Ebook Algebra and Trigonometry (Eighth edition): Part 1

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Ebook Algebra and Trigonometry (Eighth edition): Part 1 presents the following content: Equations, Inequalities, and Mathematical Modeling; Functions and Their Graphs; Polynomial Functions; Rational Functions and Conics; Exponential and Logarithmic Functions.

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Algebra and Trigonometry Eighth Edition Ron Larson The Pennsylvania State University The Behrend College With the assistance of David C. Falvo The Pennsylvania State University The Behrend College Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
Algebra and Trigonometry, Eighth Edition © 2011, 2007 Brooks/Cole, Cengage Learning Ron Larson ALL RIGHTS RESERVED. No part of this work covered by the copyright Publisher: Charlie VanWagner herein may be reproduced, transmitted, stored, or used in any form or by Acquiring Sponsoring Editor: Gary Whalen any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, Development Editor: Stacy Green information networks, or information storage and retrieval systems, Assistant Editor: Cynthia Ashton except as permitted under Section 107 or 108 of the 1976 United States Editorial Assistant: Guanglei Zhang Copyright Act, without the prior written permission of the publisher. Associate Media Editor: Lynh Pham Marketing Manager: Myriah FitzGibbon For product information and technology assistance, contact us at Marketing Coordinator: Angela Kim Cengage Learning Customer & Sales Support, 1-800-354-9706 Marketing Communications Manager: Katy Malatesta For permission to use material from this text or product, submit all requests online at www.cengage.com/permissions. Content Project Manager: Susan Miscio Further permissions questions can be emailed to Senior Art Director: Jill Ort permissionrequest@cengage.com Senior Print Buyer: Diane Gibbons Production Editor: Carol Merrigan Library of Congress Control Number: 2009930253 Text Designer: Walter Kopek Student Edition: Rights Acquiring Account Manager, Photos: Don Schlotman ISBN-13: 978-1-4390-4847-4 Photo Researcher: Prepress PMG ISBN-10: 1-4390-4847-9 Cover Designer: Harold Burch Cover Image: Richard Edelman/Woodstock Graphics Studio Compositor: Larson Texts, Inc. Brooks/Cole 10 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at: international.cengage.com/region Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit www.cengage.com Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States of America 1 2 3 4 5 6 7 13 12 11 10 09
Contents A Word from the Author (Preface) vii chapter P Prerequisites 1 P.1 Review of Real Numbers and Their Properties 2 P.2 Exponents and Radicals 15 P.3 Polynomials and Special Products 28 P.4 Factoring Polynomials 37 P.5 Rational Expressions 45 P.6 The Rectangular Coordinate System and Graphs 55 Chapter Summary 66 Review Exercises 68 Chapter Test 71 Proofs in Mathematics 72 Problem Solving 73 chapter 1 Equations, Inequalities, and Mathematical Modeling 75 1.1 Graphs of Equations 76 1.2 Linear Equations in One Variable 87 1.3 Modeling with Linear Equations 96 1.4 Quadratic Equations and Applications 107 1.5 Complex Numbers 122 1.6 Other Types of Equations 129 1.7 Linear Inequalities in One Variable 140 1.8 Other Types of Inequalities 150 Chapter Summary 160 Review Exercises 162 Chapter Test 165 Proofs in Mathematics 166 Problem Solving 167 chapter 2 Functions and Their Graphs 169 2.1 Linear Equations in Two Variables 170 2.2 Functions 185 2.3 Analyzing Graphs of Functions 200 2.4 A Library of Parent Functions 212 2.5 Transformations of Functions 219 2.6 Combinations of Functions: Composite Functions 229 2.7 Inverse Functions 238 Chapter Summary 248 Review Exercises 250 Chapter Test 253 Cumulative Test for Chapters P–2 254 Proofs in Mathematics 256 Problem Solving 257 iii
