
Invert and multiply
Factor Cancel common factors
■
Adding and Subtracting Rational Expressions Avoid making the following error: A A A BC B C
To add or subtract rational expressions, we ﬁrst ﬁnd a common denominator and then use the following property of fractions: A B AB C C C
For instance, if we let A 2, B 1, and C 1, then we see the error: 2 2 2 ⱨ 11 1 1 2 ⱨ22 2 1 ⱨ 4 Wrong!
Although any common denominator will work, it is best to use the least common denominator (LCD) as explained in Section 1.1. The LCD is found by factoring each denominator and taking the product of the distinct factors, using the highest power that appears in any of the factors.
Example 5
Adding and Subtracting Rational Expressions
Perform the indicated operations and simplify: 3 x 1 2 (a) (b) 2 x1 x2 x 1 1x 12 2 Solution (a) Here the LCD is simply the product 1x 12 1x 22 . 31x 22 x1x 12 3 x x1 x2 1x 12 1x 22 1x 12 1x 22
Write fractions using LCD
3x 6 x 2 x 1x 12 1x 22
Add fractions
x 2 2x 6 1x 12 1x 22
Combine terms in numerator
38
CHAPTER 1
Fundamentals
Mathematics in the Modern World
(b) The LCD of x 2 1 1x 12 1x 12 and 1x 12 2 is 1x 12 1x 12 2. 1 1 2 2 1x 12 1x 12 x2 1 1x 12 2 1x 12 2
1x 12 21x 12 1x 12 1x 12 2
NASA
x 1 2x 2 1x 12 1x 12 2
3x 1x 12 1x 12 2
Factor Combine fractions using LCD Distributive Property Combine terms in numerator
■
ErrorCorrecting Codes The pictures sent back by the Pathﬁnder spacecraft from the surface of Mars on July 4, 1997, were astoundingly clear. But few watching these pictures were aware of the complex mathematics used to accomplish that feat. The distance to Mars is enormous, and the background noise (or static) is many times stronger than the original signal emitted by the spacecraft. So, when scientists receive the signal, it is full of errors. To get a clear picture, the errors must be found and corrected. This same problem of errors is routinely encountered in transmitting bank records when you use an ATM machine, or voice when you are talking on the telephone. To understand how errors are found and corrected, we must ﬁrst understand that to transmit pictures, sound, or text we transform them into bits (the digits 0 or 1; see page 30). To help the receiver recognize errors, the message is “coded” by inserting additional bits. For example, suppose you want to transmit the message “10100.” A very simpleminded code is as follows: Send each digit a million times. The person receiving the message reads it in blocks of a million digits. If the ﬁrst block is mostly 1’s, he concludes that you are probably trying to transmit a 1, and so on. To say that this code is (continued)
Compound Fractions A compound fraction is a fraction in which the numerator, the denominator, or both, are themselves fractional expressions.
Example 6
Simplify:
Simplifying a Compound Fraction
x 1 y y 1 x
Solution 1 We combine the terms in the numerator into a single fraction. We do the same in the denominator. Then we invert and multiply. xy x 1 y y xy # x y xy y xy 1 x x
x1x y2 y1x y2
Solution 2 We ﬁnd the LCD of all the fractions in the expression, then multiply numerator and denominator by it. In this example the LCD of all the fractions is xy. Thus x x 1 1 y y y y 1 1 x x
#
xy xy
Multiply numerator and denominator by xy
x 2 xy xy y 2
Simplify
x1x y2 y1x y2
Factor
■
SECTION 1.4
not efﬁcient is a bit of an understatement; it requires sending a million times more data than the original message. Another method inserts “check digits.” For example, for each block of eight digits insert a ninth digit; the inserted digit is 0 if there is an even number of 1’s in the block and 1 if there is an odd number. So, if a single digit is wrong (a 0 changed to a 1, or vice versa), the check digits allow us to recognize that an error has occurred. This method does not tell us where the error is, so we can’t correct it. Modern error correcting codes use interesting mathematical algorithms that require inserting relatively few digits but which allow the receiver to not only recognize, but also correct, errors. The ﬁrst error correcting code was developed in the 1940s by Richard Hamming at MIT. It is interesting to note that the English language has a builtin error correcting mechanism; to test it, try reading this errorladen sentence: Gve mo libty ox giv ne deth.
Rational Expressions
39
The next two examples show situations in calculus that require the ability to work with fractional expressions.
Example 7
Simplify:
Simplifying a Compound Fraction
1 1 a ah h
Solution We begin by combining the fractions in the numerator using a common denominator. a 1a h2 1 1 a ah a1a h2 h h
Example 8 Simplify:
Combine fractions in the numerator
a 1a h2 1 # a1a h2 h
aah#1 a1a h2 h
Distributive Property
h # 1 a1a h2 h
Simplify
1 a1a h2
Property 5 of fractions (cancel common factors)
Property 2 of fractions (invert divisor and multiply)
■
Simplifying a Compound Fraction
11 x 2 2 1/2 x 2 11 x 2 2 1/2 1 x2
Solution 1 Factor 11 x 2 2 1/2 from the numerator. Factor out the power of 1 x 2 with the smallest exponent, in this case 11 x 2 2 1/2.
11 x 2 2 1/2 x 2 11 x 2 2 1/2 1 x2
11 x 2 2 1/2 3 11 x 2 2 x 2 4 1 x2
11 x 2 2 1/2 1x
2
1
11 x 2 2 3/2
Solution 2 Since 11 x 2 2 1/2 1/11 x 2 2 1/2 is a fraction, we can clear all fractions by multiplying numerator and denominator by 11 x 2 2 1/2. 11 x 2 2 1/2 x 2 11 x 2 2 1/2 1 x2
11 x 2 2 1/2 x 2 11 x 2 2 1/2 11 x 2 2 1/2
#
1 x2
11 x 2 2 x 2 11 x 2
2 3/2
1
11 x 2 2 3/2
11 x 2 2 1/2 ■
40
CHAPTER 1
Fundamentals
Rationalizing the Denominator or the Numerator If a fraction has a denominator of the form A B 1C, we may rationalize the denominator by multiplying numerator and denominator by the conjugate radical A B 1C. This is effective because, by Special Product Formula 1 in Section 1.3, the product of the denominator and its conjugate radical does not contain a radical: 1A B 1C 2 1A B 1C 2 A2 B2C
Example 9
Rationalizing the Denominator
Rationalize the denominator:
1 1 12
Solution We multiply both the numerator and the denominator by the conjugate radical of 1 12, which is 1 12. 1 1 # 1 12 1 12 1 12 1 12 Special Product Formula 1 1a b2 1a b2 a 2 b 2
Example 10
Multiply numerator and denominator by the conjugate radical
1 12 1 1 122 2
1 12 1 12 12 1 12 1
2
Special Product Formula 1
■
Rationalizing the Numerator
Rationalize the numerator:
14 h 2 h
Solution We multiply numerator and denominator by the conjugate radical 14 h 2. 14 h 2 14 h 2 # 14 h 2 h h 14 h 2 Special Product Formula 1 1a b2 1a b2 a 2 b 2
1 14 h2 2 22 h1 14 h 22
4h4 h1 14 h 22
1 h h1 14 h 22 14 h 2
Multiply numerator and denominator by the conjugate radical Special Product Formula 1
Property 5 of fractions (cancel common factors)
■
Avoiding Common Errors Don’t make the mistake of applying properties of multiplication to the operation of addition. Many of the common errors in algebra involve doing just that. The following table states several properties of multiplication and illustrates the error in applying them to addition.
SECTION 1.4
Correct multiplication property
Rational Expressions
41
Common error with addition
1a # b 2 2 a2 # b2
1a b 2 2 a2 b2
1a # b 1a 1b
1a, b 0 2
2a 2 # b 2 a # b
1a b 1a 1b
1a, b 0 2
2a 2 b 2 a b
1#1 1 # a b a b
1 1 1 a b ab
ab b a
ab b a
a1 # b1 1a # b 2 1
a1 b1 1a b 2 1
To verify that the equations in the righthand column are wrong, simply substitute numbers for a and b and calculate each side. For example, if we take a 2 and b 2 in the fourth error, we ﬁnd that the lefthand side is 1 1 1 1 1 a b 2 2 whereas the righthand side is 1 1 1 ab 22 4 Since 1 14, the stated equation is wrong. You should similarly convince yourself of the error in each of the other equations. (See Exercise 97.)
1.4 1–6
■
Exercises
Find the domain of the expression.
1. 4x 2 10x 3 3.
2x 1 x4
4.
5. 2x 3 7–16 7. 9. 11.
■
2. x 4 x 3 9x
6.
2t 2 5 3t 6 1
61x 1 2
2
8.
15.
y2 y
14.
y2 1 2x 3 x 2 6x 2x 2 7x 6
17–30
2x 1
■
#
x2 16x
4x 2 x 4
121x 2 2 1x 12
19.
x 2 x 12 x2 9
41x 2 1 2
16.
y 2 3y 18 2y 2 5y 3 1 x2 x3 1
Perform the multiplication or division and simplify.
17.
Simplify the rational expression.
31x 22 1x 1 2
13.
x2 x2 4
10.
x2 x 2 x2 1
21.
t3 t2 9
x 2 6x 8 x 2 5x 4
12.
x 2 x 12 x 2 5x 6
23.
x 2 7x 12 x 2 3x 2
#
3x 4x
#
t3 t2 9
#
18.
x 2 25 x 2 16
20.
x 2 2x 3 x 2 2x 3
#
22.
x2 x 6 x 2 2x
x3 x2 x 2x 3
x 2 5x 6 x 2 6x 9
#
x4 x5
#
3x 3x 2
42
24. 25. 26.
CHAPTER 1
x 2 2xy y 2 x y 2
2
#
Fundamentals
2x 2 xy y 2 x xy 2y 2
x 2 6x 5 2x 2 3x 1 2 2 x 2x 15 2x 7x 3 4y 2 9 2y 9y 18 2
2y 2 y 3 y 5y 6 2
x3 x1 27. x x 2 2x 1 29.
1 c1 53. 1 1 c1 1
2
x/y z
31–50
■
2x 2 3x 2 x2 1 28. 2 2x 5x 2 x2 x 2 30.
x y/z
Perform the addition or subtraction and simplify. x x3
32.
2x 1 1 x4
33.
1 2 x5 x3
34.
1 1 x1 x1
35.
1 1 x1 x2
36.
x 3 x4 x6
37.
x 2 2 x 1 1x 1 2
38.
5 3 2x 3 12x 3 2 2
31. 2
u 39. u 1 u1 41.
1 1 2 x2 x x
2 1 2 43. x3 x 7x 12
2 3 4 2 40. 2 ab a b 42.
1 1 1 2 3 x x x
x 1 44. 2 x2 x 4
1 1 2 45. x3 x 9
1
5 x1 x 55. x x1 x 57.
59.
1
54. 1
1 1x
ab ab a b 56. ab ab a b
2 1 1 1
x 2 y 2
58.
x 1 y 1
x 1 y 1 1x y 2 1
aa
1 m 1 n b aa b b b 60. 1 m 1 n ab b ab b a a
1 1 1 an 1 a n
61–66 ■ Simplify the fractional expression. (Expressions like these arise in calculus.) 1 1 a ah 61. h 62.
1x h2 3 x 3 h
1 1x h2
2 1x h 2 63. h 64. 65.
1x 2x
1x h2 3 71x h2 1x 3 7x 2 h
B
1 a
x 21 x
2
b
2
66.
B
1 a x3
1 2 b 4x 3
46.
x 2 2 x2 x 2 x 5x 4
67–72 ■ Simplify the expression. (This type of expression arises in calculus when using the “quotient rule.”)
47.
3 2 4 2 x x1 x x
67.
48.
x 2 1 x2 x3 x2 x 6
68.
1 1 2 49. 2 x 3x 2 x 2x 3 50.
69.
1 2 3 2 x1 1x 12 2 x 1
51–60
■
70. 71.
52. x
1x 32 4
2x1x 62 4 x 2 14 2 1x 6 2 3 1x 62 8
211 x2 1/2 x 11 x 2 1/2 11 x 2
x1
2 1/2
Simplify the compound fractional expression.
y x x y 51. 1 1 2 x2 y
31x 22 2 1x 32 2 1x 22 3 12 2 1x 32
y y x x y
72.
x 2 11 x 2 2 1/2
1 x2
311 x 2 1/3 x 11 x 2 2/3 11 x 2 2/3
17 3x2 1/2 32 x 17 3x 2 1/2 7 3x
SECTION 1.4
73–78
■
Rationalize the denominator.
73.
1 2 13
74.
2 3 15
75.
2 12 17
76.
1 1x 1
77.
y 13 1y
79–84
■
1x 1y
A
1 15 3
80.
13 15 2
81.
1r 12 5
82.
1x 1x h h 1x 1x h
83. 2x 2 1 x
(b) Complete the table by calculating the average cost per shirt for the given values of x. x
84. 1x 1 1x
16 a a 1 16 16
86.
b b 1 c bc
2 1 2 87. x 4x 2
x1 x 88. y y1
x 1 89. xy 1y
a 2a 90. 2 a b b 2b
a a b b
92.
1 1 x x2 1x x x
Applications 93. Electrical Resistance If two electrical resistors with resistances R1 and R2 are connected in parallel (see the ﬁgure), then the total resistance R is given by R
1 1 1 R1 R2
(a) Simplify the expression for R. (b) If R1 10 ohms and R2 20 ohms, what is the total resistance R? R⁄
R¤
Average cost
10 20 50 100 200 500 1000
85–92 ■ State whether the given equation is true for all values of the variables. (Disregard any value that makes a denominator zero.)
91.
500 6x 0.01x 2 x
Rationalize the numerator.
79.
85.
43
94. Average Cost A clothing manufacturer ﬁnds that the cost of producing x shirts is 500 6x 0.01x 2 dollars. (a) Explain why the average cost per shirt is given by the rational expression
21x y 2
78.
Rational Expressions
Discovery • Discussion 95. Limiting Behavior of a Rational Expression The rational expression x2 9 x3 is not deﬁned for x 3. Complete the tables and determine what value the expression approaches as x gets closer and closer to 3. Why is this reasonable? Factor the numerator of the expression and simplify to see why.
x 2.80 2.90 2.95 2.99 2.999
x2 9 x3
x
x2 9 x3
3.20 3.10 3.05 3.01 3.001
96. Is This Rationalization? In the expression 2/ 1x we would eliminate the radical if we were to square both numerator and denominator. Is this the same thing as rationalizing the denominator? 97. Algebraic Errors The lefthand column in the table lists some common algebraic errors. In each case, give an example using numbers that show that the formula is not valid. An example of this type, which shows that a
44
CHAPTER 1
Fundamentals
statement is false, is called a counterexample. Algebraic error
Counterexample
1 1 1 a b ab
1 1 1 2 2 22
98. The Form of an Algebraic Expression An algebraic expression may look complicated, but its “form” is always simple; it must be a sum, a product, a quotient, or a power. For example, consider the following expressions: 11 x 2 2 2 a
1a b2 2 a 2 b 2
11 x 2 a 1
5 x3 1 21 x
2a 2 b 2 a b
1a 3 b 3 2 1/3 a b a m/a n a m/n 1 an
3 4 (c) 2 x 14x 2 12
1.5
2
x5 b 1 x4
1x A1 x
With appropriate choices for A and B, the ﬁrst has the form A B, the second AB, the third A/B, and the fourth A1/2. Recognizing the form of an expression helps us expand, simplify, or factor it correctly. Find the form of the following algebraic expressions. 1 (a) x 1 (b) 11 x 2 2 11 x 2 3 x A
ab b a
a 1/n
x2 3 b x1
(d)
1 221 x 1 21 x 2
Equations An equation is a statement that two mathematical expressions are equal. For example, 358 is an equation. Most equations that we study in algebra contain variables, which are symbols (usually letters) that stand for numbers. In the equation 4x 7 19
x 3 is a solution of the equation 4x 7 19, because substituting x 3 makes the equation true: x3 413 2 7 19
the letter x is the variable. We think of x as the “unknown” in the equation, and our goal is to ﬁnd the value of x that makes the equation true. The values of the unknown that make the equation true are called the solutions or roots of the equation, and the process of ﬁnding the solutions is called solving the equation. Two equations with exactly the same solutions are called equivalent equations. To solve an equation, we try to ﬁnd a simpler, equivalent equation in which the variable stands alone on one side of the “equal” sign. Here are the properties that we use to solve an equation. (In these properties, A, B, and C stand for any algebraic expressions, and the symbol 3 means “is equivalent to.”)
Properties of Equality Property
Description
1. A B 3 A C B C
Adding the same quantity to both sides of an equation gives an equivalent equation. Multiplying both sides of an equation by the same nonzero quantity gives an equivalent equation.
2. A B 3 CA CB (C 0)
SECTION 1.5
Equations
45
These properties require that you perform the same operation on both sides of an equation when solving it. Thus, if we say “add 7” when solving an equation, that is just a short way of saying “add 7 to each side of the equation.”
