College Algebra and Trigonometry Seventh Edition PDF

Motivating Features to Help You Succeed! Your success in college algebra and trigonometry is important to us. To guide you to that success, we have created a textbook with features that promote learning and support various learning styles. These features are highlighted below. We encourage you to examine these features and use them to successfully complete this course. Prepare for This Section These exercises test your understanding of prerequisite skills and concepts that were covered earlier in the text. Mastery of these concepts is required for success in the following section. Motivating Applications Large selections of contemporary applications from many different disciplines demonstrate the utility of mathematics. Engaging Examples Examples are designed to capture your attention and help you master important concepts. Annotated Examples Step-by-step solutions are provided for each example. Try Exercises A reference to an exercise follows each worked example. This exercise provides you the opportunity to test your understanding by working an exercise similar to the worked example. Solutions to Try Exercises The complete solutions to the Try Exercises can be found in the Solutions to the Try Exercises appendix, starting page S1. Visualize the Solution When appropriate, both algebraic and graphical solutions are provided to help visualize the mathematics of the example and to create a link between the two. Mid-Chapter Quizzes These quizzes will help you assess your understanding of the concepts studied earlier in the chapter. They provide a mini-review of the chapter material. Chapter Test Prep This is a summary of the major concepts discussed in the chapter and will help you prepare for the chapter test. For each concept, there is a reference to a worked example illustrating how the concept is used and at least one exercise in the chapter review relating to that concept. This page intentionally left blank COLLEGE ALGEBRA AND TRIGONOMETRY This page intentionally left blank SEVENTH EDITION COLLEGE ALGEBRA AND TRIGONOMETRY Chad Ehlers/Getty Images Richard N. Aufmann Vernon C. Barker Richard D. Nation Australia Brazil Japan Korea Mexico Singapore Spain United Kingdom United States College Algebra and Trigonometry, © 2011, 2008 Brooks/Cole, Cengage Learning Seventh Edition ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may Richard N. Aufmann, Vernon C. Barker, be reproduced, transmitted, stored, or used in any form or by any means, graphic, Richard D. Nation electronic, or mechanical, including but not limited to photocopying, recording, Acquisitions Editor: Gary Whalen scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the Senior Developmental Editor: Carolyn Crockett 1976 United States Copyright Act, without the prior written permission of the Assistant Editor: Stefanie Beeck publisher. Editorial Assistant: Guanglei Zhang Associate Media Editor: Lynh Pham For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706. Marketing Manager: Myriah Fitzgibbon For permission to use material from this text or product, submit all Marketing Assistant: Angela Kim requests online at www.cengage.com/permissions. Marketing Communications Further permissions questions can be e-mailed to Manager: Katy Malatesta [email protected]. Content Project Manager: Jennifer Risden Creative Director: Rob Hugel Library of Congress Control Number: 2009938510 Art Director: Vernon Boes ISBN-13: 978-1-4390-4860-3 Print Buyer: Karen Hunt ISBN-10: 1-4390-4860-6 Rights Acquisitions Account Manager, Text: Roberta Broyer Brooks/Cole Rights Acquisitions Account Manager, 20 Davis Drive Image: Don Schlotman Belmont, CA 94002-3098 Production Service: Graphic World Inc. USA Text Designer: Diane Beasley Photo Researcher: PrepressPMG Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Copy Editor: Graphic World Inc. Mexico, Brazil, and Japan. Locate your local office at www.cengage.com/global. Illustrators: Network Graphics; Macmillan Publishing Solutions Cengage Learning products are represented in Canada by Nelson Cover Designer: Lisa Henry Education, Ltd. Cover Image: Chad Ehlers/Getty Images Compositor: MPS Limited, A Macmillan To learn more about Brooks/Cole, visit www.cengage.com/brookscole Company 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 CHAPTER P Preliminary Concepts 1 P.1 The Real Number System 2 P.2 Integer and Rational Number Exponents 17 P.3 Polynomials 32 Mid-Chapter P Quiz 39 P.4 Factoring 40 P.5 Rational Expressions 49 P.6 Complex Numbers 59 Exploring Concepts with Technology 66 Chapter P Test Prep 67 Chapter P Review Exercises 70 Chapter P Test 73 CHAPTER 1 Equations and Inequalities 75 1.1 Linear and Absolute Value Equations 76 1.2 Formulas and Applications 83 1.3 Quadratic Equations 96 Mid-Chapter 1 Quiz 109 1.4 Other Types of Equations 110 1.5 Inequalities 123 1.6 Variation and Applications 136 Exploring Concepts with Technology 144 Chapter 1 Test Prep 145 Chapter 1 Review Exercises 148 Chapter 1 Test 151 Cumulative Review Exercises 152 CHAPTER 2 Functions and Graphs 153 2.1 Two-Dimensional Coordinate System and Graphs 154 2.2 Introduction to Functions 166 2.3 Linear Functions 186 Mid-Chapter 2 Quiz 200 2.4 Quadratic Functions 200 2.5 Properties of Graphs 213 2.6 Algebra of Functions 227 2.7 Modeling Data Using Regression 237 Exploring Concepts with Technology 248 Chapter 2 Test Prep 249 Chapter 2 Review Exercises 253 Chapter 2 Test 257 Cumulative Review Exercises 258 v vi CONTENTS CHAPTER 3 Polynomial and Rational Functions 259 3.1 Remainder Theorem and Factor Theorem 260 3.2 Polynomial Functions of Higher Degree 271 3.3 Zeros of Polynomial Functions 287 Mid-Chapter 3 Quiz 299 3.4 Fundamental Theorem of Algebra 299 3.5 Graphs of Rational Functions and Their Applications 307 Exploring Concepts with Technology 323 Chapter 3 Test Prep 324 Chapter 3 Review Exercises 328 Chapter 3 Test 331 Cumulative Review Exercises 332 CHAPTER 4 Exponential and Logarithmic Functions 333 4.1 Inverse Functions 334 4.2 Exponential Functions and Their Applications 346 4.3 