iv Contents chapter 3 Polynomial Functions 259 3.1 Quadratic Functions and Models 260 3.2 Polynomial Functions of Higher Degree 270 3.3 Polynomial and Synthetic Division 284 3.4 Zeros of Polynomial Functions 293 3.5 Mathematical Modeling and Variation 308 Chapter Summary 320 Review Exercises 322 Chapter Test 326 Proofs in Mathematics 327 Problem Solving 329 chapter 4 Rational Functions and Conics 331 4.1 Rational Functions and Asymptotes 332 4.2 Graphs of Rational Functions 340 4.3 Conics 349 4.4 Translations of Conics 362 Chapter Summary 370 Review Exercises 372 Chapter Test 375 Proofs in Mathematics 376 Problem Solving 377 chapter 5 Exponential and Logarithmic Functions 379 5.1 Exponential Functions and Their Graphs 380 5.2 Logarithmic Functions and Their Graphs 391 5.3 Properties of Logarithms 401 5.4 Exponential and Logarithmic Equations 408 5.5 Exponential and Logarithmic Models 419 Chapter Summary 432 Review Exercises 434 Chapter Test 437 Cumulative Test for Chapters 3–5 438 Proofs in Mathematics 440 Problem Solving 441 chapter 6 Trigonometry 443 6.1 Angles and Their Measure 444 6.2 Right Triangle Trigonometry 456 6.3 Trigonometric Functions of Any Angle 467 6.4 Graphs of Sine and Cosine Functions 479 6.5 Graphs of Other Trigonometric Functions 490 6.6 Inverse Trigonometric Functions 501
Contents v 6.7 Applications and Models 511 Chapter Summary 522 Review Exercises 524 Chapter Test 527 Proofs in Mathematics 528 Problem Solving 529 chapter 7 Analytic Trigonometry 531 7.1 Using Fundamental Identities 532 7.2 Verifying Trigonometric Identities 540 7.3 Solving Trigonometric Equations 547 7.4 Sum and Difference Formulas 558 7.5 Multiple-Angle and Product-to-Sum Formulas 565 Chapter Summary 576 Review Exercises 578 Chapter Test 581 Proofs in Mathematics 582 Problem Solving 585 chapter 8 Additional Topics in Trigonometry 587 8.1 Law of Sines 588 8.2 Law of Cosines 597 8.3 Vectors in the Plane 605 8.4 Vectors and Dot Products 618 8.5 Trigonometric Form of a Complex Number 628 Chapter Summary 638 Review Exercises 640 Chapter Test 644 Cumulative Test for Chapters 6–8 645 Proofs in Mathematics 647 Problem Solving 651 chapter 9 Systems of Equations and Inequalities 653 9.1 Linear and Nonlinear Systems of Equations 654 9.2 Two-Variable Linear Systems 665 9.3 Multivariable Linear Systems 677 9.4 Partial Fractions 690 9.5 Systems of Inequalities 698 9.6 Linear Programming 709 Chapter Summary 718 Review Exercises 720 Chapter Test 725 Proofs in Mathematics 726 Problem Solving 727
vi Contents chapter 10 Matrices and Determinants 729 10.1 Matrices and Systems of Equations 730 10.2 Operations with Matrices 744 10.3 The Inverse of a Square Matrix 759 10.4 The Determinant of a Square Matrix 768 10.5 Applications of Matrices and Determinants 776 Chapter Summary 788 Review Exercises 790 Chapter Test 795 Proofs in Mathematics 796 Problem Solving 797 chapter 11 Sequences, Series, and Probability 799 11.1 Sequences and Series 800 11.2 Arithmetic Sequences and Partial Sums 811 11.3 Geometric Sequences and Series 821 11.4 Mathematical Induction 831 11.5 The Binomial Theorem 841 11.6 Counting Principles 849 11.7 Probability 859 Chapter Summary 872 Review Exercises 874 Chapter Test 877 Cumulative Test for Chapters 9–11 878 Proofs in Mathematics 880 Problem Solving 883 Appendix A Errors and the Algebra of Calculus A1 Answers to Odd-Numbered Exercises and Tests A9 Index A123 Index of Applications (web) Appendix B Concepts in Statistics (web) B.1 Representing Data B.2 Measures of Central Tendency and Dispersion B.3 Least Squares Regression
A Word from the Author Welcome to the Eighth Edition of Algebra and Trigonometry! We are proud to offer you a new and revised version of our textbook. With this edition, we have listened to you, our users, and have incorporated many of your suggestions for improvement. 