Linear Equations The simplest type of equation is a linear equation, or ﬁrstdegree equation, which is an equation in which each term is either a constant or a nonzero multiple of the variable.
Linear Equations A linear equation in one variable is an equation equivalent to one of the form ax b 0 where a and b are real numbers and x is the variable. Here are some examples that illustrate the difference between linear and nonlinear equations. Linear equations
Nonlinear equations
4x 5 3
x 2 2x 8
2x 12 x 7
1x 6x 0
x6
x 3
Example 1
3 2x 1 x
Not linear; contains the square of the variable Not linear; contains the square root of the variable Not linear; contains the reciprocal of the variable
Solving a Linear Equation
Solve the equation 7x 4 3x 8. Solution We solve this by changing it to an equivalent equation with all terms that have the variable x on one side and all constant terms on the other. 7x 4 3x 8
17x 42 4 13x 82 4 7x 3x 12
7x 3x 13x 122 3x 4x 12
1 4
Check Your Answer x 3:
LHS RHS
Add 4 Simplify Subtract 3x Simplify
# 4x 14 # 12
Multiply by 41
x3 Because it is important to CHECK YOUR ANSWER, we do this in many of our examples. In these checks, LHS stands for “lefthand side” and RHS stands for “righthand side” of the original equation.
Given equation
■
Simplify x3
x3
LHS 7132 4
RHS 313 2 8
17
17
46
CHAPTER 1
Fundamentals
Many formulas in the sciences involve several variables, and it is often necessary to express one of the variables in terms of the others. In the next example we solve for a variable in Newton’s Law of Gravity.
Example 2 This is Newton’s Law of Gravity. It gives the gravitational force F between two masses m and M that are a distance r apart. The constant G is the universal gravitational constant.
Solving for One Variable in Terms of Others
Solve for the variable M in the equation FG
Solution Although this equation involves more than one variable, we solve it as usual by isolating M on one side and treating the other variables as we would numbers. F a a
Gm bM r2
Factor M from RHS
r2 r2 Gm bF a b a 2 bM Gm Gm r r 2F M Gm
The solution is M
Example 3 l
mM r2
Multiply by reciprocal of
Gm r2
Simplify
r 2F . Gm
■
Solving for One Variable in Terms of Others
The surface area A of the closed rectangular box shown in Figure 1 can be calculated from the length l, the width „, and the height h according to the formula A 2l„ 2„h 2lh
h
Solve for „ in terms of the other variables in this equation. „ Figure 1 A closed rectangular box
Solution Although this equation involves more than one variable, we solve it as usual by isolating „ on one side, treating the other variables as we would numbers. A 12l„ 2„h2 2lh A 2lh 2l„ 2„h
A 2lh 12l 2h2„ A 2lh „ 2l 2h The solution is „
A 2lh . 2l 2h
Collect terms involving w Subtract 2lh Factor w from RHS Divide by 2l 2h ■
Quadratic Equations Linear equations are ﬁrstdegree equations like 2x 1 5 or 4 3x 2. Quadratic equations are seconddegree equations like x 2 2x 3 0 or 2x 2 3 5x.
SECTION 1.5
Quadratic Equations x 2x 8 0
Equations
47
Quadratic Equations
2
3x 10 4x 2 1 2 2x
13 x 16 0
A quadratic equation is an equation of the form ax 2 bx c 0 where a, b, and c are real numbers with a 0.
Some quadratic equations can be solved by factoring and using the following basic property of real numbers.
ZeroProduct Property AB 0
if and only if
A 0 or B 0
This means that if we can factor the lefthand side of a quadratic (or other) equation, then we can solve it by setting each factor equal to 0 in turn. This method works only when the righthand side of the equation is 0.
Example 4
Solving a Quadratic Equation by Factoring
Solve the equation x 2 5x 24. Solution We must ﬁrst rewrite the equation so that the righthand side is 0. x 2 5x 24
Check Your Answers
x 2 5x 24 0
x 3:
132 5132 9 15 24 2
1x 32 1x 82 0
x 8:
x30
2
x3
18 2 5182 64 40 24
Subtract 24
or
Factor
x80 x 8
ZeroProduct Property Solve
The solutions are x 3 and x 8.
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Do you see why one side of the equation must be 0 in Example 4? Factoring the equation as x1x 52 24 does not help us ﬁnd the solutions, since 24 can be factored in inﬁnitely many ways, such as 6 # 4, 12 # 48, A25 B # 1602 , and so on. A quadratic equation of the form x 2 c 0, where c is a positive constant, factors as 1x 1c 2 1x 1c 2 0, and so the solutions are x 1c and x 1c. We often abbreviate this as x 1c.
Solving a Simple Quadratic Equation The solutions of the equation x 2 c are x 1c and x 1c.
48
CHAPTER 1
Fundamentals
Example 5
Solving Simple Quadratics
Solve each equation. (a) x 2 5 (b) 1x 42 2 5 Solution (a) From the principle in the preceding box, we get x 15. (b) We can take the square root of each side of this equation as well. 1x 42 2 5 x 4 15 x 4 15
Take the square root Add 4
The solutions are x 4 15 and x 4 15. See page 30 for how to recognize when a quadratic expression is a perfect square. Completing the Square Area of blue region is b x 2 2 a b x x 2 bx 2 Add a small square of area 1b/22 2 to “complete” the square. b x 2
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As we saw in Example 5, if a quadratic equation is of the form 1x a2 2 c, then we can solve it by taking the square root of each side. In an equation of this form the lefthand side is a perfect square: the square of a linear expression in x. So, if a quadratic equation does not factor readily, then we can solve it using the technique of completing the square. This means that we add a constant to an expression to make it a perfect square. For example, to make x 2 6x a perfect square we must add 9, since x 2 6x 9 1x 32 2.
Completing the Square b 2 To make x 2 bx a perfect square, add a b , the square of half the 2 coefﬁcient of x. This gives the perfect square
x
b 2 b 2 x 2 bx a b a x b 2 2
b 2
Example 6
Solving Quadratic Equations by Completing the Square
Solve each equation. (a) x 2 8x 13 0 When completing the square, make sure the coefﬁcient of x 2 is 1. If it isn’t, you must factor this coefﬁcient from both terms that contain x: b ax bx a a x x b a 2
2
Then complete the square inside the parentheses. Remember that the term added inside the parentheses is multiplied by a.
(b) 3x 2 12x 6 0
Solution (a) x 2 8x 13 0 x 8x 13 2
x 2 8x 16 13 16 1x 42 2 3
x 4 13 x 4 13
Given equation Subtract 13 Complete the square: add a Perfect square Take square root Add 4
8 2 b 16 2
SECTION 1.5
Equations
49
(b) After subtracting 6 from each side of the equation, we must factor the coefﬁcient of x 2 (the 3) from the left side to put the equation in the correct form for completing the square. 3x 2 12x 6 0
Given equation
3x 12x 6 2
Subtract 6
31x 2 4x2 6
Factor 3 from LHS
Now we complete the square by adding 122 4 inside the parentheses. Since everything inside the parentheses is multiplied by 3, this means that we are actually adding 3 # 4 12 to the left side of the equation. Thus, we must add 12 to the right side as well. 2
François Viète (1540–1603) had a successful political career before taking up mathematics late in life. He became one of the most famous French mathematicians of the 16th century. Viète introduced a new level of abstraction in algebra by using letters to stand for known quantities in an equation. Before Viète’s time, each equation had to be solved on its own. For instance, the quadratic equations 3x 2 2x 8 0
31x 2 4x 42 6 3 # 4 31x 22 6 2
Perfect square
1x 22 2 2
Divide by 3
x 2 12 x 2 12
Take square root ■
Add 2
We can use the technique of completing the square to derive a formula for the roots of the general quadratic equation ax 2 bx c 0.
5x 2 6x 4 0
The Quadratic Formula
had to be solved separately by completing the square. Viète’s idea was to consider all quadratic equations at once by writing
The roots of the quadratic equation ax 2 bx c 0, where a 0, are x
ax 2 bx c 0 where a, b, and c are known quantities. Thus, he made it possible to write a formula (in this case, the quadratic formula) involving a, b, and c that can be used to solve all such equations in one fell swoop. Viète’s mathematical genius proved quite valuable during a war between France and Spain. To communicate with their troops, the Spaniards used a complicated code that Viète managed to decipher. Unaware of Viète’s accomplishment, the Spanish king, Philip II, protested to the Pope, claiming that the French were using witchcraft to read his messages.
Complete the square: add 4
b 2b2 4ac 2a
■
Proof First, we divide each side of the equation by a and move the constant to the right side, giving b c x2 x a a
Divide by a
We now complete the square by adding 1b/2a2 2 to each side of the equation: b 2 b b 2 c x2 x a b a b a a 2a 2a ax
b 2 4ac b 2 b 2a 4a 2
x
2b 2 4ac b 2a 2a x
b 2b 2 4ac 2a
Complete the square: Add a
b 2 b 2a
Perfect square
Take square root
Subtract
b 2a
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The quadratic formula could be used to solve the equations in Examples 4 and 6. You should carry out the details of these calculations.
50
CHAPTER 1
Fundamentals
Example 7
Using the Quadratic Formula
Find all solutions of each equation. (a) 3x 2 5x 1 0 (b) 4x 2 12x 9 0
(c) x 2 2x 2 0
Solution (a) In this quadratic equation a 3, b 5, and c 1. b 5
3x 2 5x 1 0 a3
c 1
By the quadratic formula, x
152 2152 2 4132 112 5 137 2132 6
If approximations are desired, we can use a calculator to obtain x Another Method 4x 2 12x 9 0
and
x
5 137 ⬇ 0.1805 6
(b) Using the quadratic formula with a 4, b 12, and c 9 gives
12x 3 2 2 0
x
2x 3 0 x 32
5 137 ⬇ 1.8471 6
12 21122 2 4 # 4 # 9 12 0 3 2#4 8 2
This equation has only one solution, x 32. (c) Using the quadratic formula with a 1, b 2, and c 2 gives x
2 222 4 # 2 2 14 2 211 1 11 2 2 2
Since the square of any real number is nonnegative, 11 is undeﬁned in the real number system. The equation has no real solution.
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In Section 3.4 we study the complex number system, in which the square roots of negative numbers do exist. The equation in Example 7(c) does have solutions in the complex number system. The quantity b 2 4ac that appears under the square root sign in the quadratic formula is called the discriminant of the equation ax 2 bx c 0 and is given the symbol D. If D 0, then 2b 2 4ac is undeﬁned, and the quadratic equation has no real solution, as in Example 7(c). If D 0, then the equation has only one real solution, as in Example 7(b). Finally, if D 0, then the equation has two distinct real solutions, as in Example 7(a). The following box summarizes these observations.
The Discriminant The discriminant of the general quadratic ax 2 bx c 0 1a 02 is D b 2 4ac. 1. If D 0, then the equation has two distinct real solutions. 2. If D 0, then the equation has exactly one real solution. 3. If D 0, then the equation has no real solution.
SECTION 1.5
Example 8
Equations
51
Using the Discriminant
Use the discriminant to determine how many real solutions each equation has. (a) x 2 4x 1 0 (b) 4x 2 12x 9 0 (c) 13 x 2 2x 4 0 Solution (a) The discriminant is D 42 4112 112 20 0, so the equation has two distinct real solutions. (b) The discriminant is D 1122 2 4 # 4 # 9 0, so the equation has exactly one real solution. (c) The discriminant is D 122 2 4A 13 B4 43 0, so the equation has no real solution. ■ Now let’s consider a reallife situation that can be modeled by a quadratic equation.
Example 9 This formula depends on the fact that acceleration due to gravity is constant near the earth’s surface. Here we neglect the effect of air resistance.
descent ascent h
The Path of a Projectile
An object thrown or ﬁred straight upward at an initial speed of √ 0 ft/s will reach a height of h feet after t seconds, where h and t are related by the formula h 16t 2 √ 0 t Suppose that a bullet is shot straight upward with an initial speed of 800 ft/s. Its path is shown in Figure 2. (a) (b) (c) (d)
When does the bullet fall back to ground level? When does it reach a height of 6400 ft? When does it reach a height of 2 mi? How high is the highest point the bullet reaches?
Solution Since the initial speed in this case is √0 800 ft/s, the formula is h 16t 2 800t
Figure 2
(a) Ground level corresponds to h 0, so we must solve the equation 0 16t 2 800t
Set h 0
0 16t1t 502
Factor
Thus, t 0 or t 50. This means the bullet starts 1t 02 at ground level and returns to ground level after 50 s. (b) Setting h 6400 gives the equation 6400 16t 2 800t 16t 800t 6400 0 2
All terms to LHS
t 2 50t 400 0
Divide by 16
1t 102 1t 402 0 6400 ft
t 10
or
Set h 6400
Factor
t 40
Solve
The bullet reaches 6400 ft after 10 s (on its ascent) and again after 40 s (on its descent to earth).
52
CHAPTER 1
Fundamentals
(c) Two miles is 2 5280 10,560 ft. 10,560 16t 2 800t 16t 2 800t 10,560 0 2 mi
Set h 10,560 All terms to LHS
t 50t 660 0 2
Divide by 16
The discriminant of this equation is D 1502 2 416602 140, which is negative. Thus, the equation has no real solution. The bullet never reaches a height of 2 mi. (d) Each height the bullet reaches is attained twice, once on its ascent and once on its descent. The only exception is the highest point of its path, which is reached only once. This means that for the highest value of h, the following equation has only one solution for t: 10,000 ft
h 16t 2 800t 16t 2 800t h 0
All terms to LHS
This in turn means that the discriminant D of the equation is 0, and so D 18002 2 41162h 0 640,000 64h 0 h 10,000 The maximum height reached is 10,000 ft.
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Other Types of Equations So far we have learned how to solve linear and quadratic equations. Now we study other types of equations, including those that involve higher powers, fractional expressions, and radicals.
Example 10 Check Your Answers
Solve the equation
x 3: LHS
3 5 3 32
112 RHS 2 LHS RHS
5 3 2. x x2
Solution We eliminate the denominators by multiplying each side by the lowest common denominator. a
5 3 b x1x 22 2x 1x 22 x x2 31x 22 5x 2x 2 4x 8x 6 2x 4x 2
x 1: LHS
An Equation Involving Fractional Expressions
5 3 1 1 2
RHS 2 LHS RHS
x3
Expand LHS Subtract 8x 6
0 x 2x 3
Divide both sides by 2
0 1x 32 1x 12 x30
Expand
0 2x 2 4x 6 2
3 5 2
Multiply by LCD x(x 2)
or
x10 x 1
Factor ZeroProduct Property Solve
SECTION 1.5
Equations
53