Logarithmic Functions and Their Applications 358 4.4 Properties of Logarithms and Logarithmic Scales 369 Mid-Chapter 4 Quiz 380 4.5 Exponential and Logarithmic Equations 380 4.6 Exponential Growth and Decay 390 4.7 Modeling Data with Exponential and Logarithmic Functions 404 Exploring Concepts with Technology 416 Chapter 4 Test Prep 418 Chapter 4 Review Exercises 421 Chapter 4 Test 424 Cumulative Review Exercises 425 CHAPTER 5 Trigonometric Functions 427 5.1 Angles and Arcs 428 5.2 Right Triangle Trigonometry 442 5.3 Trigonometric Functions of Any Angle 454 5.4 Trigonometric Functions of Real Numbers 461 Mid-Chapter 5 Quiz 472 5.5 Graphs of the Sine and Cosine Functions 473 5.6 Graphs of the Other Trigonometric Functions 481 5.7 Graphing Techniques 491 5.8 Harmonic Motion—An Application of the Sine and Cosine Functions 500 Exploring Concepts with Technology 505 Chapter 5 Test Prep 506 Chapter 5 Review Exercises 510 Chapter 5 Test 512 Cumulative Review Exercises 513 CONTENTS vii CHAPTER 6 Trigonometric Identities and Equations 515 6.1 Verification of Trigonometric Identities 516 6.2 Sum, Difference, and Cofunction Identities 522 6.3 Double- and Half-Angle Identities 532 Mid-Chapter 6 Quiz 540 6.4 Identities Involving the Sum of Trigonometric Functions 541 6.5 Inverse Trigonometric Functions 548 6.6 Trigonometric Equations 560 Exploring Concepts with Technology 572 Chapter 6 Test Prep 573 Chapter 6 Review Exercises 576 Chapter 6 Test 578 Cumulative Review Exercises 579 CHAPTER 7 Applications of Trigonometry 581 7.1 Law of Sines 582 7.2 Law of Cosines and Area 592 7.3 Vectors 601 Mid-Chapter 7 Quiz 615 7.4 Trigonometric Form of Complex Numbers 616 7.5 De Moivre’s Theorem 622 Exploring Concepts with Technology 626 Chapter 7 Test Prep 627 Chapter 7 Review Exercises 630 Chapter 7 Test 631 Cumulative Review Exercises 632 CHAPTER 8 Topics in Analytic Geometry 633 8.1 Parabolas 634 8.2 Ellipses 645 8.3 Hyperbolas 658 8.4 Rotation of Axes 670 Mid-Chapter 8 Quiz 678 8.5 Introduction to Polar Coordinates 678 8.6 Polar Equations of the Conics 691 8.7 Parametric Equations 696 Exploring Concepts with Technology 705 Chapter 8 Test Prep 706 Chapter 8 Review Exercises 711 Chapter 8 Test 713 Cumulative Review Exercises 714 viii CONTENTS CHAPTER 9 Systems of Equations and Inequalities 717 9.1 Systems of Linear Equations in Two Variables 718 9.2 Systems of Linear Equations in Three Variables 728 9.3 Nonlinear Systems of Equations 740 Mid-Chapter 9 Quiz 747 9.4 Partial Fractions 748 9.5 Inequalities in Two Variables and Systems of Inequalities 755 9.6 Linear Programming 762 Exploring Concepts with Technology 772 Chapter 9 Test Prep 773 Chapter 9 Review Exercises 775 Chapter 9 Test 777 Cumulative Review Exercises 777 CHAPTER 10 Matrices 779 10.1 Gaussian Elimination Method 780 10.2 Algebra of Matrices 791 10.3 Inverse of a Matrix 813 Mid-Chapter 10 Quiz 823 10.4 Determinants 824 10.5 Cramer’s Rule 833 Exploring Concepts with Technology 837 Chapter 10 Test Prep 839 Chapter 10 Review Exercises 840 Chapter 10 Test 844 Cumulative Review Exercises 845 CHAPTER 11 Sequences, Series, and Probability 847 11.1 Infinite Sequences and Summation Notation 848 11.2 Arithmetic Sequences and Series 854 11.3 Geometric Sequences and Series 860 Mid-Chapter 11 Quiz 871 11.4 Mathematical Induction 871 11.5 Binomial Theorem 878 11.6 Permutations and Combinations 883 11.7 Introduction to Probability 890 Exploring Concepts with Technology 899 Chapter 11 Test Prep 900 Chapter 11 Review Exercises 902 Chapter 11 Test 905 Cumulative Review Exercises 906 Solutions to the Try Exercises S1 Answers to Selected Exercises A1 Index I1 PREFACE We are proud to offer the seventh edition of College Algebra and Trigonometry. Your success in college algebra and trigonometry is important to us. To guide you to that success, we have created a textbook with features that promote learning and support various learning styles. These features are highlighted below. We encourage you to examine the features and use them to help you successfully complete this course. Features Chapter Openers Each Chapter Opener demonstrates a contemporary application of a mathematical concept developed in that chapter. Related Exercise References Each Chapter Opener ends with a reference to a particular exercise within the chapter that requires you to solve a problem related to that chapter opener topic. Listing of Major Concepts A list of major concepts in each section is provided in the margin of the first page of each section. Prepare for This Section Each section (after the first section) of a chapter opens with review exercises titled Prepare for This Section. These exercises give you a chance to test your understanding of prerequisite skills and concepts before proceeding to the new topics presented in the section. ix x PREFACE Thoughtfully Designed Exercise Sets We have thoroughly reviewed each exercise set to ensure a smooth progression from routine exercises to exercises that are more challenging. The exercises illustrate the many facets of topics discussed in the text. The exercise sets emphasize skill building, skill maintenance, conceptual understanding, and, as appropriate, applications. Each chapter includes a Chapter Review Exercise set and each chapter, except Chapter P, includes a Cumulative Review Exercise set. Contemporary Applications Carefully developed mathematics is complemented by abundant, relevant, and contemporary applications, many of which feature real data, tables, graphs, and charts. Applications demonstrate the value of algebra and cover topics from a wide variety of disciplines. Besides providing motivation to study mathematics, the applications will help you develop good problem-solving skills. PREFACE xi By incorporating many interactive learning techniques, including the key features outlined below, College Algebra and Trigonometry uses the proven Aufmann Interactive Method (AIM) to help you understand concepts and obtain greater mathematical fluency. The AIM consists of Annotated Examples followed by Try Exercises (and solutions) and a conceptual