8th 7th 6th 5th 4th 3rd 2nd 1st In the Eighth Edition, we continue to offer instructors and students a text that is pedagogically sound, mathematically precise, and still comprehensible. There are many changes in the mathematics, art, and design; the more significant changes are noted here. • New Chapter Openers Each Chapter Opener has three parts, In Mathematics, In Real Life, and In Careers. In Mathematics describes an important mathematical topic taught in the chapter. In Real Life tells students where they will encounter this topic in real-life situations. In Careers relates application exercises to a variety of careers. • New Study Tips and Warning/Cautions Insightful information is given to students in two new features. The Study Tip provides students with useful information or suggestions for learning the topic. The Warning/Caution points out common mathematical errors made by students. • New Algebra Helps Algebra Help directs students to sections of the textbook where they can review algebra skills needed to master the current topic. • New Side-by-Side Examples Throughout the text, we present solutions to many examples from multiple perspectives—algebraically, graphically, and numerically. The side-by-side format of this pedagogical feature helps students to see that a problem can be solved in more than one way and to see that different methods yield the same result. The side-by-side format also addresses many different learning styles. vii
viii A Word from the Author • New Capstone Exercises Capstones are conceptual problems that synthesize key topics and provide students with a better understanding of each section’s concepts. Capstone exercises are excellent for classroom discussion or test prep, and teachers may find value in integrating these problems into their reviews of the section. • New Chapter Summaries The Chapter Summary now includes an explanation and/or example of each objective taught in the chapter. • Revised Exercise Sets The exercise sets have been carefully and extensively examined to ensure they are rigorous and cover all topics suggested by our users. Many new skill-building and challenging exercises have been added. For the past several years, we’ve maintained an independent website— CalcChat.com—that provides free solutions to all odd-numbered exercises in the text. Thousands of students using our textbooks have visited the site for practice and help with their homework. For the Eighth Edition, we were able to use information from CalcChat.com, including which solutions students accessed most often, to help guide the revision of the exercises. I hope you enjoy the Eighth Edition of Algebra and Trigonometry. As always, I welcome comments and suggestions for continued improvements.
Acknowledgments I would like to thank the many people who have helped me prepare the text and the supplements package. Their encouragement, criticisms, and suggestions have been invaluable. Thank you to all of the instructors who took the time to review the changes in this edition and to provide suggestions for improving it. Without your help, this book would not be possible. Reviewers Chad Pierson, University of Minnesota-Duluth; Sally Shao, Cleveland State University; Ed Stumpf, Central Carolina Community College; Fuzhen Zhang, Nova Southeastern University; Dennis Shepherd, University of Colorado, Denver; Rhonda Kilgo, Jacksonville State University; C. Altay Özgener, Manatee Community College Bradenton; William Forrest, Baton Rouge Community College; Tracy Cook, University of Tennessee Knoxville; Charles Hale, California State Poly University Pomona; Samuel Evers, University of Alabama; Seongchun Kwon, University of Toledo; Dr. Arun K. Agarwal, Grambling State University; Hyounkyun Oh, Savannah State University; Michael J. McConnell, Clarion University; Martha Chalhoub, Collin County Community College; Angela Lee Everett, Chattanooga State Tech Community College; Heather Van Dyke, Walla Walla Community College; Gregory Buthusiem, Burlington County Community College; Ward Shaffer, College of Coastal Georgia; Carmen Thomas, Chatham University; Emily J. Keaton My thanks to David Falvo, The Behrend College, The Pennsylvania State University, for his contributions to this project. My thanks also to Robert Hostetler, The Behrend College, The Pennsylvania State University, and Bruce Edwards, University of Florida, for their significant contributions to previous editions of this text. I would also like to thank the staff at Larson Texts, Inc. who assisted with proof- reading the manuscript, preparing and proofreading the art package, and checking and typesetting the supplements. On a personal level, I am grateful to my spouse, Deanna Gilbert Larson, for her love, patience, and support. Also, a special thanks goes to R. Scott O’Neil. If you have suggestions for improving this text, please feel free to write to me. Over the past two decades I have received many useful comments from both instructors and students, and I value these comments very highly. Ron Larson ix