We must check our answers because multiplying by an expression that contains the variable can introduce extraneous solutions. From Check Your Answers we see that the solutions are x 3 and 1. ■ Check Your Answers x 14:
LHS 2A 14 B 12
RHS 1 22
A 14 B
1 294 1 32 12
When you solve an equation that involves radicals, you must be especially careful to check your ﬁnal answers. The next example demonstrates why.
Example 11
An Equation Involving a Radical
Solve the equation 2x 1 12 x. Solution To eliminate the square root, we ﬁrst isolate it on one side of the equal sign, then square. 2x 1 12 x
LHS RHS
12x 12 2 2 x
x 1:
Subtract 1 Square each side
LHS 211 2 2
4x 4x 1 2 x
Expand LHS
RHS 1 12 1
4x 2 3x 1 0
Add 2 x
110 LHS RHS
2
14x 12 1x 12 0 4x 1 0 x
14
or
Factor
x10 x1
ZeroProduct Property Solve
The values x 14 and x 1 are only potential solutions. We must check them to see if they satisfy the original equation. From Check Your Answers we see that x 14 is a solution but x 1 is not. The only solution is x 14.
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When we solve an equation, we may end up with one or more extraneous solutions, that is, potential solutions that do not satisfy the original equation. In Example 11, the value x 1 is an extraneous solution. Extraneous solutions may be introduced when we square each side of an equation because the operation of squaring can turn a false equation into a true one. For example, 1 1, but 112 2 12. Thus, the squared equation may be true for more values of the variable than the original equation. That is why you must always check your answers to make sure that each satisﬁes the original equation. An equation of the form aW 2 bW c 0, where W is an algebraic expression, is an equation of quadratic type. We solve equations of quadratic type by substituting for the algebraic expression, as we see in the next two examples.
Example 12
A FourthDegree Equation of Quadratic Type
Find all solutions of the equation x 4 8x 2 8 0. Solution If we set W x 2, then we get a quadratic equation in the new variable W: 1x 2 2 2 8x 2 8 0
W 2 8W 8 0
W
182 2182 2 4 # 8 4 212 2
x 2 4 2 12 x 24 2 12
Write x4 as 1x 2 2 2 Let W x 2 Quadratic formula W x2 Take square roots
54
CHAPTER 1
Fundamentals
So, there are four solutions: Pythagoras (circa 580–500 B.C.) founded a school in Croton in southern Italy, which was devoted to the study of arithmetic, geometry, music, and astronomy. The Pythagoreans, as they were called, were a secret society with peculiar rules and initiation rites. They wrote nothing down, and were not to reveal to anyone what they had learned from the Master. Although women were barred by law from attending public meetings, Pythagoras allowed women in his school, and his most famous student was Theano (whom he later married). According to Aristotle, the Pythagoreans were convinced that “the principles of mathematics are the principles of all things.” Their motto was “Everything is Number,” by which they meant whole numbers. The outstanding contribution of Pythagoras is the theorem that bears his name: In a right triangle the area of the square on the hypotenuse is equal to the sum of the areas of the square on the other two sides.
24 2 12,
24 2 12,
24 2 12,
24 2 12
Using a calculator, we obtain the approximations x ⬇ 2.61, 1.08, 2.61, 1.08.
Example 13
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An Equation Involving Fractional Powers
Find all solutions of the equation x 1/3 x 1/6 2 0. Solution This equation is of quadratic type because if we let W x 1/6, then W 2 1x 1/6 2 2 x 1/3. x 1/3 x 1/6 2 0 Let W x 1/6
W2 W 2 0
1W 12 1W 22 0 W10
Factor
W20
or
W1
W 2
x 1/6 1
x 1/6 2
ZeroProduct Property Solve W x 1/6
x 122 6 64
x 16 1
Take the 6th power
From Check Your Answers we see that x 1 is a solution but x 64 is not. The only solution is x 1.
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Check Your Answers x 1:
x 64:
LHS 11/3 11/6 2 0
LHS 641/3 641/6 2 4224
c
a
b
RHS 0
RHS 0
LHS RHS
LHS RHS
When solving equations that involve absolute values, we usually take cases. c™=a™+b™ The converse of Pythagoras’s Theorem is also true: A triangle whose sides a, b, and c satisfy a2 b2 c2 is a right triangle.
Example 14
An Absolute Value Equation
Solve the equation 0 2x 5 0 3. Solution By the deﬁnition of absolute value, 0 2x 5 0 3 is equivalent to 2x 5 3
or
2x 5 3
2x 8
2x 2
x4
x1
The solutions are x 1, x 4.
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SECTION 1.5
1.5
1. 4x 7 9x 3 (a) x 2 (b) x 2
34. A P a 1
37–44
5–22 ■ The given equation is either linear or equivalent to a linear equation. Solve the equation. 6. 5x 3 4
9. 7„ 15 2„ 11.
1 2y
2 13 y
13. 211 x 2 311 2x 2 5 14.
17.
10. 5t 13 12 5t z 3 z 7 12. 5 10
4 1 1 x 3x
18.
x x1 6x 2 4
2x 1 4 x2 5
4 2 35 2 x1 x1 x 1 x5 21. 1t 4 2 2 1t 42 2 32 22. 13x 112 13 ■ Solve the equation for the indicated variable. 23–36 19.
1 1 3 x1 2 3x 3
23. PV nRT;
for R
1 1 1 ; 25. R R1 R2 ax b 2; 27. cx d
for R1
20.
24. F G
mM ; r2
26. P 2l 2„;
for m for „
for i 36. S
n1n 12 2
;
for n
Solve the equation by factoring. 38. x 2 3x 4 0
39. x 2 7x 12 0
40. x 2 8x 12 0
41. 4x 2 4x 15 0
42. 2y 2 7y 3 0
43. 3x 2 5x 2
44. 6x1x 12 21 x
45–52
■
Solve the equation by completing the square.
45. x 2x 5 0
46. x 2 4x 2 0
47. x 2 3x 74 0
48. x 2 34 x 18
49. 2x 2 8x 1 0
50. 3x 2 6x 1 0
51. 4x 2 x 0
52. 2x 2 6x 3 0
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Find all real solutions of the quadratic equation.
53. x 2x 15 0
54. x 2 30x 200 0
55. x 2 3x 1 0
56. x 2 6x 1 0
57. 2x 2 x 3 0
58. 3x 2 7x 4 0
59. 2y 2 y 12 0
60. u 2 32 u 169 0
61. 4x 2 16x 9 0
62. „ 2 31„ 12
63. 3 5z z2 0
64. x 2 15x 1 0
65. 16x 2 2x 23/2 0 66. 3x 2 2x 2 0 67. 25x 2 70x 49 0
68. 5x 2 7x 5 0
69–74 ■ Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. 69. x 2 6x 1 0
70. 3x 2 6x 9
71. x 2 2.20x 1.21 0
72. x 2 2.21x 1.21 0
73. 4x 2 5x 138 0
74. x 2 rx s 0
1s 0 2
for x
28. a 2 3b 31c x 2 4 6;
29. a 2x 1a 1 2 1a 1 2 x ; 30.
for t
2
16. 2x
for r
37. x x 12 0
53–68
y1 2 1 y 1 y 32 3 2 4
15. x 13 x 12 x 5 0
i 2 b ; 100
2
8. 3 13 x 5
mM ; r2
for b
2
x 3/2 4. x8 x6 (a) x 4 (b) x 8
81
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32. F G
for r
35. h 12 gt 2 √ 0 t;
1 1 3. 1 x x4 (a) x 2 (b) x 4
5. 2x 7 31
31. V 13 pr 2h; 33. a 2 b 2 c 2;
2. 1 3 2 13 x 2 4 4x 16 x 2 (a) x 2 (b) x 4
1 2x
55
Exercises
1–4 ■ Determine whether the given value is a solution of the equation.
7.
Equations
a1 b1 a1 ; a b b
75–98 for x for x for a
■
Find all real solutions of the equation.
75.
1 1 5 x1 x2 4
76.
12 10 40 x x3
77.
x2 50 x 100
78.
2 1 20 x1 x
56
79.
CHAPTER 1
Fundamentals
x x5 5 28 x1 80. 2 1 x2 x2 2x 7 x3 x 4
81. 12x 1 1 x
82. 15 x 1 x 2
83. 2x 1x 1 8
84. 2 1x 5 x 5
85. x 13x 40 0
86. x 4 5x 2 4 0
87. 2x 4 4x 2 1 0
88. x 6 2x 3 3 0
89. x 4/3 5x 2/3 6 0
4 90. 1x 3 1 x40
4
2
where S is the fraction of the original beam length that disappears due to shrinkage. (a) A beam 12.025 m long is cast in concrete that contains 250 kg/m3 water. What is the shrinkage factor S? How long will the beam be when it has dried? (b) A beam is 10.014 m long when wet. We want it to shrink to 10.009 m, so the shrinkage factor should be S 0.00050. What water content will provide this amount of shrinkage?
91. 41x 12 1/2 51x 12 3/2 1x 1 2 5/2 0 92. x 1/2 3x1/2 10x3/2
93. x 1/2 3x 1/3 3x 1/6 9
94. x 51x 6 0
97. 0 x 4 0 0.01
98. 0 x 6 0 1
95. 0 2x 0 3
96. 0 3x 5 0 1
Applications 99–100 ■ FallingBody Problems Suppose an object is dropped from a height h0 above the ground. Then its height after t seconds is given by h 16t 2 h0, where h is measured in feet. Use this information to solve the problem. 99. If a ball is dropped from 288 ft above the ground, how long does it take to reach ground level? 100. A ball is dropped from the top of a building 96 ft tall. (a) How long will it take to fall half the distance to ground level? (b) How long will it take to fall to ground level? 101–102 ■ FallingBody Problems Use the formula h 16t 2 √ 0 t discussed in Example 9. 101. A ball is thrown straight upward at an initial speed of √ 0 40 ft /s. (a) When does the ball reach a height of 24 ft? (b) When does it reach a height of 48 ft? (c) What is the greatest height reached by the ball? (d) When does the ball reach the highest point of its path? (e) When does the ball hit the ground? 102. How fast would a ball have to be thrown upward to reach a maximum height of 100 ft? [Hint: Use the discriminant of the equation 16t 2 √ 0 t h 0.] 103. Shrinkage in Concrete Beams As concrete dries, it shrinks—the higher the water content, the greater the shrinkage. If a concrete beam has a water content of „ kg/m3, then it will shrink by a factor S
0.032„ 2.5 10,000
104. The Lens Equation If F is the focal length of a convex lens and an object is placed at a distance x from the lens, then its image will be at a distance y from the lens, where F, x, and y are related by the lens equation 1 1 1 x y F Suppose that a lens has a focal length of 4.8 cm, and that the image of an object is 4 cm closer to the lens than the object itself. How far from the lens is the object? 105. Fish Population The ﬁsh population in a certain lake rises and falls according to the formula F 1000130 17t t 2 2 Here F is the number of ﬁsh at time t, where t is measured in years since January 1, 2002, when the ﬁsh population was ﬁrst estimated. (a) On what date will the ﬁsh population again be the same as on January 1, 2002? (b) By what date will all the ﬁsh in the lake have died? 106. Fish Population A large pond is stocked with ﬁsh. The ﬁsh population P is modeled by the formula P 3t 10 1t 140, where t is the number of days since the ﬁsh were ﬁrst introduced into the pond. How many days will it take for the ﬁsh population to reach 500? 107. Proﬁt A smallappliance manufacturer ﬁnds that the proﬁt P (in dollars) generated by producing x microwave ovens per week is given by the formula P 101 x 1300 x2 provided that 0 x 200. How many ovens must be manufactured in a given week to generate a proﬁt of $1250? 108. Gravity If an imaginary line segment is drawn between the centers of the earth and the moon, then the net
SECTION 1.5
gravitational force F acting on an object situated on this line segment is K 0.012K x2 1239 x2 2
F
where K 0 is a constant and x is the distance of the object from the center of the earth, measured in thousands of miles. How far from the center of the earth is the “dead spot” where no net gravitational force acts upon the object? (Express your answer to the nearest thousand miles.)
111. Proof That 0 ⴝ 1? The following steps appear to give equivalent equations, which seem to prove that 1 0. Find the error. Given
x x 2
Multiply by x
x x0 2
Subtract x
x1x 12 0 x1x 12 x1
109. Depth of a Well One method for determining the depth of a well is to drop a stone into it and then measure the time it takes until the splash is heard. If d is the depth of the well (in feet) and t1 the time (in seconds) it takes for the stone to fall, then d 16t 21, so t 1 1d/4. Now if t2 is the time it takes for the sound to travel back up, then d 1090t2 because the speed of sound is 1090 ft/s. So t2 d/1090. Thus, the total time elapsed between dropping the stone and hearing the splash is d 1d t1 t2 4 1090
57
is really a family of equations, because for each value of k, we get a different equation with the unknown x. The letter k is called a parameter for this family. What value should we pick for k to make the given value of x a solution of the resulting equation? (a) x 0 (b) x 1 (c) x 2
x1
x
Equations
Factor
0 x1
Divide by x 1
x0
Simplify
10
Given x 1
112. Volumes of Solids The sphere, cylinder, and cone shown here all have the same radius r and the same volume V. (a) Use the volume formulas given on the inside front cover of this book, to show that 4 3 3 pr
pr 2h 1
4 3 3 pr
and
13 pr 2h 2
(b) Solve these equations for h1 and h2.
How deep is the well if this total time is 3 s? r r
h¤ h⁄ r
Time
Time
stone falls:
sound rises:
t⁄=
œ∑ d 4
d t¤= 1090
113. Relationship between Roots and Coefﬁcients The quadratic formula gives us the roots of a quadratic equation from its coefﬁcients. We can also obtain the coefﬁcients from the roots. For example, ﬁnd the roots of the equation x 2 9x 20 0 and show that the product of the roots is the constant term 20 and the sum of the roots is 9, the negative of the coefﬁcient of x. Show that the same relationship between roots and coefﬁcients holds for the following equations: x 2 2x 8 0
Discovery ● Discussion 110. A Family of Equations The equation 3x k 5 kx k 1
x 2 4x 2 0 Use the quadratic formula to prove that in general, if the equation x 2 bx c 0 has roots r1 and r2, then c r1r2 and b 1r1 r2 2 .
58
CHAPTER 1
Fundamentals
114. Solving an Equation in Different Ways We have learned several different ways to solve an equation in this section. Some equations can be tackled by more than one method. For example, the equation x 1x 2 0 is of quadratic type: We can solve it by letting 1x u and x u 2, and factoring. Or we could solve for 1x, square each side, and then solve the resulting quadratic equation.
1.6
Solve the following equations using both methods indicated, and show that you get the same ﬁnal answers. (a) x 1x 2 0 quadratic type; solve for the radical, and square 12 10 (b) 1 0 quadratic type; multiply x3 1x 32 2 by LCD
Modeling with Equations Many problems in the sciences, economics, ﬁnance, medicine, and numerous other ﬁelds can be translated into algebra problems; this is one reason that algebra is so useful. In this section we use equations as mathematical models to solve reallife problems.
Guidelines for Modeling with Equations We will use the following guidelines to help us set up equations that model situations described in words. To show how the guidelines can help you set up equations, we note them in the margin as we work each example in this section.