Question/Answer follow-up. See the samples below: Engaging Examples Examples are designed to capture your attention and help you master important concepts. Annotated Examples Step-by-step solutions are provided for most numbered examples. Try Exercises A reference to an exercise follows all worked examples. This exercise provides you with an opportunity to test your understanding of concepts by working an exercise related to the worked example. Solutions to Try Exercises Complete solutions to the Try Exercises can be found in the Solutions to the Try Exercises appendix on page S1. Question/Answer In each section, we have posed at least one question that encourages you to pause and think about the concepts presented in the current discussion. To ensure that you do not miss this important information, the answer is provided as a footnote on the same page. xii PREFACE Immediate Examples of Definitions and Concepts Immediate examples of many definitions and concepts are provided to enhance your understanding of new topics. Margin Notes alert you to a point requiring special attention or are used to provide study tips. To Review Notes in the margin will help you recognize the prerequisite skills needed to understand new concepts. These notes direct you to the appropriate page or section for review. Calculus Connection Icons identify topics that will be revisited in a subsequent calculus course. Visualize the Solution When appropriate, both algebraic and graphical solutions are provided to help you visualize the mathematics of an example and to create a link between the algebraic and visual components of a solution. PREFACE xiii Integrating Technology Integrating Technology boxes show how technology can be used to illustrate concepts and solve many mathematical problems. Examples and exercises that require a calculator or a computer to find a solution are identified by the graphing calculator icon. Exploring Concepts with Technology The optional Exploring Concepts with Technology feature appears after the last section in each chapter and provides you the opportunity to use a calculator or a computer to solve computationally difficult problems. In addition, you are challenged to think about pitfalls that can be produced when using technology to solve mathematical problems. Modeling Modeling sections and exercises rely on the use of a graphing calculator or a computer. These optional sections and exercises introduce the idea of a mathematical model and help you see the relevance of mathematical concepts. xiv PREFACE NEW Mid-Chapter Quizzes New to this edition, these quizzes help you assess your understanding of the concepts studied earlier in the chapter. The answers for all exercises in the Mid-Chapter Quizzes are provided in the Answers to Selected Exercises appendix along with a reference to the section in which a particular concept was presented. NEW Chapter Test Preps The Chapter Test Preps summarize the major concepts discussed in each chapter. These Test Preps help you prepare for a chapter test. For each concept there is a reference to a worked example illustrating the concept and at least one exercise in the Chapter Review Exercise set relating to that concept. Chapter Review Exercise Sets and Chapter Tests The Chapter Review Exercise sets and the Chapter Tests at the end of each chapter are designed to provide you with another opportunity to assess your understanding of the concepts presented in a chapter. The answers for all exercises in the Chapter Review Exercise sets and the Chapter Tests are provided in the Answers to Selected Exercises appendix along with a reference to the section in which the concept was presented. PREFACE xv In addition to the New! Mid-Chapter Quizzes and New! Chapter Test Preps, the following changes appear in this seventh edition of College Algebra and Trigonometry: Chapter P Preliminary Concepts P.1 This section has been reorganized. The Order of Operations Agreement has been given more prominence to ensure that students understand this important concept. P.2 Additional examples have been added to illustrate more situations with radicals and rational exponents. P.3 Another example has been added, new exercises have been added, and some of the existing exercises have been rearranged. P.4 This section has been reorganized, and new examples have been added. The exercise set has been reorganized, and new exercises have been added. P.5 New examples have been added to show operations on rational expressions. Chapter 1 Equations and Inequalities 1.1 Two examples were added for solving first-degree equations. 1.2 This section has been reorganized, and new applications have been added. The exercise set has been changed to include new applications. 1.3 New examples showing how to solve quadratic equations were added. The exercise set has been extensively revised. 1.4 Much of this section has been rewritten and reorganized, and new application problems have been added. The exercise set has been reorganized, and many new exercises have been added. 1.5 The critical-value method of solving polynomial inequalities has been expanded, and new exercises have been added. Chapter 2 Functions and Graphs 2.2 This section has been reorganized so that appropriate emphasis is given to the various aspects of working with functions. We introduced the connection of x-intercepts to real zeros of a function to better prepare students for a full discussion of zeros in Chapter 3. The exercise set has been reorganized, and many new exercises were added. 