Supplements Supplements for the Instructor Annotated Instructor’s Edition This AIE is the complete student text plus point-of- use annotations for the instructor, including extra projects, classroom activities, teaching strategies, and additional examples. Answers to even-numbered text exercises, Vocabulary Checks, and Explorations are also provided. Complete Solutions Manual This manual contains solutions to all exercises from the text, including Chapter Review Exercises and Chapter Tests. Instructor’s Companion Website This free companion website contains an abundance of instructor resources. PowerLecture™ with ExamView® The CD-ROM provides the instructor with dynamic media tools for teaching college algebra. PowerPoint® lecture slides and art slides of the figures from the text, together with electronic files for the test bank and a link to the Solution Builder, are available. The algorithmic ExamView allows you to create, deliver, and customize tests (both print and online) in minutes with this easy-to-use assessment system. Enhance how your students interact with you, your lecture, and each other. Solutions Builder This is an electronic version of the complete solutions manual available via the PowerLecture and Instructor’s Companion Website. It provides instructors with an efficient method for creating solution sets to homework or exams that can then be printed or posted. x
Supplements xi Supplements for the Student Student Companion Website This free companion website contains an abundance of student resources. Instructional DVDs Keyed to the text by section, these DVDs provide comprehensive coverage of the course—along with additional explanations of concepts, sample problems, and applications—to help students review essential topics. Student Study and Solutions Manual This guide offers step-by-step solutions for all odd-numbered text exercises, Chapter and Cumulative Tests, and Practice Tests with solutions. Premium eBook The Premium eBook offers an interactive version of the textbook with search features, highlighting and note-making tools, and direct links to videos or tutorials that elaborate on the text discussions. Enhanced WebAssign Enhanced WebAssign is designed for you to do your homework online. This proven and reliable system uses pedagogy and content found in Larson’s text, and then enhances it to help you learn Algebra and Trigonometry more effectively. Automatically graded homework allows you to focus on your learning and get interactive study assistance outside of class.
Prerequisites P.1 Review of Real Numbers and Their Properties P P.2 Exponents and Radicals P.3 Polynomials and Special Products P.4 Factoring Polynomials P.5 Rational Expressions P.6 The Rectangular Coordinate System and Graphs In Mathematics Real numbers, exponents, radicals, and polynomials are used in many different branches of mathematics. In Real Life The concepts in this chapter are used to model compound interest, volumes, rates of change, and other real-life applications. Darren McCollester/ Getty Images News /Getty Images For instance, polynomials can be used to model the stopping distance of an automobile. (See Exercise 116, page 36.) IN CAREERS There are many careers that use prealgebra concepts. Several are listed below. • Engineer • Financial Analyst Exercise 115, page 35 Exercises 99 and 100, page 54 • Chemist • Meteorologist Exercise 148, page 44 Exercise 114, page 70 1