Guidelines for Modeling with Equations 1. Identify the Variable. Identify the quantity that the problem asks you to ﬁnd. This quantity can usually be determined by a careful reading of the question posed at the end of the problem. Then introduce notation for the variable (call it x or some other letter). 2. Express All Unknown Quantities in Terms of the Variable. Read each sentence in the problem again, and express all the quantities mentioned in the problem in terms of the variable you deﬁned in Step 1. To organize this information, it is sometimes helpful to draw a diagram or make a table. 3. Set Up the Model. Find the crucial fact in the problem that gives a relationship between the expressions you listed in Step 2. Set up an equation (or model) that expresses this relationship. 4. Solve the Equation and Check Your Answer. Solve the equation, check your answer, and express it as a sentence that answers the question posed in the problem.
The following example illustrates how these guidelines are used to translate a “word problem” into the language of algebra.
SECTION 1.6
Example 1
Modeling with Equations
59
Renting a Car
A car rental company charges $30 a day and 15¢ a mile for renting a car. Helen rents a car for two days and her bill comes to $108. How many miles did she drive? Solution We are asked to ﬁnd the number of miles Helen has driven. So we let Identify the variable
x number of miles driven Then we translate all the information given in the problem into the language of algebra.
Express all unknown quantities in terms of the variable
In Words
In Algebra
Number of miles driven Mileage cost (at $0.15 per mile) Daily cost (at $30 per day)
x 0.15x 21302
Now we set up the model. Set up the model
mileage cost
daily cost
total cost
0.15x 21302 108 0.15x 48
Solve
x
Check Your Answer total cost mileage cost daily cost
x 320
0.15 13202 2 1302 108
48 0.15
Subtract 60 Divide by 0.15 Calculator
Helen drove her rental car 320 miles.
■
Constructing Models In the examples and exercises that follow, we construct equations that model problems in many different reallife situations.
Example 2 Interest on an Investment Mary inherits $100,000 and invests it in two certiﬁcates of deposit. One certiﬁcate pays 6% and the other pays 4 12 % simple interest annually. If Mary’s total interest is $5025 per year, how much money is invested at each rate? Solution The problem asks for the amount she has invested at each rate. So we let Identify the variable
x the amount invested at 6% Since Mary’s total inheritance is $100,000, it follows that she invested 100,000 x at 4 12 %. We translate all the information given into the language of algebra.
60
CHAPTER 1
Fundamentals
Express all unknown quantities in terms of the variable
In Words
In Algebra
Amount invested at 6% Amount invested at 4 12 % Interest earned at 6% Interest earned at 4 12 %
x 100,000 x 0.06x 0.0451100,000 x2
We use the fact that Mary’s total interest is $5025 to set up the model. Set up the model
interest at 6% interest at 4 12 % total interest 0.06x 0.0451100,000 x2 5025
Solve
0.06x 4500 0.045x 5025 0.015x 4500 5025 0.015x 525 x
525 35,000 0.015
Multiply Combine the xterms Subtract 4500 Divide by 0.015
So Mary has invested $35,000 at 6% and the remaining $65,000 at 4 12 %.
■
Check Your Answer total interest 6% of $35,000 4 12 % of $65,000 $2100 $2925 $5025
Example 3 In a problem such as this, which involves geometry, it is essential to draw a diagram like the one shown in Figure 1.
Dimensions of a Poster
A poster has a rectangular printed area 100 cm by 140 cm, and a blank strip of uniform width around the four edges. The perimeter of the poster is 1 12 times the perimeter of the printed area. What is the width of the blank strip, and what are the dimensions of the poster? Solution We are asked to ﬁnd the width of the blank strip. So we let x the width of the blank strip
Identify the variable
Then we translate the information in Figure 1 into the language of algebra:
Express all unknown quantities in terms of the variable
In Words
In Algebra
Width of blank strip Perimeter of printed area Width of poster Length of poster Perimeter, of poster
x 211002 2 11402 480 100 2x 140 2x 21100 2x2 2 1140 2x2
SECTION 1.6
Modeling with Equations
61
Now we use the fact that the perimeter of the poster is 112 times the perimeter of the printed area to set up the model. perimeter of poster 32 ⴢ perimeter of printed area
Set up the model
21100 2x2 21140 2x2 32 # 480 480 8x 720
Solve
8x 240 x 30
Expand and combine like terms on LHS Subtract 480 Divide by 8
The blank strip is 30 cm wide, so the dimensions of the poster are 100 30 30 160 cm wide 140 30 30 200 cm long
by
100 cm x
140 cm
x
■
Figure 1
Example 4
Dimensions of a Building Lot
A rectangular building lot is 8 ft longer than it is wide and has an area of 2900 ft2. Find the dimensions of the lot. Solution We are asked to ﬁnd the width and length of the lot. So let „ width of lot
Identify the variable
Then we translate the information given in the problem into the language of algebra (see Figure 2 on page 62).
Express all unknown quantities in terms of the variable
In Words
In Algebra
Width of lot Length of lot
„ „8
Now we set up the model.
62
CHAPTER 1
Fundamentals
width of lot
Set up the model
ⴢ
length of lot
area of lot
„ 1„ 82 2900 Solve
„ 2 8„ 2900
Expand
„ 2 8„ 2900 0
Subtract 2900
1„ 502 1„ 582 0 „ 50
or
Factor
„ 58
ZeroProduct Property
Since the width of the lot must be a positive number, we conclude that „ 50 ft. The length of the lot is „ 8 50 8 58 ft.
w
w+8
Figure 2
Example 5
■
Determining the Height of a Building Using Similar Triangles
A man 6 ft tall wishes to ﬁnd the height of a certain fourstory building. He measures its shadow and ﬁnds it to be 28 ft long, while his own shadow is 3 12 ft long. How tall is the building? Solution The problem asks for the height of the building. So let h the height of the building
Identify the variable
We use the fact that the triangles in Figure 3 are similar. Recall that for any pair of similar triangles the ratios of corresponding sides are equal. Now we translate these observations into the language of algebra.
Express all unknown quantities in terms of the variable
In Words
In Algebra
Height of building
h
Ratio of height to base in large triangle Ratio of height to base in small triangle
h 28 6 3.5
Since the large and small triangles are similar, we get the equation Set up the model
ratio of height to base in large triangle
ratio of height to base in small triangle
h 6 28 3.5 Solve
h
6 # 28 48 3.5
SECTION 1.6
Modeling with Equations
63
The building is 48 ft tall.
h
6 ft 28 ft
3 12 ft
Figure 3
Example 6
■
Mixtures and Concentration
A manufacturer of soft drinks advertises their orange soda as “naturally ﬂavored,” although it contains only 5% orange juice. A new federal regulation stipulates that to be called “natural” a drink must contain at least 10% fruit juice. How much pure orange juice must this manufacturer add to 900 gal of orange soda to conform to the new regulation? Solution The problem asks for the amount of pure orange juice to be added. So let x the amount 1in gallons2 of pure orange juice to be added
Identify the variable
In any problem of this type—in which two different substances are to be mixed—drawing a diagram helps us organize the given information (see Figure 4).
Volume Amount of orange juice Figure 4
5% juice
100% juice
900 gallons
x gallons
5% of 900 gallons =45 gallons
10% juice
900+x gallons
100% of x gallons 10% of 900+x gallons =0.1(900+x) gallons =x gallons
64
CHAPTER 1
Fundamentals
We now translate the information in the ﬁgure into the language of algebra.
Express all unknown quantities in terms of the variable
In Words
In Algebra
Amount of orange juice to be added Amount of the mixture Amount of orange juice in the ﬁrst vat Amount of orange juice in the second vat Amount of orange juice in the mixture
x 900 x 0.05 19002 = 45 1 xx 0.101900 + x2
#
To set up the model, we use the fact that the total amount of orange juice in the mixture is equal to the orange juice in the ﬁrst two vats.
Set up the model
amount of orange juice in ﬁrst vat
amount of orange juice in second vat
amount of orange juice in mixture
45 x 0.11900 x2
From Figure 4
45 x 90 0.1x
Multiply
0.9x 45
Solve
x
45 50 0.9
Subtract 0.1x and 45 Divide by 0.9
The manufacturer should add 50 gal of pure orange juice to the soda.
Check Your Answer amount of juice before mixing 5% of 900 gal 50 gal pure juice 45 gal 50 gal 95 gal amount of juice after mixing 10% of 950 gal 95 gal Amounts are equal.
Example 7 B A
Time Needed to Do a Job
Because of an anticipated heavy rainstorm, the water level in a reservoir must be lowered by 1 ft. Opening spillway A lowers the level by this amount in 4 hours, whereas opening the smaller spillway B does the job in 6 hours. How long will it take to lower the water level by 1 ft if both spillways are opened? Solution We are asked to ﬁnd the time needed to lower the level by 1 ft if both spillways are open. So let
Identify the variable
x the time 1in hours 2 it takes to lower the water level by 1 ft if both spillways are open Finding an equation relating x to the other quantities in this problem is not easy.
■
SECTION 1.6
Modeling with Equations
65
Certainly x is not simply 4 6, because that would mean that together the two spillways require longer to lower the water level than either spillway alone. Instead, we look at the fraction of the job that can be done in one hour by each spillway.
Express all unknown quantities in terms of the variable
In Words
In Algebra
Time it takes to lower level 1 ft with A and B together Distance A lowers level in 1 h Distance B lowers level in 1 h Distance A and B together lower levels in 1 h
xh 1 4 ft 1 6 ft 1 x ft
Now we set up the model. fraction done by A fraction done by B fraction done by both
Set up the model
1 1 1 x 4 6 3x 2x 12
Solve
5x 12 x
12 5
Multiply by the LCD, 12x Add Divide by 5
It will take 2 25 hours, or 2 h 24 min to lower the water level by 1 ft if both spillways are open. ■ The next example deals with distance, rate (speed), and time. The formula to keep in mind here is distance rate time where the rate is either the constant speed or average speed of a moving object. For example, driving at 60 mi/h for 4 hours takes you a distance of 60 ⴢ 4 240 mi.
Example 8
A DistanceSpeedTime Problem
A jet ﬂew from New York to Los Angeles, a distance of 4200 km. The speed for the return trip was 100 km/h faster than the outbound speed. If the total trip took 13 hours, what was the jet’s speed from New York to Los Angeles? Solution We are asked for the speed of the jet from New York to Los Angeles. So let s speed from New York to Los Angeles
Identify the variable
Then
s 100 speed from Los Angeles to New York
Now we organize the information in a table. We ﬁll in the “Distance” column ﬁrst, since we know that the cities are 4200 km apart. Then we ﬁll in the “Speed” column, since we have expressed both speeds (rates) in terms of the variable s. Finally, we calculate the entries for the “Time” column, using time
distance rate
66
CHAPTER 1
Fundamentals
Express all unknown quantities in terms of the variable
Distance (km)
Speed (km /h)
N.Y. to L.A.
4200
s
L.A. to N.Y.
4200
s 100
Time (h)
4200 s 4200 s 100
The total trip took 13 hours, so we have the model time from N.Y. to L.A.
Set up the model
time from L.A. to N.Y.
total time
4200 4200 13 s s 100 Multiplying by the common denominator, s1s 1002 , we get 42001s 1002 4200s 13s1s 1002 8400s 420,000 13s 2 1300s 0 13s 2 7100s 420,000 Although this equation does factor, with numbers this large it is probably quicker to use the quadratic formula and a calculator. s
Solve
7100 2171002 2 41132 1420,0002 21132 7100 8500 26
s 600
or
s
1400 ⬇ 53.8 26
Since s represents speed, we reject the negative answer and conclude that the jet’s speed from New York to Los Angeles was 600 km/h.
Example 9 A
Island
5 mi B
C x 12 mi
Figure 5
D Nesting area
■
Energy Expended in Bird Flight
Ornithologists have determined that some species of birds tend to avoid ﬂights over large bodies of water during daylight hours, because air generally rises over land and falls over water in the daytime, so ﬂying over water requires more energy. A bird is released from point A on an island, 5 mi from B, the nearest point on a straight shoreline. The bird ﬂies to a point C on the shoreline and then ﬂies along the shoreline to its nesting area D, as shown in Figure 5. Suppose the bird has 170 kcal of energy reserves. It uses 10 kcal/mi ﬂying over land and 14 kcal/mi ﬂying over water. (a) Where should the point C be located so that the bird uses exactly 170 kcal of energy during its ﬂight? (b) Does the bird have enough energy reserves to ﬂy directly from A to D?
SECTION 1.6
Modeling with Equations
67
Solution (a) We are asked to ﬁnd the location of C. So let x distance from B to C
Identify the variable
From the ﬁgure, and from the fact that energy used energy per mile miles flown we determine the following: In Words
Express all unknown quantities in terms of the variable
In Algebra
Distance from B to C Distance ﬂown over water (from A to C ) Distance ﬂown over land (from C to D) Energy used over water Energy used over land
x 2x 2 25 12 x 14 2x 2 25 10112 x2
Pythagorean Theorem
Now we set up the model. Set up the model
total energy used
energy used over water
energy used over land
170 142x 2 25 10112 x2 To solve this equation, we eliminate the square root by ﬁrst bringing all other terms to the left of the equal sign and then squaring each side. Solve
Isolate squareroot term on RHS
170 10112 x2 142x 2 25 50 10x 142x 2 25
Simplify LHS
150 10x2 2 1142 2 1x 2 252
Square each side
2500 1000x 100x 196x 4900 2
2
Expand
0 96x 2 1000x 2400
All terms to RHS
This equation could be factored, but because the numbers are so large it is easier to use the quadratic formula and a calculator: x
1000 2110002 2 41962 124002 21962 1000 280 6 23 192
or
3 34
Point C should be either 6 23 mi or 3 34 mi from B so that the bird uses exactly 170 kcal of energy during its ﬂight. (b) By the Pythagorean Theorem (see page 54), the length of the route directly from A to D is 252 122 13 mi, so the energy the bird requires for that route is 14 13 182 kcal. This is more energy than the bird has available, so it can’t use this route.
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68
CHAPTER 1
1.6
Fundamentals
Exercises