2.3 The introduction to slope has been expanded. New examples on finding the equation of a line were added to give students models of the various types of problems found in the exercise set. 2.5 New examples were added to illustrate various transformations. The effect was to slow the pace of this section so students could better understand these important concepts. Chapter 3 Polynomial and Rational Functions 3.2 A new example on modeling data with a cubic function was added. This example is followed by a discussion concerning the strengths and weaknesses of modeling data from an application with cubic and quartic regression functions. Five new application exercises involving the use of cubic and quartic models were added to the exercise set. 3.5 A new example on using a rational function to solve an application was added. Two new exercises that make use of a rational function to solve an application were added. Three exercises that involve creating a rational function whose graph has given properties were added. The definition of a slant asymptote was included in this section. Several new exercises were added to the Chapter Review Exercises. A new application exercise was added to the Chapter Test. xvi PREFACE Chapter 4 Exponential and Logarithmic Functions 4.1 Two new application exercises were added to the exercise set. 4.2 Two new applications were created to introduce increasing and decreasing exponential functions. Additional expository material was inserted to better explain the concept of the expression bx where x is an irrational number. 4.6 Examples and application exercises involving dates were updated or replaced. New application exercises involving the concept of a declining logistic model were added to the exercise set. 4.7 Examples and application exercises involving dates were updated or replaced. New application exercises were added to the exercise set. New exercises were added to the Chapter Review Exercises. Chapter 5 Trigonometric Functions 5.2 New application exercises were added to the exercise set. 5.3 The reference angle evaluation procedure and the accompanying example were revised to simplify the evaluation process. Several exercises were added to the Chapter Review Exercises. Chapter 6 Trigonometric Identities and Equations 6.6 Application examples and exercises that involve dates were updated. Several exercises were added to the Chapter Review Exercise set. Chapter 7 Applications of Trigonometry 7.1 Three new examples were added to better illustrate the ambiguous case of the Law of Sines. New application exercises were added to the Chapter Review Exercises. A new application exercise was added to the Chapter Test. Chapter 8 Topics in Analytic Geometry 8.4 A new art piece was included in Example 4 to better illustrate the procedure for graphing a rotated conic section with a graphing utility. 8.5 New art pieces were inserted in Example 3 to better illustrate the procedure for graphing a polar equation with a graphing utility. A new example and exercises concerning the graphs of lemniscates were added. The example concerning the transformation from rectangular to polar coordinates was revised to better illustrate the multiple representation of a point in the polar coordinate system. Several exercises were added to the exercise set. New application exercises were added to the Chapter Review Exercises. New application exercises were added to the Chapter Test. Chapter 9 Systems of Equations and Inequalities A new chapter opener page was written to introduce some of the concepts in this chapter. 9.5 New art pieces were included to better illustrate the concept of finding the solution set of a system of inequalities by graphing. The targeted exercise heart rate formula was updated in an example and in the application exercises concerning physical fitness. 9.6 New illustrations were added to an example and two application exercises. A new application exercise on maximizing profit was added. PREFACE xvii Chapter 10 Matrices 10.1 A new example on augmented matrices was added. Some new exercises were added to show different row-reduced forms and systems of equations with no solution. 10.2 A new example on finding a power of a matrix was added. A graph theory application involving multiplication of matrices was added. Chapter 11 Sequences, Series, and Probability 11.1 A new example on finding the sum of a series was added. 11.2 Example 2 was rewritten to better illustrate that a series is the sum of the terms of a sequence. 11.5 Two examples were added that demonstrate the Binomial Theorem. New exercises were added to more gradually move from easier to more difficult applications of the Binomial Theorem. 11.7 This section has been reorganized and a new example that shows the use of the probability addition rules has been added. SUPPLEMENTS For the Instructor Complete Solutions Manual for Aufmann/Barker/Nation’s College Algebra and Trigonometry, 7e ISBN: 0-538-73927-4 The complete solutions manual provides worked-out solutions to all of the problems in the text. (Print) *online version available; see description for Solution Builder below Text Specific DVDs for Aufmann/Barker/Nation’s College Algebra and Trigonometry Series, 7e ISBN: 0-538-79788-6 Available to adopting instructors, these DVDs, which cover all sections in the text, are hosted by Dana Mosely and captioned for the hearing-impaired. Ideal for promoting individual study and review, these comprehensive DVDs also support students in online courses or may be checked out by a student who may have missed a lecture. 12 DVDs contain over 40 hours of video. (Media) Enhanced WebAssign® (access code packaged with student edition at request of instructor; instructor access obtained by request of instructor to Cengage Learning representative) Enhanced WebAssign® allows instructors to assign, collect, grade, and record homework assignments online, minimizing workload and streamlining the grading process. EWA also gives students the ability to stay organized with assignments and have up-to-date grade information. For your convenience, the exercises available in EWA are indicated in the instructor’s edition by a blue triangle. (Online) PowerLecture ISBN: 0-538-73906-1 PowerLecture contains PowerPoint ® lecture outlines, a database of all art in the text, ExamView®, and a link to the Solution Builder. (CD) ExamView® (included on PowerLecture CD) Create, deliver, and customize tests (both print and online) in minutes with this easy-to-use assessment system. (CD) Solution Builder (included on PowerLecture CD and available online at http://academic.cengage.com/solutionbuilder/) The Solution