2 Chapter P Prerequisites P.1 REVIEW OF REAL NUMBERS AND THEIR PROPERTIES What you should learn Real Numbers • Represent and classify real numbers. • Order real numbers and use Real numbers are used in everyday life to describe quantities such as age, miles per inequalities. gallon, and population. Real numbers are represented by symbols such as • Find the absolute values of real 4 numbers and find the distance Ϫ5, 9, 0, , 0.666 . . . , 28.21, Ί2, ␲, and ΊϪ32. 3 3 between two real numbers. • Evaluate algebraic expressions. Here are some important subsets (each member of subset B is also a member of set A) • Use the basic rules and of the real numbers. The three dots, called ellipsis points, indicate that the pattern properties of algebra. continues indefinitely. Why you should learn it ͭ1, 2, 3, 4, . . .ͮ Set of natural numbers Real numbers are used to represent ͭ0, 1, 2, 3, 4, . . .ͮ Set of whole numbers many real-life quantities. For example, ͭ. . . , Ϫ3, Ϫ2, Ϫ1, 0, 1, 2, 3, . . .ͮ Set of integers in Exercises 83–88 on page 13, you will use real numbers to represent the A real number is rational if it can be written as the ratio p͞q of two integers, where federal deficit. q 0. For instance, the numbers 1 1 125 ϭ 0.3333 . . . ϭ 0.3, ϭ 0.125, and ϭ 1.126126 . . . ϭ 1.126 3 8 111 are rational. The decimal representation of a rational number either repeats ͑as in 55 ϭ 3.145 ͒ or terminates ͑as in 2 ϭ 0.5͒. A real number that cannot be written as the 173 1 ratio of two integers is called irrational. Irrational numbers have infinite nonrepeating decimal representations. For instance, the numbers Ί2 ϭ 1.4142135 . . . Ϸ 1.41 and ␲ ϭ 3.1415926 . . . Ϸ 3.14 are irrational. (The symbol Ϸ means “is approximately equal to.”) Figure P.1 shows subsets of real numbers and their relationships to each other. Real numbers Example 1 Classifying Real Numbers Determine which numbers in the set Irrational Rational numbers numbers ΆϪ13, Ϫ 1 5 Ί5, Ϫ1, Ϫ , 0, , Ί2, ␲, 7 3 8 · are (a) natural numbers, (b) whole numbers, (c) integers, (d) rational numbers, and Integers Noninteger fractions (e) irrational numbers. (positive and negative) Solution a. Natural numbers: ͭ7ͮ Negative Whole b. Whole numbers: ͭ0, 7ͮ integers numbers c. Integers: ͭϪ13, Ϫ1, 0, 7ͮ Natural Zero Ά 1 5 d. Rational numbers: Ϫ13, Ϫ1, Ϫ , 0, , 7 3 8 · numbers e. Irrational numbers: ͭ Ϫ Ί5, Ί2, ␲ͮ FIGURE P.1 Subsets of real numbers Now try Exercise 11.
Section P.1 Review of Real Numbers and Their Properties 3 Real numbers are represented graphically on the real number line. When you draw a point on the real number line that corresponds to a real number, you are plotting the real number. The point 0 on the real number line is the origin. Numbers to the right of 0 are positive, and numbers to the left of 0 are negative, as shown in Figure P.2. The term nonnegative describes a number that is either positive or zero. Origin Negative Positive direction −4 −3 −2 −1 0 1 2 3 4 direction FIGURE P.2 The real number line As illustrated in Figure P.3, there is a one-to-one correspondence between real numbers and points on the real number line. −5 −2.4 2 3 0.75 π −3 −2 −1 0 1 2 3 −3 −2 −1 0 1 2 3 Every real number corresponds to exactly Every point on the real number line one point on the real number line. corresponds to exactly one real number. FIGURE P.3 One-to-one correspondence Example 2 Plotting Points on the Real Number Line Plot the real numbers on the real number line. 7 a. Ϫ 4 b. 2.3 2 c. 3 d. Ϫ1.8 Solution All four points are shown in Figure P.4. − 1.8 − 7 4 2 3 2.3 −2 −1 0 1 2 3 FIGURE P.4 a. The point representing the real number Ϫ 7 ϭ Ϫ1.75 lies between Ϫ2 and Ϫ1, but 4 closer to Ϫ2, on the real number line. b. The point representing the real number 2.3 lies between 2 and 3, but closer to 2, on the real number line. c. The point representing the real number 2 ϭ 0.666 . . . lies between 0 and 1, but 3 closer to 1, on the real number line. d. The point representing the real number Ϫ1.8 lies between Ϫ2 and Ϫ1, but closer to Ϫ2, on the real number line. Note that the point representing Ϫ1.8 lies slightly to the left of the point representing Ϫ 7. 4 Now try Exercise 17.