1–12 ■ Express the given quantity in terms of the indicated variable. 1. The sum of three consecutive integers; n ﬁrst integer of the three 2. The sum of three consecutive integers; n middle integer of the three 3. The average of three test scores if the ﬁrst two scores are 78 and 82; s third test score 4. The average of four quiz scores if each of the ﬁrst three scores is 8; q fourth quiz score 5. The interest obtained after one year on an investment at 2 12 % simple interest per year; x number of dollars invested 6. The total rent paid for an apartment if the rent is $795 a month; n number of months 7. The area (in ft 2) of a rectangle that is three times as long as it is wide; „ width of the rectangle (in ft) 8. The perimeter (in cm) of a rectangle that is 5 cm longer than it is wide; „ width of the rectangle (in cm) 9. The distance (in mi) that a car travels in 45 min; s speed of the car (in mi/h) 10. The time (in hours) it takes to travel a given distance at 55 mi/h; d given distance (in mi) 11. The concentration (in oz/gal) of salt in a mixture of 3 gal of brine containing 25 oz of salt, to which some pure water has been added; x volume of pure water added (in gal) 12. The value (in cents) of the change in a purse that contains twice as many nickels as pennies, four more dimes than nickels, and as many quarters as dimes and nickels combined; p number of pennies
Applications 13. Number Problem Find three consecutive integers whose sum is 156. 14. Number Problem Find four consecutive odd integers whose sum is 416. 15. Number Problem Find two numbers whose sum is 55 and whose product is 684.
on these investments was $525. How much money did she invest at each rate? 18. Investments If Ben invests $4000 at 4% interest per year, how much additional money must he invest at 5 12 % annual interest to ensure that the interest he receives each year is 4 12 % of the total amount invested? 19. Investments What annual rate of interest would you have to earn on an investment of $3500 to ensure receiving $262.50 interest after one year? 20. Investments Jack invests $1000 at a certain annual interest rate, and he invests another $2000 at an annual rate that is onehalf percent higher. If he receives a total of $190 interest in one year, at what rate is the $1000 invested? 21. Salaries An executive in an engineering ﬁrm earns a monthly salary plus a Christmas bonus of $8500. If she earns a total of $97,300 per year, what is her monthly salary? 22. Salaries A woman earns 15% more than her husband. Together they make $69,875 per year. What is the husband’s annual salary? 23. Inheritance Craig is saving to buy a vacation home. He inherits some money from a wealthy uncle, then combines this with the $22,000 he has already saved and doubles the total in a lucky investment. He ends up with $134,000, just enough to buy a cabin on the lake. How much did he inherit? 24. Overtime Pay Helen earns $7.50 an hour at her job, but if she works more than 35 hours in a week she is paid 1 12 times her regular salary for the overtime hours worked. One week her gross pay was $352.50. How many overtime hours did she work that week? 25. Labor Costs A plumber and his assistant work together to replace the pipes in an old house. The plumber charges $45 an hour for his own labor and $25 an hour for his assistant’s labor. The plumber works twice as long as his assistant on this job, and the labor charge on the ﬁnal bill is $4025. How long did the plumber and his assistant work on this job? 26. Career Home Runs During his major league career, Hank Aaron hit 41 more home runs than Babe Ruth hit during his career. Together they hit 1469 home runs. How many home runs did Babe Ruth hit?
16. Number Problem The sum of the squares of two consecutive even integers is 1252. Find the integers.
27. A Riddle A movie star, unwilling to give his age, posed the following riddle to a gossip columnist. “Seven years ago, I was eleven times as old as my daughter. Now I am four times as old as she is.” How old is the star?
17. Investments Phyllis invested $12,000, a portion earning a simple interest rate of 4 12 % per year and the rest earning a rate of 4% per year. After one year the total interest earned
28. A Riddle A father is four times as old as his daughter. In 6 years, he will be three times as old as she is. How old is the daughter now?
SECTION 1.6
29. Value of Coins A change purse contains an equal number of pennies, nickels, and dimes. The total value of the coins is $1.44. How many coins of each type does the purse contain? 30. Value of Coins Mary has $3.00 in nickels, dimes, and quarters. If she has twice as many dimes as quarters and ﬁve more nickels than dimes, how many coins of each type does she have? 31. Law of the Lever The ﬁgure shows a lever system, similar to a seesaw that you might ﬁnd in a children’s playground. For the system to balance, the product of the weight and its distance from the fulcrum must be the same on each side; that is „1x 1 „2x 2 This equation is called the law of the lever, and was ﬁrst discovered by Archimedes (see page 748). A woman and her son are playing on a seesaw. The boy is at one end, 8 ft from the fulcrum. If the son weighs 100 lb and the mother weighs 125 lb, where should the woman sit so that the seesaw is balanced?
Modeling with Equations
33. Length and Area Find the length x in the ﬁgure. The area of the shaded region is given. (a)
(b)
x
x 14 in.
10 cm
6 cm
13 in. x
x area=160 in2
area=144 cm2
34. Length and Area Find the length y in the ﬁgure. The area of the shaded region is given. (a)
(b) y
y
y
y y
area=120 in2
x⁄
x¤
32. Law of the Lever A plank 30 ft long rests on top of a ﬂatroofed building, with 5 ft of the plank projecting over the edge, as shown in the ﬁgure. A worker weighing 240 lb sits on one end of the plank. What is the largest weight that can be hung on the projecting end of the plank if it is to remain in balance? (Use the law of the lever stated in Exercise 31.)
1 cm area=1200 cm2
„¤ „⁄
69
35. Length of a Garden A rectangular garden is 25 ft wide. If its area is 1125 ft 2, what is the length of the garden?
x ft 25 ft
36. Width of a Pasture A pasture is twice as long as it is wide. Its area is 115,200 ft 2. How wide is the pasture? 5 ft
37. Dimensions of a Lot A square plot of land has a building 60 ft long and 40 ft wide at one corner. The rest of the land outside the building forms a parking lot. If the parking lot has area 12,000 ft 2, what are the dimensions of the entire plot of land? 38. Dimensions of a Lot A halfacre building lot is ﬁve times as long as it is wide. What are its dimensions? [Note: 1 acre 43,560 ft 2.] 39. Dimensions of a Garden A rectangular garden is 10 ft longer than it is wide. Its area is 875 ft 2. What are its dimensions?
70
CHAPTER 1
Fundamentals
40. Dimensions of a Room A rectangular bedroom is 7 ft longer than it is wide. Its area is 228 ft 2. What is the width of the room? 41. Dimensions of a Garden A farmer has a rectangular garden plot surrounded by 200 ft of fence. Find the length and width of the garden if its area is 2400 ft 2. perimeter=200 ft
46. Width of a Lawn A factory is to be built on a lot measuring 180 ft by 240 ft. A local building code speciﬁes that a lawn of uniform width and equal in area to the factory must surround the factory. What must the width of this lawn be, and what are the dimensions of the factory? 47. Reach of a Ladder A 19 21 foot ladder leans against a building. The base of the ladder is 7 12 ft from the building. How high up the building does the ladder reach?
19 12 ft 42. Dimensions of a Lot A parcel of land is 6 ft longer than it is wide. Each diagonal from one corner to the opposite corner is 174 ft long. What are the dimensions of the parcel? 43. Dimensions of a Lot A rectangular parcel of land is 50 ft wide. The length of a diagonal between opposite corners is 10 ft more than the length of the parcel. What is the length of the parcel? 44. Dimensions of a Track A running track has the shape shown in the ﬁgure, with straight sides and semicircular ends. If the length of the track is 440 yd and the two straight parts are each 110 yd long, what is the radius of the semicircular parts (to the nearest yard)?
7 12 ft
48. Height of a Flagpole A ﬂagpole is secured on opposite sides by two guy wires, each of which is 5 ft longer than the pole. The distance between the points where the wires are ﬁxed to the ground is equal to the length of one guy wire. How tall is the ﬂagpole (to the nearest inch)?
110 yd
r
45. Framing a Painting Al paints with watercolors on a sheet of paper 20 in. wide by 15 in. high. He then places this sheet on a mat so that a uniformly wide strip of the mat shows all around the picture. The perimeter of the mat is 102 in. How wide is the strip of the mat showing around the picture?
15 in.
x
49. Length of a Shadow A man is walking away from a lamppost with a light source 6 m above the ground. The man is 2 m tall. How long is the man’s shadow when he is 10 m from the lamppost? [Hint: Use similar triangles.]
6m 2m
20 in.
10 m
x
SECTION 1.6
50. Height of a Tree A woodcutter determines the height of a tall tree by ﬁrst measuring a smaller one 125 ft away, then moving so that his eyes are in the line of sight along the tops of the trees, and measuring how far he is standing from the small tree (see the ﬁgure). Suppose the small tree is 20 ft tall, the man is 25 ft from the small tree, and his eye level is 5 ft above the ground. How tall is the taller tree?
20 ft 5 ft 25 ft
125 ft
51. Buying a Cottage A group of friends decides to buy a vacation home for $120,000, sharing the cost equally. If they can ﬁnd one more person to join them, each person’s contribution will drop by $6000. How many people are in the group? 52. Mixture Problem What quantity of a 60% acid solution must be mixed with a 30% solution to produce 300 mL of a 50% solution? 53. Mixture Problem A jeweler has ﬁve rings, each weighing 18 g, made of an alloy of 10% silver and 90% gold. He decides to melt down the rings and add enough silver to reduce the gold content to 75%. How much silver should he add? 54. Mixture Problem A pot contains 6 L of brine at a concentration of 120 g/L. How much of the water should be boiled off to increase the concentration to 200 g/L? 55. Mixture Problem The radiator in a car is ﬁlled with a solution of 60% antifreeze and 40% water. The manufacturer of the antifreeze suggests that, for summer driving, optimal cooling of the engine is obtained with only 50% antifreeze. If the capacity of the radiator is 3.6 L, how much coolant should be drained and replaced with water to reduce the antifreeze concentration to the recommended level? 56. Mixture Problem A health clinic uses a solution of bleach to sterilize petri dishes in which cultures are grown. The sterilization tank contains 100 gal of a solution of 2% ordinary household bleach mixed with pure distilled water. New research indicates that the concentration of bleach should be 5% for complete sterilization. How much of the solution should be drained and replaced with bleach to increase the bleach content to the recommended level?
Modeling with Equations
71
57. Mixture Problem A bottle contains 750 mL of fruit punch with a concentration of 50% pure fruit juice. Jill drinks 100 mL of the punch and then reﬁlls the bottle with an equal amount of a cheaper brand of punch. If the concentration of juice in the bottle is now reduced to 48%, what was the concentration in the punch that Jill added? 58. Mixture Problem A merchant blends tea that sells for $3.00 a pound with tea that sells for $2.75 a pound to produce 80 lb of a mixture that sells for $2.90 a pound. How many pounds of each type of tea does the merchant use in the blend? 59. Sharing a Job Candy and Tim share a paper route. It takes Candy 70 min to deliver all the papers, and it takes Tim 80 min. How long does it take the two when they work together? 60. Sharing a Job Stan and Hilda can mow the lawn in 40 min if they work together. If Hilda works twice as fast as Stan, how long does it take Stan to mow the lawn alone? 61. Sharing a Job Betty and Karen have been hired to paint the houses in a new development. Working together the women can paint a house in twothirds the time that it takes Karen working alone. Betty takes 6 h to paint a house alone. How long does it take Karen to paint a house working alone? 62. Sharing a Job Nextdoor neighbors Bob and Jim use hoses from both houses to ﬁll Bob’s swimming pool. They know it takes 18 h using both hoses. They also know that Bob’s hose, used alone, takes 20% less time than Jim’s hose alone. How much time is required to ﬁll the pool by each hose alone? 63. Sharing a Job Henry and Irene working together can wash all the windows of their house in 1 h 48 min. Working alone, it takes Henry 1 12 h more than Irene to do the job. How long does it take each person working alone to wash all the windows? 64. Sharing a Job Jack, Kay, and Lynn deliver advertising ﬂyers in a small town. If each person works alone, it takes Jack 4 h to deliver all the ﬂyers, and it takes Lynn 1 h longer than it takes Kay. Working together, they can deliver all the ﬂyers in 40% of the time it takes Kay working alone. How long does it take Kay to deliver all the ﬂyers alone? 65. Distance, Speed, and Time Wendy took a trip from Davenport to Omaha, a distance of 300 mi. She traveled part of the way by bus, which arrived at the train station just in time for Wendy to complete her journey by train. The bus averaged 40 mi/h and the train 60 mi/h. The entire trip took 5 12 h. How long did Wendy spend on the train? 66. Distance, Speed, and Time Two cyclists, 90 mi apart, start riding toward each other at the same time. One cycles
72
CHAPTER 1
Fundamentals
twice as fast as the other. If they meet 2 h later, at what average speed is each cyclist traveling? 67. Distance, Speed, and Time A pilot ﬂew a jet from Montreal to Los Angeles, a distance of 2500 mi. On the return trip the average speed was 20% faster than the outbound speed. The roundtrip took 9 h 10 min. What was the speed from Montreal to Los Angeles?
eastbound boat travels at a speed 3 mi/h faster than the southbound boat. After two hours the boats are 30 mi apart. Find the speed of the southbound boat.
N W
68. Distance, Speed, and Time A woman driving a car 14 ft long is passing a truck 30 ft long. The truck is traveling at 50 mi/h. How fast must the woman drive her car so that she can pass the truck completely in 6 s, from the position shown in ﬁgure (a) to the position shown in ﬁgure (b)? [Hint: Use feet and seconds instead of miles and hours.]