Builder is an electronic version of the Complete Solutions Manual, providing instructors with an efficient method for creating solution sets to homework and exams that can be printed or posted. (CD and online) xviii PREFACE Syllabus Creator (included on PowerLecture CD) Quickly and easily create your course syllabus with Syllabus Creator, which was created by the authors. For the Student Study Guide with Student Solutions Manual for Aufmann/Barker/Nation’s College Algebra and Trigonometry, 7e ISBN: 0-538-73908-8 The student solutions manual reinforces student understanding and aids in test preparation with detailed explanations, worked-out examples, and practice problems. Lists key ideas to master and builds problem-solving skills. Includes worked solutions to the odd-numbered problems in the text. (Print) Enhanced WebAssign® (access code packaged with student edition at request of instructor) Enhanced WebAssign® allows instructors to assign, collect, grade, and record homework assignments online, minimizing workload and streamlining the grading process. EWA also gives students the ability to stay organized with assignments and have up-to-date grade information. (Online) ACKNOWLEDGMENTS We would like to thank the wonderful team of editors, accuracy checkers, proofreaders, and solutions manual authors. Special thanks to Cindy Harvey, Helen Medley, and Christi Verity. Cindy Harvey was a very valuable asset during development and production of the manuscript. Helen Medley was the accuracy reviewer for both College Algebra and College Algebra and Trigonometry, and Christi Verity wrote the solutions for the Complete Solutions Manual and the Student Solutions Manual for College Algebra and College Algebra and Trigonometry. Both Helen and Christi have improved the accuracy of the texts and provided valuable suggestions for improving the texts. We are grateful to the users of the previous edition for their helpful suggestions on improving the text. Also, we sincerely appreciate the time, effort, and suggestions of the reviewers of this edition: Robin Anderson—Southwestern Illinois College Richard Bailey—Midlands Tech College Cecil J. Coone—Southwest Tennessee Community College Kyle Costello—Salt Lake Community College Thomas English—College of the Mainland Celeste Hernandez—Richland College Magdalen Ivanovska—New York University Skopje Rose Jenkins—Midlands Tech College Stefan C. Mancus—Embry-Riddle Aeronautical University Jamie Whittimore McGill—East Tennessee State University Zephyrinus Okonkwo—Albany State University Mike Shirazi—Germanna Community College Lalitha Subramanian—Potomac State College of West Virginia University Tan Zhang—Murray State University CHAPTER P PRELIMINARY CONCEPTS P.1 The Real Number System P.2 Integer and Rational Number Exponents P.3 Polynomials P.4 Factoring P.5 Rational Expressions P.6 Complex Numbers AFP/Getty Images Albert Einstein proposed relativity theory more than 100 years ago, in 1905. Relativity Is More Than 100 Years Old Positron emission tomography (PET) scans, the temperature of Earth’s crust, smoke detectors, neon signs, carbon dating, and the warmth we Martial Trezzini/epa/CORBIS receive from the sun may seem to be disparate concepts. However, they have a common theme: Albert Einstein’s Theory of Special Relativity. When Einstein was asked about his innate curiosity, he replied: The important thing is not to stop questioning. Curiosity has its own reason for The Large Hadron Collider (LHC). Atomic particles are accelerated existing. One cannot help but be in awe when he contemplates the mysteries of eternity, to high speeds inside the long of life, of the marvelous structure of reality. It is enough if one tries merely to structure in the photo above. By comprehend a little of this mystery every day. studying particles moving at speeds that approach the speed of Today, relativity theory is used in conjunction with other concepts of light, physicists can confirm some physics to study ideas ranging from the structure of an atom to the of the tenets of relativity theory. structure of the universe. Some of Einstein’s equations require working with radical expressions, such as the expression given in Exercise 139 on page 31; other equations use rational expressions, such as the expression given in Exercise 64 on page 59. 1 2 CHAPTER P PRELIMINARY CONCEPTS SECTION P.1 The Real Number System Sets Union and Intersection of Sets Sets Interval Notation Human beings share the desire to organize and classify. Ancient astronomers classified Absolute Value and Distance stars into groups called constellations. Modern astronomers continue to classify stars Exponential Expressions by such characteristics as color, mass, size, temperature, and distance from Earth. In Order of Operations Agreement mathematics it is useful to place numbers with similar characteristics into sets. The Simplifying Variable Expressions following sets of numbers are used extensively in the study of algebra. Natural numbers 51, 2, 3, 4, Á 6 Integers 5 Á , -3, -2, - 1, 0, 1, 2, 3, Á 6 Rational numbers 5all terminating or repeating decimals6 Irrational numbers 5all nonterminating, nonrepeating decimals6 Real numbers 5all rational or irrational numbers6 If a number in decimal form terminates or repeats a block of digits, then the number is a rational number. Here are two examples of rational numbers. 0.75 is a terminating decimal. 0.245 is a repeating decimal. The bar over the 45 means that the digits 45 repeat without end. That is, 0.245 = 0.24545454 Á. p Rational numbers also can be written in the form , where p and q are inte- q gers and q Z 0. Examples of rational numbers written in this form are 3 27 5 7 -4 - 4 110 2 1 3 7 n Note that = 7, and, in general, = n for any integer n. Therefore, all integers are rational 1 1 Math Matters numbers. Archimedes (c. 287–212 B.C.) was p the first to calculate p with any When a rational number is written in the form , the decimal form of the rational q degree of precision. He was able to show that number can be found by dividing the numerator by the denominator. 