4 Chapter P Prerequisites Ordering Real Numbers One important property of real numbers is that they are ordered. Definition of Order on the Real Number Line If a and b are real numbers, a is less than b if b Ϫ a is positive. The order of a and b is denoted by the inequality a < b. This relationship can also be described by saying that b is greater than a and writing b > a. The inequality a ≤ b means that a is less than or equal to b, and the inequality b ≥ a means that b is greater than or equal to a. The symbols , Յ, and Ն are inequality symbols. a b −1 0 1 2 FIGURE P.5 a < b if and only if a lies to Geometrically, this definition implies that a < b if and only if a lies to the left of the left of b. b on the real number line, as shown in Figure P.5. Example 3 Ordering Real Numbers −4 −3 −2 −1 0 Place the appropriate inequality symbol ͑< or >͒ between the pair of real numbers. FIGURE P.6 1 1 1 1 a. Ϫ3, 0 b. Ϫ2, Ϫ4 c. , d. Ϫ , Ϫ 4 3 5 2 −4 −3 −2 −1 0 Solution FIGURE P.7 a. Because Ϫ3 lies to the left of 0 on the real number line, as shown in Figure P.6, you 1 1 can say that Ϫ3 is less than 0, and write Ϫ3 < 0. 4 3 b. Because Ϫ2 lies to the right of Ϫ4 on the real number line, as shown in Figure P.7, 0 1 you can say that Ϫ2 is greater than Ϫ4, and write Ϫ2 > Ϫ4. c. Because 1 lies to the left of 3 on the real number line, as shown in Figure P.8, you 1 FIGURE P.8 4 can say that 1 is less than 1, and write 1 < 1. 4 3 4 3 −1 −1 2 5 d. Because Ϫ 1 lies to the right of Ϫ 1 on the real number line, as shown in Figure P.9, 5 2 −1 0 you can say that Ϫ 1 is greater than Ϫ 1, and write Ϫ 1 > Ϫ 1. 5 2 5 2 FIGURE P.9 Now try Exercise 25. Example 4 Interpreting Inequalities Describe the subset of real numbers represented by each inequality. a. x Յ 2 b. Ϫ2 Յ x < 3 x≤2 x Solution 0 1 2 3 4 a. The inequality x ≤ 2 denotes all real numbers less than or equal to 2, as shown in FIGURE P.10 Figure P.10. −2 ≤ x < 3 b. The inequality Ϫ2 ≤ x < 3 means that x ≥ Ϫ2 and x < 3. This “double inequality” denotes all real numbers between Ϫ2 and 3, including Ϫ2 but not including 3, as x −2 −1 0 1 2 3 shown in Figure P.11. FIGURE P.11 Now try Exercise 31.
Section P.1 Review of Real Numbers and Their Properties 5 Inequalities can be used to describe subsets of real numbers called intervals. In the bounded intervals below, the real numbers a and b are the endpoints of each interval. The endpoints of a closed interval are included in the interval, whereas the endpoints of an open interval are not included in the interval. Bounded Intervals on the Real Number Line Notation Interval Type Inequality Graph ͓a, b͔ Closed a Յ x Յ b x a b ͑a, b͒ Open a < x < b x The reason that the four types a b of intervals at the right are called ͓a, b͒ a Յ x < b x bounded is that each has a finite a b length. An interval that does not have a finite length is unbounded ͑a, b͔ a < x Յ b x (see below). a b The symbols ϱ, positive infinity, and Ϫ ϱ, negative infinity, do not represent WARNING / CAUTION real numbers. They are simply convenient symbols used to describe the unboundedness Whenever you write an interval of an interval such as ͑1, ϱ͒ or ͑Ϫ ϱ, 3͔. containing ϱ or Ϫ ϱ, always use a parenthesis and never a Unbounded Intervals on the Real Number Line bracket. This is because ϱ and Ϫ ϱ are never an endpoint of an Notation Interval Type Inequality Graph interval and therefore are not ͓a, ϱ͒ x Ն a x included in the interval. a ͑a, ϱ͒ Open x > a x a ͑Ϫ ϱ, b͔ x Յ b x b ͑Ϫ ϱ, b͒ Open x < b x b ͑Ϫ ϱ, ϱ͒ Entire real line Ϫϱ < x < ϱ x Example 5 Using Inequalities to Represent Intervals Use inequality notation to describe each of the following. a. c is at most 2. b. m is at least Ϫ3. c. All x in the interval ͑Ϫ3, 5͔ Solution a. The statement “c is at most 2” can be represented by c ≤ 2. b. The statement “m is at least Ϫ3” can be represented by m ≥ Ϫ3. c. “All x in the interval ͑Ϫ3, 5͔” can be represented by Ϫ3 < x ≤ 5. Now try Exercise 45.