E S
30
mi
50 mi/ h 73. Dimensions of a Box A large plywood box has a volume of 180 ft3. Its length is 9 ft greater than its height, and its width is 4 ft less than its height. What are the dimensions of the box?
(a)
x+9 50 mi/ h x (b) x4 69. Distance, Speed, and Time A salesman drives from Ajax to Barrington, a distance of 120 mi, at a steady speed. He then increases his speed by 10 mi/h to drive the 150 mi from Barrington to Collins. If the second leg of his trip took 6 min more time than the ﬁrst leg, how fast was he driving between Ajax and Barrington? 70. Distance, Speed, and Time Kiran drove from Tortula to Cactus, a distance of 250 mi. She increased her speed by 10 mi/h for the 360mi trip from Cactus to Dry Junction. If the total trip took 11 h, what was her speed from Tortula to Cactus? 71. Distance, Speed, and Time It took a crew 2 h 40 min to row 6 km upstream and back again. If the rate of ﬂow of the stream was 3 km/h, what was the rowing speed of the crew in still water? 72. Speed of a Boat Two ﬁshing boats depart a harbor at the same time, one traveling east, the other south. The
74. Radius of a Sphere A jeweler has three small solid spheres made of gold, of radius 2 mm, 3 mm, and 4 mm. He decides to melt these down and make just one sphere out of them. What will the radius of this larger sphere be? 75. Dimensions of a Box A box with a square base and no top is to be made from a square piece of cardboard by cutting 4in. squares from each corner and folding up the sides, as shown in the ﬁgure. The box is to hold 100 in3. How big a piece of cardboard is needed? 4 in. 4 in.
SECTION 1.6
76. Dimensions of a Can A cylindrical can has a volume of 40p cm3 and is 10 cm tall. What is its diameter? [Hint: Use the volume formula listed on the inside back cover of this book.]
Modeling with Equations
73
standing on the boardwalk, exactly 750 ft across the sand from his beach umbrella, which is right at the shoreline. The man walks 4 ft/s on the boardwalk and 2 ft/s on the sand. How far should he walk on the boardwalk before veering off onto the sand if he wishes to reach his umbrella in exactly 4 min 45 s?
10 cm
77. Radius of a Tank A spherical tank has a capacity of 750 gallons. Using the fact that one gallon is about 0.1337 ft3, ﬁnd the radius of the tank (to the nearest hundredth of a foot).
750 ft 210 ft
Boardwalk 78. Dimensions of a Lot A city lot has the shape of a right triangle whose hypotenuse is 7 ft longer than one of the other sides. The perimeter of the lot is 392 ft. How long is each side of the lot? 79. Construction Costs The town of Foxton lies 10 mi north of an abandoned eastwest road that runs through Grimley, as shown in the ﬁgure. The point on the abandoned road closest to Foxton is 40 mi from Grimley. County ofﬁcials are about to build a new road connecting the two towns. They have determined that restoring the old road would cost $100,000 per mile, whereas building a new road would cost $200,000 per mile. How much of the abandoned road should be used (as indicated in the ﬁgure) if the ofﬁcials intend to spend exactly $6.8 million? Would it cost less than this amount to build a new road connecting the towns directly?
81. Volume of Grain Grain is falling from a chute onto the ground, forming a conical pile whose diameter is always three times its height. How high is the pile (to the nearest hundredth of a foot) when it contains 1000 ft3 of grain?
Foxton
New road Grimley
10 mi
Abandoned road 40 mi
80. Distance, Speed, and Time A boardwalk is parallel to and 210 ft inland from a straight shoreline. A sandy beach lies between the boardwalk and the shoreline. A man is
82. TV Monitors Two television monitors sitting beside each other on a shelf in an appliance store have the same screen height. One has a conventional screen, which is 5 in. wider than it is high. The other has a wider, highdeﬁnition screen, which is 1.8 times as wide as it is high. The diagonal measure of the wider screen is 14 in. more than the diagonal measure of the smaller. What is the height of the screens, correct to the nearest 0.1 in.?
74
CHAPTER 1
Fundamentals
83. Dimensions of a Structure A storage bin for corn consists of a cylindrical section made of wire mesh, surmounted by a conical tin roof, as shown in the ﬁgure. The height of the roof is onethird the height of the entire structure. If the total volume of the structure is 1400p ft3 and its radius is 10 ft, what is its height? [Hint: Use the volume formulas listed on the inside front cover of this book.]
A 10ftlong stem of bamboo is broken in such a way that its tip touches the ground 3 ft from the base of the stem, as shown in the ﬁgure. What is the height of the break? [Hint: Use the Pythagorean Theorem.]
1 3h
h
3 ft 10 ft
Discovery • Discussion 84. Comparing Areas A wire 360 in. long is cut into two pieces. One piece is formed into a square and the other into a circle. If the two ﬁgures have the same area, what are the lengths of the two pieces of wire (to the nearest tenth of an inch)?
86. Historical Research Read the biographical notes on Pythagoras (page 54), Euclid (page 532), and Archimedes (page 748). Choose one of these mathematicians and ﬁnd out more about him from the library or on the Internet. Write a short essay on your ﬁndings. Include both biographical information and a description of the mathematics for which he is famous. 87. A Babylonian Quadratic Equation The ancient Babylonians knew how to solve quadratic equations. Here is a problem from a cuneiform tablet found in a Babylonian school dating back to about 2000 B.C.
85. An Ancient Chinese Problem This problem is taken from a Chinese mathematics textbook called Chuichang suanshu, or Nine Chapters on the Mathematical Art, which was written about 250 B.C.
I have a reed, I know not its length. I broke from it one cubit, and it ﬁt 60 times along the length of my ﬁeld. I restored to the reed what I had broken off, and it ﬁt 30 times along the width of my ﬁeld. The area of my ﬁeld is 375 square nindas. What was the original length of the reed? Solve this problem. Use the fact that 1 ninda 12 cubits.
SECTION 1.6
Modeling with Equations
Equations through the Ages DISCOVERY PROJECT
Equations have been used to solve problems throughout recorded history, in every civilization. (See, for example, Exercise 85 on page 74.) Here is a problem from ancient Babylon (ca. 2000 B.C.). I found a stone but did not weigh it. After I added a seventh, and then added an eleventh of the result, I weighed it and found it weighed 1 mina. What was the original weight of the stone?
The British Museum
The answer given on the cuneiform tablet is 23 mina, 8 sheqel, and 22 12 se, where 1 mina 60 sheqel, and 1 sheqel 180 se. In ancient Egypt, knowing how to solve word problems was a highly prized secret. The Rhind Papyrus (ca. 1850 B.C.) contains many such problems (see page 716). Problem 32 in the Papyrus states A quantity, its third, its quarter, added together become 2. What is the quantity?
The answer in Egyptian notation is 1 4 76, where the bar indicates “reciprocal,” much like our notation 41. The Greek mathematician Diophantus (ca. 250 A.D., see page 20) wrote the book Arithmetica, which contains many word problems and equations. The Indian mathematician Bhaskara (12th century A.D., see page 144) and the Chinese mathematician Chang Ch’iuChien (6th century A.D.) also studied and wrote about equations. Of course, equations continue to be important today. 1. Solve the Babylonian problem and show that their answer is correct. 2. Solve the Egyptian problem and show that their answer is correct. 3. The ancient Egyptians and Babylonians used equations to solve practical problems. From the examples given here, do you think that they may have enjoyed posing and solving word problems just for fun? 4. Solve this problem from 12thcentury India. 15 x 45
A peaco*ck is perched at the top of a 15cubit pillar, and a snake’s hole is at the foot of the pillar. Seeing the snake at a distance of 45 cubits from its hole, the peaco*ck pounces obliquely upon the snake as it slithers home. At how many cubits from the snake’s hole do they meet, assuming that each has traveled an equal distance?
5. Consider this problem from 6thcentury China. If a rooster is worth 5 coins, a hen 3 coins, and three chicks together one coin, how many roosters, hens, and chicks, totaling 100, can be bought for 100 coins?
This problem has several answers. Use trial and error to ﬁnd at least one answer. Is this a practical problem or more of a riddle? Write a short essay to support your opinion. 6. Write a short essay explaining how equations affect your own life in today’s world.
75
76
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Fundamentals
1.7
Inequalities Some problems in algebra lead to inequalities instead of equations. An inequality looks just like an equation, except that in the place of the equal sign is one of the symbols, , , , or . Here is an example of an inequality: 4x 7 19
4x 7 19
x
11 19 15 19 19 19 23 19 27 19
1 2 3 4 5
The table in the margin shows that some numbers satisfy the inequality and some numbers don’t. To solve an inequality that contains a variable means to ﬁnd all values of the variable that make the inequality true. Unlike an equation, an inequality generally has inﬁnitely many solutions, which form an interval or a union of intervals on the real line. The following illustration shows how an inequality differs from its corresponding equation: Solution
Graph
Equation:
4x 7 19
x3
3
Inequality:
4x 7 19
x 3
3
To solve inequalities, we use the following rules to isolate the variable on one side of the inequality sign. These rules tell us when two inequalities are equivalent (the symbol 3 means “is equivalent to”). In these rules the symbols A, B, and C stand for real numbers or algebraic expressions. Here we state the rules for inequalities involving the symbol , but they apply to all four inequality symbols.
Rules for Inequalities Rule
Description
1. A B
3
AC BC
2. A B
3
AC BC
3. If C 0,
then
A B
3
CA CB
4. If C 0,
then
A B
3
CA CB
5. If A 0
and
then
A B
B 0, 3
1 1 A B
6. If A B and C D, then A C B D
Adding the same quantity to each side of an inequality gives an equivalent inequality. Subtracting the same quantity from each side of an inequality gives an equivalent inequality. Multiplying each side of an inequality by the same positive quantity gives an equivalent inequality. Multiplying each side of an inequality by the same negative quantity reverses the direction of the inequality. Taking reciprocals of each side of an inequality involving positive quantities reverses the direction of the inequality. Inequalities can be added.
SECTION 1.7
Inequalities
77
Pay special attention to Rules 3 and 4. Rule 3 says that we can multiply (or divide) each side of an inequality by a positive number, but Rule 4 says that if we multiply each side of an inequality by a negative number, then we reverse the direction of the inequality. For example, if we start with the inequality 35 and multiply by 2, we get 6 10 but if we multiply by 2, we get 6 10
Linear Inequalities An inequality is linear if each term is constant or a multiple of the variable.
Example 1
Solving a Linear Inequality
Solve the inequality 3x 9x 4 and sketch the solution set. Solution 3x 9x 4 3x 9x 9x 4 9x Multiplying by the negative number 16 reverses the direction of the inequality.
6x 4
A 16 B16x2
A16 B142
x 23 _ 23
Subtract 9x Simplify Multiply by 61 (or divide by 6) Simplify
The solution set consists of all numbers greater than 23 . In other words the solution of the inequality is the interval A 23, qB . It is graphed in Figure 1.
■
Figure 1
Example 2
Solving a Pair of Simultaneous Inequalities
Solve the inequalities 4 3x 2 13. Solution The solution set consists of all values of x that satisfy both of the inequalities 4 3x 2 and 3x 2 13. Using Rules 1 and 3, we see that the following inequalities are equivalent: 4 3x 2 13
0 Figure 2
2
5
6 3x 15
Add 2
2 x5
Divide by 3
Therefore, the solution set is 32, 52 , as shown in Figure 2.
■
Nonlinear Inequalities To solve inequalities involving squares and other powers of the variable, we use factoring, together with the following principle.
78
CHAPTER 1
Fundamentals
The Sign of a Product or Quotient If a product or a quotient has an even number of negative factors, then its value is positive. If a product or a quotient has an odd number of negative factors, then its value is negative.
Example 3
A Quadratic Inequality
Solve the inequality x 2 5x 6 0. Solution First we factor the left side. 1x 22 1x 32 0
(_`, 2) 0
(2, 3) 2
(3, `) 3
Figure 3
We know that the corresponding equation 1x 22 1x 32 0 has the solutions 2 and 3. As shown in Figure 3, the numbers 2 and 3 divide the real line into three intervals: 1q, 22 , 12, 3 2 , and 13, q 2 . On each of these intervals we determine the signs of the factors using test values. We choose a number inside each interval and check the sign of the factors x 2 and x 3 at the value selected. For instance, if we use the test value x 1 for the interval 1q, 22 shown in Figure 4, then substitution in the factors x 2 and x 3 gives x 2 1 2 1 0 x 3 1 3 2 0
and Test value x=1
0 Figure 4
Test value x = 2 21
2
3
Test value x=4
So both factors are negative on this interval. (The factors x 2 and x 3 change sign only at 2 and 3, respectively, so they maintain their signs over the length of each interval. That is why using a single test value on each interval is sufﬁcient.) Using the test values x 2 12 and x 4 for the intervals 12, 32 and 13, q 2 (see Figure 4), respectively, we construct the following sign table. The ﬁnal row of the table is obtained from the fact that the expression in the last row is the product of the two factors.
1q, 22
12, 32
13, q2
Sign of x 2
Sign of x 3
Sign of Óx 2ÔÓx 3Ô
Interval
If you prefer, you can represent this information on a real number line, as in the following sign diagram. The vertical lines indicate the points at which the real line is divided into intervals:
SECTION 1.7
Inequalities
79
3
2 Sign of x2