10 1 3 27 3 6 p 6 3 = 0.75 = 0.245 71 7 4 110 from which we get the approxi- In its decimal form, an irrational number neither terminates nor repeats. For mation example, 0.272272227 Á is a nonterminating, nonrepeating decimal and thus is an 1 22 irrational number. One of the best-known irrational numbers is pi, denoted by the 3 = L p 7 7 Greek symbol p. The number p is defined as the ratio of the circumference of a circle to its diameter. Often in applications the rational number 3.14 or the rational The use of the symbol p for this 22 quantity was introduced by number is used as an approximation of the irrational number p. Leonhard Euler (1707–1783) in 7 1739, approximately 2000 years Every real number is either a rational number or an irrational number. If a real num- after Archimedes. ber is written in decimal form, it is a terminating decimal, a repeating decimal, or a nonterminating and nonrepeating decimal. P.1 THE REAL NUMBER SYSTEM 3 Math Matters The relationships among the various sets of numbers are shown in Figure P.1. Sophie Germain (1776–1831) was born in Paris, France. Because Positive integers enrollment in the university she (natural numbers) wanted to attend was available only 7 1 103 to men, Germain attended under the name of Antoine-August Le Integers Rational numbers Real numbers Zero Blanc. Eventually her ruse was dis- 0 3 3 −201 7 0 −5 4 3.1212 −1.34 −5 4 3.1212 −1.34 covered, but not before she came 7 0 −5 to the attention of Pierre Lagrange, Negative integers Irrational numbers 1 103 −201 one of the best mathematicians of the time. He encouraged her work −201 −8 −5 −0.101101110... √7 π −0.101101110... √7 π and became a mentor to her. A cer- tain type of prime number is Figure P.1 named after her, called a Germain prime number. It is a number p such Prime numbers and composite numbers play an important role in almost every branch that p and 2p + 1 are both prime. of mathematics. A prime number is a positive integer greater than 1 that has no positive- For instance, 11 is a Germain prime integer factors1 other than itself and 1. The 10 smallest prime numbers are 2, 3, 5, 7, 11, because 2(11) + 1 = 23 and 11 and 23 are both prime numbers. 13, 17, 19, 23, and 29. Each of these numbers has only itself and 1 as factors. Germain primes are used in public A composite number is a positive integer greater than 1 that is not a prime number. key cryptography, a method used For example, 10 is a composite number because 10 has both 2 and 5 as factors. The 10 small- to send secure communications est composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, and 18. over the Internet. EXAMPLE 1 Classify Real Numbers Determine which of the following numbers are a. integers b. rational numbers c. irrational numbers d. real numbers e. prime numbers f. composite numbers -0.2, 0, 0.3, 0.71771777177771 Á , p, 6, 7, 41, 51 Solution a. Integers: 0, 6, 7, 41, 51 b. Rational numbers: - 0.2, 0, 0.3, 6, 7, 41, 51 c. Irrational numbers: 0.71771777177771... , p d. Real numbers: - 0.2, 0, 0.3, 0.71771777177771 Á , p, 6, 7, 41, 51 e. Prime numbers: 7, 41 f. Composite numbers: 6, 51 Try Exercise 2, page 14 Each member of a set is called an element of the set. For instance, if C = 52, 3, 56, then the elements of C are 2, 3, and 5. The notation 2 C is read “2 is an element of C.” 1 A factor of a number divides the number evenly. For instance, 3 and 7 are factors of 21; 5 is not a factor of 21. 4 CHAPTER P PRELIMINARY CONCEPTS Set A is a subset of set B if every element of A is also an element of B, and we write A 8 B. For instance, the set of negative integers { -1, -2, - 3, - 4, Á } is a subset of the set of integers. The set of positive integers 51, 2, 3, 4, Á 6 (the natural numbers) is also a subset of the set of integers. Question Are the integers a subset of the rational numbers? Note The empty set, or null set, is the set that contains no elements. The symbol is The order of the elements of a set used to represent the empty set. The set of people who have run a 2-minute mile is the is not important. For instance, the empty set. set of natural numbers less than The set of natural numbers less than 6 is 51, 2, 3, 4, 56. This is an example of a finite 6 given at the right could have set; all the elements of the set can be listed. The set of all natural numbers is an example been written 53, 5, 2, 1, 46. It is of an infinite set. There is no largest natural number, so all the elements of the set of natu- customary, however, to list ele- ral numbers cannot be listed. ments of a set in numerical order. Sets are often written using set-builder notation. Set-builder notation can be used to describe almost any set, but it is especially useful when writing infinite sets. For instance, the set Math Matters 52n ƒ n natural numbers6 A fuzzy set is one in which each is read as “the set of elements 2n such that n is a natural number.” By replacing n with each of the natural numbers, this becomes the set of positive even integers: 52, 4, 6, 8,...6. element is given a “degree” of membership. The concepts behind fuzzy sets are used in a The set of real numbers greater than 2 is written wide variety of applications such 5x ƒ x 7 2, x real numbers6 as traffic lights, washing machines, and computer speech and is read “the set of x such that x is greater than 2 and x is an element of the real recognition programs. numbers.” Much of the work we do in this text uses the real numbers. With this in mind, we will frequently write, for instance, 5x ƒ x 7 2, x real numbers6 in a shortened form as 5x ƒ x 7 26, where we assume that x is a real number. EXAMPLE 2 Use Set-Builder Notation List the four smallest elements in 5n3 ƒ n natural numbers6. Solution Because we want the four smallest elements, we choose the four smallest natural numbers. Thus n = 1, 2, 3, and 4. Therefore, the four smallest elements of the set 5n3 ƒ n natural numbers6 are 1, 8, 27, and 64. Try Exercise 6, page 14 Union and Intersection of Sets Just as operations such as addition and multiplication are performed on real numbers, operations are performed on