6 Chapter P Prerequisites Example 6 Interpreting Intervals Give a verbal description of each interval. a. ͑Ϫ1, 0͒ b. ͓ 2, ϱ͒ c. ͑Ϫ ϱ, 0͒ Solution a. This interval consists of all real numbers that are greater than Ϫ1 and less than 0. b. This interval consists of all real numbers that are greater than or equal to 2. c. This interval consists of all negative real numbers. Now try Exercise 41. Absolute Value and Distance The absolute value of a real number is its magnitude, or the distance between the origin and the point representing the real number on the real number line. Definition of Absolute Value If a is a real number, then the absolute value of a is ԽaԽ ϭ ΆϪa, a, if a ≥ 0 . if a < 0 Notice in this definition that the absolute value of a real number is never negative. Խ Խ For instance, if a ϭ Ϫ5, then Ϫ5 ϭ Ϫ ͑Ϫ5͒ ϭ 5. The absolute value of a real number is either positive or zero. Moreover, 0 is the only real number whose absolute ԽԽ value is 0. So, 0 ϭ 0. Example 7 Finding Absolute Values Խ Խ a. Ϫ15 ϭ 15 b. ԽԽ 2 3 ϭ 2 3 c. ԽϪ4.3Խ ϭ 4.3 Խ Խ d. Ϫ Ϫ6 ϭ Ϫ ͑6͒ ϭ Ϫ6 Now try Exercise 51. Example 8 Evaluating the Absolute Value of a Number Evaluate ԽxԽ for (a) x > 0 and (b) x < 0. x Solution ԽxԽ ϭ x ϭ 1. ԽԽ a. If x > 0, then x ϭ x and x x ԽxԽ ϭ Ϫx ϭ Ϫ1. ԽԽ b. If x < 0, then x ϭ Ϫx and x x Now try Exercise 59.
Section P.1 Review of Real Numbers and Their Properties 7 The Law of Trichotomy states that for any two real numbers a and b, precisely one of three relationships is possible: a ϭ b, a < b, or a > b. Law of Trichotomy Example 9 Comparing Real Numbers Place the appropriate symbol (, or =) between the pair of real numbers. Խ Խ᭿Խ3Խ a. Ϫ4 Խ b. Ϫ10 Խ᭿Խ10Խ c. Ϫ Ϫ7 Խ Խ᭿ԽϪ7Խ Solution Խ Խ ԽԽ Խ Խ ԽԽ a. Ϫ4 > 3 because Ϫ4 ϭ 4 and 3 ϭ 3, and 4 is greater than 3. Խ Խ Խ Խ Խ Խ Խ Խ b. Ϫ10 ϭ 10 because Ϫ10 ϭ 10 and 10 ϭ 10. Խ Խ Խ Խ Խ Խ Խ Խ c. Ϫ Ϫ7 < Ϫ7 because Ϫ Ϫ7 ϭ Ϫ7 and Ϫ7 ϭ 7, and Ϫ7 is less than 7. Now try Exercise 61. Properties of Absolute Values ԽԽ 1. a Ն 0 Խ Խ ԽԽ 2. Ϫa ϭ a Խ Խ Խ ԽԽ Խ 3. ab ϭ a b 4. ԽԽ a b ϭ ԽaԽ, b ԽbԽ 0 Absolute value can be used to define the distance between two points on the real 7 number line. For instance, the distance between Ϫ3 and 4 is −3 −2 −1 0 1 2 3 4 ԽϪ3 Ϫ 4Խ ϭ ԽϪ7Խ ϭ7 FIGURE P.12 The distance between Ϫ3 and 4 is 7. as shown in Figure P.12. Distance Between Two Points on the Real Number Line Let a and b be real numbers. The distance between a and b is Խ Խ Խ d͑a, b͒ ϭ b Ϫ a ϭ a Ϫ b . Խ Example 10 Finding a Distance Find the distance between Ϫ25 and 13. Solution The distance between Ϫ25 and 13 is given by ԽϪ25 Ϫ 13Խ ϭ ԽϪ38Խ ϭ 38. Distance between Ϫ25 and 13 The distance can also be found as follows. Խ13 Ϫ ͑Ϫ25͒Խ ϭ Խ38Խ ϭ 38 Distance between Ϫ25 and 13 Now try Exercise 67.