+
+
Sign of x3


+
Sign of (x2)(x3)
+

+
We read from the table or the diagram that 1x 22 1x 32 is negative on the interval 12, 3 2 . Thus, the solution of the inequality 1x 22 1x 32 0 is 0 Figure 5
2
3
5x 0 2 x 36 32, 3 4 We have included the endpoints 2 and 3 because we seek values of x such that the product is either less than or equal to zero. The solution is illustrated in Figure 5.
■
Example 3 illustrates the following guidelines for solving an inequality that can be factored.
Guidelines for Solving Nonlinear Inequalities 1. Move All Terms to One Side. If necessary, rewrite the inequality so that all nonzero terms appear on one side of the inequality sign. If the nonzero side of the inequality involves quotients, bring them to a common denominator. 2. Factor. Factor the nonzero side of the inequality. 3. Find the Intervals. Determine the values for which each factor is zero. These numbers will divide the real line into intervals. List the intervals determined by these numbers. 4. Make a Table or Diagram. Use test values to make a table or diagram of the signs of each factor on each interval. In the last row of the table determine the sign of the product (or quotient) of these factors. 5. Solve. Determine the solution of the inequality from the last row of the sign table. Be sure to check whether the inequality is satisﬁed by some or all of the endpoints of the intervals (this may happen if the inequality involves
or ).
The factoring technique described in these guidelines works only if all nonzero terms appear on one side of the inequality symbol. If the inequality is not written in this form, ﬁrst rewrite it, as indicated in Step 1. This technique is illustrated in the examples that follow.
80
CHAPTER 1
Fundamentals
It is tempting to multiply both sides of the inequality by 1 x (as you would if this were an equation). But this doesn’t work because we don’t know if 1 x is positive or negative, so we can’t tell if the inequality needs to be reversed. (See Exercise 110.)
Terms to one side
Example 4 Solve:
An Inequality Involving a Quotient
1x 1 1x
Solution First we move all nonzero terms to the left side, and then we simplify using a common denominator. 1x 1 1x 1x Subtract 1 (to move all 10 terms to LHS) 1x 1x 1x 0 Common denominator 1 x 1x 1x 1x1x Combine the fractions 0 1x 2x Simplify 0 1x The numerator is zero when x 0 and the denominator is zero when x 1, so we construct the following sign diagram using these values to deﬁne intervals on the real line. 1
Make a diagram
Solve
1
Sign of 2x

+
+
Sign of 1x 2x Sign of 1x
+
+


+

From the diagram we see that the solution set is 5x 0 0 x 16 30,1 2 . We include the endpoint 0 because the original inequality requires the quotient to be greater than or equal to 1. However, we do not include the other endpoint 1, since the quotient in the inequality is not deﬁned at 1. Always check the endpoints of solution intervals to determine whether they satisfy the original inequality. The solution set 30, 12 is illustrated in Figure 6. ■
Example 5
Figure 6
Solving an Inequality with Three Factors
Solve the inequality x
Terms to one side
Factor
2 . x1
Solution After moving all nonzero terms to one side of the inequality, we use a common denominator to combine the terms. 2 2 x 0 Subtract x1 x1 x1x 12 2 Common denominator x 1 0 x1 x1 x2 x 2 Combine fractions 0 x1 1x 12 1x 22 Factor numerator 0 x1
SECTION 1.7
Inequalities
81
The factors in this quotient change sign at 1, 1, and 2, so we must examine the intervals 1q, 12 , 11, 1 2 , 11, 2 2 , and 12, q 2 . Using test values, we get the following sign diagram.
Find the intervals
1
_1
Make a diagram
2
Sign of x+1

+
+
+
Sign of x2



+
Sign of x1 (x+1)(x2) Sign of x1


+
+

+

+
Since the quotient must be negative, the solution is _1
1
1q, 1 2 傼 11, 2 2
2
as illustrated in Figure 7.
Figure 7
■
Absolute Value Inequalities We use the following properties to solve inequalities that involve absolute value.
Properties of Absolute Value Inequalities These properties hold when x is replaced by any algebraic expression. (In the ﬁgures we assume that c 0.)
c _c
c c
x x
Inequality
Equivalent form
1. 앚x 앚 c
c x c
2. 앚x 앚 c
c x c
3. 앚x 앚 c
x c
or
4. 앚x 앚 c
x c
or
c x
_c
c
_c
c
_c
c
_c
c
These properties can be proved using the deﬁnition of absolute value. To prove Property 1, for example, note that the inequality 0 x 0 c says that the distance from x to 0 is less than c, and from Figure 8 you can see that this is true if and only if x is between c and c.
Example 6
Figure 8
cx
Graph
Solving an Absolute Value Inequality
Solve the inequality 0 x 5 0 2. Solution 1 The inequality 0 x 5 0 2 is equivalent to 2 x 5 2 2 0 Figure 9
3
3x7
2 5
The solution set is the open interval 13, 72 . 7
Property 1 Add 5
Solution 2 Geometrically, the solution set consists of all numbers x whose dis■ tance from 5 is less than 2. From Figure 9 we see that this is the interval 13,72 .
82
CHAPTER 1
Fundamentals
Example 7
Solving an Absolute Value Inequality
Solve the inequality 0 3x 2 0 4.
Solution By Property 4 the inequality 0 3x 2 0 4 is equivalent to 3x 2 4
3x 2 4
or
3x 2
3x 6
Subtract 2
x
x 2
Divide by 3
2 3
So the solution set is _2
5x 0 x 2
2 3
or
x 23 6 1q, 24 傼 3 23, q 2
The set is graphed in Figure 10.
■
Figure 10
Modeling with Inequalities Modeling reallife problems frequently leads to inequalities because we are often interested in determining when one quantity is more (or less) than another.
Example 8
Carnival Tickets
A carnival has two plans for tickets. Plan A:
$5 entrance fee and 25¢ each ride
Plan B:
$2 entrance fee and 50¢ each ride
How many rides would you have to take for plan A to be less expensive than plan B? Solution We are asked for the number of rides for which plan A is less expensive than plan B. So let x number of rides
Identify the variable
The information in the problem may be organized as follows.
Express all unknown quantities in terms of the variable
In Words
In Algebra
Number of rides Cost with plan A Cost with plan B
x 5 0.25x 2 0.50x
Now we set up the model. Set up the model
cost with cost with plan A plan B 5 0.25x 2 0.50x
Solve
3 0.25x 0.50x 3 0.25x 12 x
Subtract 2 Subtract 0.25x Divide by 0.25
So if you plan to take more than 12 rides, plan A is less expensive.
■
SECTION 1.7
Example 9
86
30
Inequalities
83
Fahrenheit and Celsius Scales
The instructions on a box of ﬁlm indicate that the box should be stored at a temperature between 5 °C and 30°C. What range of temperatures does this correspond to on the Fahrenheit scale? Solution The relationship between degrees Celsius (C ) and degrees Fahrenheit (F ) is given by the equation C 59 1F 322 . Expressing the statement on the box in terms of inequalities, we have
41
5 *C
*F
5 C 30 So the corresponding Fahrenheit temperatures satisfy the inequalities 5 59 1F 322 30
9 5
# 5 F 32 95 # 30
Multiply by 95
9 F 32 54
Simplify
9 32 F 54 32
Add 32
41 F 86
Simplify
The ﬁlm should be stored at a temperature between 41°F and 86°F.
Example 10
Concert Tickets
A group of students decide to attend a concert. The cost of chartering a bus to take them to the concert is $450, which is to be shared equally among the students. The concert promoters offer discounts to groups arriving by bus. Tickets normally cost $50 each but are reduced by 10¢ per ticket for each person in the group (up to the maximum capacity of the bus). How many students must be in the group for the total cost per student to be less than $54? Solution We are asked for the number of students in the group. So let Identify the variable
x number of students in the group The information in the problem may be organized as follows.
Express all unknown quantities in terms of the variable
In Words
In Algebra
Number of students in group
x 450 x 50 0.10x
Bus cost per student Ticket cost per student
Now we set up the model. Set up the model
bus cost per student
ticket cost per student
450 150 0.10x2 54 x
54
■
84
CHAPTER 1
Fundamentals
450 4 0.10x 0 x
Solve
Subtract 54
450 4x 0.10x 2 0 x
Common denominator
4500 40x x 2 0 x
Multiply by 10
190 x2 150 x2 0 x
Factor numerator 50
_90 Sign of 90+x