sets. Two operations performed on sets are union and intersection. The union of two sets A and B is the set of elements that belong to A or to B, or to both A and B. Answer Yes. P.1 THE REAL NUMBER SYSTEM 5 Definition of the Union of Two Sets The union of two sets, written A ´ B, is the set of all elements that belong to either A or B. In set-builder notation, this is written A ´ B = 5x ƒ x A or x B6 EXAMPLE Given A = 52, 3, 4, 56 and B = 50, 1, 2, 3, 46, find A ´ B. A ´ B = 50, 1, 2, 3, 4, 56 Note that an element that belongs to both sets is listed only once. The intersection of the two sets A and B is the set of elements that belong to both A and B. Definition of the Intersection of Two Sets The intersection of two sets, written A ¨ B, is the set of all elements that are common to both A and B. In set-builder notation, this is written A ¨ B = 5x ƒ x A and x B6 EXAMPLE Given A = 52, 3, 4, 56 and B = 50, 1, 2, 3, 46, find A ¨ B. A ¨ B = 52, 3, 46 The intersection of two sets contains the elements common to both sets. If the intersection of two sets is the empty set, the two sets are said to be disjoint. For example, if A = 52, 3, 46 and B = 57, 86, then A ¨ B = and A and B are disjoint sets. EXAMPLE 3 Find the Union and Intersection of Sets Find each intersection or union given A = 50, 2, 4, 6, 10, 126, B = 50, 3, 6, 12, 156, and C = 51, 2, 3, 4, 5, 6, 76. a. A´C b. B ¨ C c. A ¨ (B ´ C) d. B ´ (A ¨ C ) Solution a. A ´ C = 50, 1, 2, 3, 4, 5, 6, 7, 10, 126 The elements that belong to A or C b. B ¨ C = 53, 66 The elements that belong to B and C c. First determine B ´ C = 50, 1, 2, 3, 4, 5, 6, 7, 12, 156. Then A ¨ (B ´ C) = 50, 2, 4, 6, 126 The elements that belong to A and (B ´ C ) d. First determine A ¨ C = 52, 4, 66. Then B ´ (A ¨ C) = 50, 2, 3, 4, 6, 12, 156 The elements that belong to B or (A ¨ C ) Try Exercise 16, page 14 6 CHAPTER P PRELIMINARY CONCEPTS Interval Notation The graph of 5x ƒ x 7 26 is shown in Figure P.2. The set is the real numbers greater than 2. −5 −4 −3 −2 −1 0 1 2 3 4 5 The parenthesis at 2 indicates that 2 is not included in the set. Rather than write this set Figure P.2 of real numbers using set-builder notation, we can write the set in interval notation as (2, q ). In general, the interval notation (a, b) represents all real numbers between a and b, not including a and b. This is an open interval. In set-builder notation, we write 5x ƒ a 6 x 6 b6. −5 −4 −3 −2 −1 0 1 2 3 4 5 Figure P.3 The graph of ( -4, 2) is shown in Figure P.3. 3a, b4 represents all real numbers between a and b, including a and b. This is a closed interval. In set-builder notation, we write 5x ƒ a … x … b6. −5 −4 −3 −2 −1 0 1 2 3 4 5 Figure P.4 The graph of 30, 44 is shown in Figure P.4. The brackets at 0 and 4 indicate that those numbers are included in the graph. (a, b4 represents all real numbers between a and b, not including a but −5 −4 −3 −2 −1 0 1 2 3 4 5 including b. This is a half-open interval. In set-builder notation, we Figure P.5 write 5x ƒ a 6 x … b6. The graph of ( -1, 34 is shown in Figure P.5. 3a, b) represents all real numbers between a and b, including a but not −5 −4 −3 −2 −1 0 1 2 3 4 5 including b. This is a half-open interval. In set-builder notation, we Figure P.6 write 5x ƒ a … x 6 b6. The graph of 3 - 4, - 1) is shown in Figure P.6. Subsets of the real numbers whose graphs extend forever in one or both directions can be represented by interval notation using the infinity symbol q or the negative infinity symbol - q. ( - q , a) represents all real numbers less than a. a (b, q ) represents all real numbers greater than b. b ( - q , a4 represents all real numbers less than or a equal to a. 3b, q ) represents all real numbers greater than b or equal to b. ( - q, q) represents all real numbers. 0 EXAMPLE 4 Graph a Set Given in Interval Notation Graph (- q , 34. Write the interval in set-builder notation. Solution Caution The set is the real numbers less than or equal to 3. In set-builder notation, this is the set It is never correct to use a bracket 5x ƒ x … 36. Draw a right bracket at 3, and darken the number line to the left of 3, as when using the infinity symbol. shown in Figure P.7. For instance, [- q , 3] is not correct. Nor is [2, q ] correct. Neither − 5 − 4 −3 −2 −1 0 1 2 3 4 5 negative infinity nor positive infinity is a real number and therefore Figure P.7 cannot be contained in an interval. Try Exercise 40, page 15 P.1 THE REAL NUMBER SYSTEM 7 The set 5x ƒ x … - 26 ´ 5x ƒ x 7 36 is the set of real numbers that are either less than −5 −4 −3 −2 −1 0 1 2 3 4 5 or equal to -2 or greater than 3. We also could write this in interval notation as Figure P.8 (- q , -24 ´ (3, q ). The graph of the set is shown in Figure P.8. The set 5x ƒ x 7 - 46 ¨ 5x ƒ x 6 16 is the set of real numbers that are greater than -4 −5 −4 −3 −2 −1 0 1 2 3 4 5 and less than 1. Note from Figure P.9 that this set is the interval ( - 4, 1), which can be writ- Figure P.9 ten in set-builder notation as 5x ƒ -4 6 x 6 16. EXAMPLE 5 Graph Intervals Graph the following. Write a. and b. using interval notation. Write c. and d. using set- builder notation. a. 5x ƒ x … - 16 ´ 5x ƒ x Ú 26 b. 5x ƒ x Ú - 16 ¨ 5x ƒ x 6 56 c. (- q , 0) ´ 31, 34 d. 3 - 1, 34 ¨ (1, 5) Solution a. ( - q , - 14 ´ 32, q ) − 5 − 4 −3 −2 −1 0 1 2 3 4 5 b. 3 - 1, 5) − 5 − 4 −3 −2 −1 0 1 2 3 4 5 c. 5x ƒ x 6 06 ´ 5x ƒ 1 … x … 36 − 5 − 4 −3 −2 −1 0 1 2 3 4 5 d. The graphs of 3 -1, 34, in red, and (1, 5), in blue, are shown below. −5 −4 −3 −2 −1 0 1 2 3 4 5 Note that the intersection of the sets occurs where the graphs intersect. Although 1 3 - 1, 34, 1 > (1, 5). Therefore, 1 does not belong to the intersection of the sets. On the other hand, 3 3- 1, 34 and 3 (1, 5). Therefore, 3 belongs to the intersection of the sets. Thus we have the following. 5x ƒ 1 6 x … 36 − 5 − 4 −3 −2 −1 0 1 2 3 4 5 Try Exercise 50, page 15 Absolute Value and Distance − 4.25 − 52 1 π √29 The real numbers can be represented geometrically by a coordinate axis called a real −5 −4 −3 −2 −1 0 1 2 3 4 5 number line. Figure P.10 shows a portion of a real number line. The number associated Figure P.10 with a point on a real number line is called the coordinate of the point. The point corre- sponding to zero is called the origin. Every real number corresponds to a point on the number line, and every point on the number line corresponds to a real number. 