+
+
+
Sign of 50x
+
+
+

Sign of x (90+x)(50x) Sign of x


+
+
+

+

The sign diagram shows that the solution of the inequality is 190, 02 傼 150, q 2 . Because we cannot have a negative number of students, it follows that the group must have more than 50 students for the total cost per person to be less than $54. ■
1.7
Exercises
1–6 ■ Let S 52, 1, 0, 12, 1, 12, 2, 46 . Determine which elements of S satisfy the inequality.
27.
1 2x 13 2
6 12 3
28.
4 3x 1 1
2 5 4
1. 3 2x 12
2. 2x 1 x
3. 1 2x 4 7
4. 2 3 x 2
29–62 ■ Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set.
6. x 2 2 4
29. 1x 2 2 1x 32 0
30. 1x 52 1x 4 2 0
33. x 2 3x 18 0
34. x 2 5x 6 0
35. 2x 2 x 1
36. x 2 x 2
37. 3x 2 3x 2x 2 4
38. 5x 2 3x 3x 2 2
39. x 2 31x 62
40. x 2 2x 3
41. x 2 4
42. x 2 9
5.
1 1
x 2
7–28 ■ Solve the linear inequality. Express the solution using interval notation and graph the solution set. 7. 2x 5 3
8. 3x 11 5
9. 7 x 5
10. 5 3x 16
11. 2x 1 0
12. 0 5 2x
13. 3x 11 6x 8
14. 6 x 2x 9
15. 12 x 23 2
16. 25 x 1 15 2x
17. 13 x 2 16 x 1
18.
2 3
12 x 16 x
31. x12x 7 2 0
32. x12 3x 2 0
43. 2x 2 4
44. 1x 22 1x 1 2 1x 3 2 0
19. 4 3x 11 8x2
20. 217x 3 2 12x 16
45. x 3 4x 0
46. 16x x 3
21. 2 x 5 4
22. 5 3x 4 14
47.
x3 0 x1
48.
23. 1 2x 5 7
24. 1 3x 4 16
25. 2 8 2x 1
26. 3 3x 7 12
49.
4x 2 2x 3
50. 2
2x 6 0 x2 x1 x3
SECTION 1.7
Inequalities
85
51.
2x 1
3 x5
52.
3x 1 3x
87–90 ■ Determine the values of the variable for which the expression is deﬁned as a real number.
53.
4 x x
54.
x 3x x1
87. 216 9x 2
88. 23x 2 5x 2
89. a
90.
2 2
55. 1 x x1
3 4 1 56. x x1
6 6 1 57. x x1
x 5 4 58. 2 x1
x2 x1 59. x3 x2
1 1
0 60. x1 x2
61. x 4 x 2
62. x 5 x 2
63–76 ■ Solve the absolute value inequality. Express the answer using interval notation and graph the solution set. 63. 0 x 0 4
64. 0 3x 0 15
65. 0 2x 0 7
66.
67. 0 x 5 0 3
0x0 1
68. 0 x 1 0 1
69. 0 2x 3 0 0.4 71. `
1 2
70. 0 5x 2 0 6
x2 ` 2 3
75. 8 0 2x 1 0 6
76. 7 0 x 2 0 5 4
80. All real numbers x at most 4 units from 2
a c b d Show that
ac c a b bd d
Applications 93. Temperature Scales Use the relationship between C and F given in Example 9 to ﬁnd the interval on the Fahrenheit scale corresponding to the temperature range 20 C 30.
Plan A: Plan B:
$30 per day and 10¢ per mile $50 per day with free unlimited mileage
For what range of miles will plan B save you money? 96. LongDistance Cost A telephone company offers two longdistance plans. $25 per month and 5¢ per minute $5 per month and 12¢ per minute
For how many minutes of longdistance calls would plan B be ﬁnancially advantageous?
81–86 ■ A set of real numbers is graphed. Find an inequality involving an absolute value that describes the set. _5 _4 _3 _2 _1 0
1
2
3
4
5
_5 _4 _3 _2 _1 0
1
2
3
4
5
_5 _4 _3 _2 _1 0
1
2
3
4
5
_5 _4 _3 _2 _1 0
1
2
3
4
5
_5 _4 _3 _2 _1 0
1
2
3
4
5
_5 _4 _3 _2 _1 0
1
2
3
4
5
82.
86.
92. Suppose that a, b, c, and d are positive numbers such that
Plan A: Plan B:
79. All real numbers x at least 5 units from 7
85.
(b) a bx c 2a
95. Car Rental Cost A car rental company offers two plans for renting a car.
78. All real numbers x more than 2 units from 0
84.
(a) a1bx c 2 bc
74. 3 0 2x 4 0 1
77. All real numbers x less than 3 units from 0
83.
91. Solve the inequality for x, assuming that a, b, and c are positive constants.
94. Temperature Scales What interval on the Celsius scale corresponds to the temperature range 50 F 95?
77–80 ■ A phrase describing a set of real numbers is given. Express the phrase as an inequality involving an absolute value.
81.
4 1 x B2 x
x1 ` 4 2
72. `
73. 0 x 6 0 0.001
1/2 1 b x 5x 14 2
97. Driving Cost It is estimated that the annual cost of driving a certain new car is given by the formula C 0.35m 2200 where m represents the number of miles driven per year and C is the cost in dollars. Jane has purchased such a car, and decides to budget between $6400 and $7100 for next year’s driving costs. What is the corresponding range of miles that she can drive her new car? 98. Gas Mileage The gas mileage g (measured in mi/gal) for a particular vehicle, driven at √ mi/h, is given by the formula g 10 0.9√ 0.01√ 2, as long as √ is between 10 mi/h and 75 mi/h. For what range of speeds is the vehicle’s mileage 30 mi/gal or better?
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CHAPTER 1
Fundamentals
99. Gravity The gravitational force F exerted by the earth on an object having a mass of 100 kg is given by the equation F
4,000,000 d2
where d is the distance (in km) of the object from the center of the earth, and the force F is measured in newtons (N). For what distances will the gravitational force exerted by the earth on this object be between 0.0004 N and 0.01 N? 100. Bonﬁre Temperature In the vicinity of a bonﬁre, the temperature T in C at a distance of x meters from the center of the ﬁre was given by T
600,000 x 2 300
At what range of distances from the ﬁre’s center was the temperature less than 500C?
101. Stopping Distance For a certain model of car the distance d required to stop the vehicle if it is traveling at √ mi/h is given by the formula d√
√2 20
where d is measured in feet. Kerry wants her stopping distance not to exceed 240 ft. At what range of speeds can she travel?
103. Air Temperature As dry air moves upward, it expands and in so doing cools at a rate of about 1 °C for each 100meter rise, up to about 12 km. (a) If the ground temperature is 20°C, write a formula for the temperature at height h. (b) What range of temperatures can be expected if a plane takes off and reaches a maximum height of 5 km? 104. Airline Ticket Price A charter airline ﬁnds that on its Saturday ﬂights from Philadelphia to London, all 120 seats will be sold if the ticket price is $200. However, for each $3 increase in ticket price, the number of seats sold decreases by one. (a) Find a formula for the number of seats sold if the ticket price is P dollars. (b) Over a certain period, the number of seats sold for this ﬂight ranged between 90 and 115. What was the corresponding range of ticket prices? 105. Theater Tour Cost A riverboat theater offers bus tours to groups on the following basis. Hiring the bus costs the group $360, to be shared equally by the group members. Theater tickets, normally $30 each, are discounted by 25¢ times the number of people in the group. How many members must be in the group so that the cost of the theater tour (bus fare plus theater ticket) is less than $39 per person? 106. Fencing a Garden A determined gardener has 120 ft of deerresistant fence. She wants to enclose a rectangular vegetable garden in her backyard, and she wants the area enclosed to be at least 800 ft2. What range of values is possible for the length of her garden? 107. Thickness of a Laminate A company manufactures industrial laminates (thin nylonbased sheets) of thickness 0.020 in, with a tolerance of 0.003 in. (a) Find an inequality involving absolute values that describes the range of possible thickness for the laminate. (b) Solve the inequality you found in part (a).
240 ft 0.020 in. 102. Manufacturer’s Proﬁt If a manufacturer sells x units of a certain product, his revenue R and cost C (in dollars) are given by: R 20x C 2000 8x 0.0025x 2 Use the fact that profit revenue cost to determine how many units he should sell to enjoy a proﬁt of at least $2400.
108. Range of Height The average height of adult males is 68.2 in, and 95% of adult males have height h that satisﬁes the inequality `
h 68.2 ` 2 2.9
Solve the inequality to ﬁnd the range of heights.
SECTION 1.8
110. What’s Wrong Here? It is tempting to try to solve an inequality like an equation. For instance, we might try to solve 1 3/x by multiplying both sides by x, to get x 3, so the solution would be 1q, 3 2 . But that’s wrong; for
1.8
87
example, x 1 lies in this interval but does not satisfy the original inequality. Explain why this method doesn’t work (think about the sign of x). Then solve the inequality correctly.
Discovery • Discussion 109. Do Powers Preserve Order? If a b, is a 2 b 2? (Check both positive and negative values for a and b.) If a b, is a 3 b 3 ? Based on your observations, state a general rule about the relationship between a n and b n when a b and n is a positive integer.
Coordinate Geometry
111. Using Distances to Solve Absolute Value Inequalities Recall that 0 a b 0 is the distance between a and b on the number line. For any number x, what do 0 x 1 0 and 0 x 3 0 represent? Use this interpretation to solve the inequality 0 x 1 0 0 x 3 0 geometrically. In general, if a b, what is the solution of the inequality 0 x a 0 0 x b 0?
Coordinate Geometry The coordinate plane is the link between algebra and geometry. In the coordinate plane we can draw graphs of algebraic equations. The graphs, in turn, allow us to “see” the relationship between the variables in the equation. In this section we study the coordinate plane.
The Coordinate Plane The Cartesian plane is named in honor of the French mathematician René Descartes (1596–1650), although another Frenchman, Pierre Fermat (1601–1665), also invented the principles of coordinate geometry at the same time. (See their biographies on pages 112 and 652.)
Just as points on a line can be identiﬁed with real numbers to form the coordinate line, points in a plane can be identiﬁed with ordered pairs of numbers to form the coordinate plane or Cartesian plane. To do this, we draw two perpendicular real lines that intersect at 0 on each line. Usually one line is horizontal with positive direction to the right and is called the xaxis; the other line is vertical with positive direction upward and is called the yaxis. The point of intersection of the xaxis and the yaxis is the origin O, and the two axes divide the plane into four quadrants, labeled I, II, III, and IV in Figure 1. (The points on the coordinate axes are not assigned to any quadrant.) y
y P (a, b)
b
II
I
(1,3))
(_2,2) 1
O
III Figure 1 Although the notation for a point 1a, b 2 is the same as the notation for an open interval 1a, b2 , the context should make clear which meaning is intended.
a
IV
x
(5,0)) x
1
(_3,_2) (2,_4)
Figure 2
Any point P in the coordinate plane can be located by a unique ordered pair of numbers 1a, b2 , as shown in Figure 1. The ﬁrst number a is called the xcoordinate of P; the second number b is called the ycoordinate of P. We can think of the coordinates of P as its “address,” because they specify its location in the plane. Several points are labeled with their coordinates in Figure 2.
88
CHAPTER 1
Fundamentals
Example 1
Coordinates as Addresses The coordinates of a point in the xyplane uniquely determine its location. We can think of the coordinates as the “address” of the point. In Salt Lake City, Utah, the addresses of most buildings are in fact expressed as coordinates. The city is divided into quadrants with Main Street as the vertical (NorthSouth) axis and S. Temple Street as the horizontal (EastWest) axis. An address such as 1760 W
2100 S
indicates a location 17.6 blocks west of Main Street and 21 blocks south of S. Temple Street. (This is the address of the main post ofﬁce in Salt Lake City.) With this logical system it is possible for someone unfamiliar with the city to locate any address immediately, as easily as one locates a point in the coordinate plane.
Graphing Regions in the Coordinate Plane
Describe and sketch the regions given by each set. (a) 51x, y2 0 x 06 (b) 51x, y2 0 y 16
(c) 51x, y2 @ 0 y 0 16
Solution (a) The points whose xcoordinates are 0 or positive lie on the yaxis or to the right of it, as shown in Figure 3(a). (b) The set of all points with ycoordinate 1 is a horizontal line one unit above the xaxis, as in Figure 3(b). (c) Recall from Section 1.7 that 0y0 1
if and only if
1 y 1
So the given region consists of those points in the plane whose ycoordinates lie between 1 and 1. Thus, the region consists of all points that lie between (but not on) the horizontal lines y 1 and y 1. These lines are shown as broken lines in Figure 3(c) to indicate that the points on these lines do not lie in the set. y
y
y
y=1 x
x
x
y=_1 500 North St.
(a) x≥0
(c) y 