3 3 The absolute value of a real number a, denoted ƒ a ƒ , is the distance between a and 0 on −5 −4 −3 −2 −1 0 1 2 3 4 5 the number line. For instance, ƒ 3 ƒ = 3 and ƒ - 3 ƒ = 3 because both 3 and - 3 are 3 units Figure P.11 from zero. See Figure P.11. 8 CHAPTER P PRELIMINARY CONCEPTS In general, if a Ú 0, then ƒ a ƒ = a; however, if a 6 0, then ƒ a ƒ = - a because - a is positive when a 6 0. This leads to the following definition. Note Definition of Absolute Value The second part of the definition of absolute value states that The absolute value of the real number a is defined by if a 6 0, then ƒ a ƒ = - a.For a if a Ú 0 instance, if a = - 4, then ƒaƒ = e ƒ a ƒ = ƒ - 4 ƒ = - (-4) = 4. -a if a 6 0 EXAMPLE ƒ5ƒ = 5 ƒ -4 ƒ = 4 ƒ0ƒ = 0 EXAMPLE 6 Simplify an Absolute Value Expression Simplify ƒ x - 3 ƒ + ƒ x + 2 ƒ given that - 1 … x … 2. Solution Recall that ƒ a ƒ = - a when a 6 0 and ƒ a ƒ = a when a Ú 0. When - 1 … x … 2, x - 3 6 0 and x + 2 7 0. Therefore, ƒ x - 3 ƒ = - (x - 3) and ƒ x + 2 ƒ = x + 2. Thus ƒ x - 3 ƒ + ƒ x + 2 ƒ = - (x - 3) + (x + 2) = 5. Try Exercise 60, page 15 The definition of distance between two points on a real number line makes use of absolute value. Definition of the Distance Between Points on a Real Number Line If a and b are the coordinates of two points on a real number line, the distance between the graph of a and the graph of b, denoted by d(a, b), is given by d(a, b) = ƒ a - b ƒ. EXAMPLE Find the distance between a point whose coordinate on the real number line is -2 and a point whose coordinate is 5. d( -2, 5) = ƒ - 2 - 5 ƒ = ƒ - 7 ƒ = 7 7 Note in Figure P.12 that there are 7 units between -2 and 5. Also note that the −5 −4 − 3 −2 −1 0 1 2 3 4 5 order of the coordinates in the formula does not matter. Figure P.12 d(5, - 2) = ƒ 5 - ( -2) ƒ = ƒ 7 ƒ = 7 EXAMPLE 7 Use Absolute Value to Express the Distance Between Two Points Express the distance between a and - 3 on the number line using absolute value notation. Solution d(a, - 3) = ƒ a - ( -3) ƒ = ƒ a + 3 ƒ Try Exercise 70, page 15 P.1 THE REAL NUMBER SYSTEM 9 Math Matters Exponential Expressions A compact method of writing 5 # 5 # 5 # 5 is 5 4. The expression 5 4 is written in exponential The expression 10100 is called a googol. The term was coined by the 9-year-old nephew of the notation. Similarly, we can write American mathematician Edward 2x # 2x # 2x 2x 3 Kasner. Many calculators do not as a b provide for numbers of this mag- 3 3 3 3 nitude, but it is no serious loss. To appreciate the magnitude of a Exponential notation can be used to express the product of any expression that is used googol, consider that if all the repeatedly as a factor. atoms in the known universe were counted, the number would not even be close to a googol. But if a Definition of Natural Number Exponents googol is too small for you, try 10googol, which is called a googol- If b is any real number and n is a natural number, then plex. As a final note, the name of the Internet site Google.com is a b$''%''& is a factor n times takeoff on the word googol. bn = b # b # b # Á # b where b is the base and n is the exponent. EXAMPLE a b = # # = 3 3 3 3 3 27 4 4 4 4 64 -54 = - (5 # 5 # 5 # 5) = - 625 (-5)4 = ( -5)( - 5)( - 5)( -5) = 625 Pay close attention to the difference between - 54 (the base is 5) and ( -5)4 (the base is - 5). EXAMPLE 8 Evaluate an Exponential Expression Evaluate. -4 4 a. (-3 4)( - 4) 2 b. ( - 4) 4 Solution a. (-3 4)( - 4)2 = - (3 # 3 # 3 # 3) # ( -4)( - 4) = - 81 # 16 = -1296 -44 -(4 # 4 # 4 # 4) -256 b. = = = -1 (-4)4 ( - 4)( - 4)( -4)( - 4) 256 Try Exercise 76, page 15 Order of Operations Agreement The approximate pressure p, in pounds per square inch, on a scuba diver x feet below the water’s surface is given by p = 15 + 0.5x 10 CHAPTER P PRELIMINARY CONCEPTS The pressure on the diver at various depths is given below. 10 feet 15 + 0.5(10) = 15 + 5 = 20 pounds 20 feet 15 + 0.5(20) = 15 + 10 = 25 pounds 40 feet 15 + 0.5(40) = 15 + 20 = 35 pounds 70 feet 15 + 0.5(70) = 15 + 35 = 50 pounds Note that the expression 15 + 0.5(70) has two operations, addition and multiplication. When an expression contains more than one operation, the operations must be performed in a specified order, as given by the Order of Operations Agreement. The Order of Operations Agreement If grouping symbols are present, evaluate by first performing the operations within the grouping symbols, innermost grouping symbols first, while observing the order given in steps 1 to 3. Step 1 Evaluate exponential expressions. Step 2 Do multiplication and division as they occur from left to right. Step 3 Do addition and subtraction as they occur from left to right. EXAMPLE 5 - 7(23 - 5 2) - 16 , 2 3 = 5 - 7(23 - 25) - 16 , 23 Begin inside the parentheses and evaluate 52 = 25. = 5 - 7( -2) - 16 , 23 Continue inside the parentheses and evaluate 23 - 25 = - 2. = 5 - 7( -2) - 16 , 8 Evaluate 23 = 8. = 5 - (- 14) - 2 Perform multiplication and division from left to right. = 17 Perform addition and subtraction from left to right. EXAMPLE 9 Use the Order of Operations Agreement Evaluate: 3 # 5 2 - 6( - 3 2 - 4 2) , ( -15) Solution 3 # 5 2 - 6(-3 2 - 4 2) , ( -15) = 3 # 5 2 - 6( -9 - 16) , ( - 15) Begin inside the parentheses. = 3 # 5 2 - 6( -25) , ( - 15) Simplify - 9 - 16. = 3 # 25 - 6( - 25) , ( -15) Evaluate 5 2. = 75 + 150 , (- 15) Do mulipltication and division from left to right. = 75 + (-10) = 65 Do addition. Try Exercise 80, page 15 P.1 THE REAL NUMBER SYSTEM 11 Recall One of the ways in which the Order of Operations Subtraction can be rewritten as Agreement is used is to evaluate variable expressions. The addition of the opposite. Therefore, addends of a variable expression are called terms. The 3x 2 - 4xy + 5x - y - 7 3x2 - 4xy + 5x - y - 7 terms for the expression at the right are 3x , -4xy, 5x, -y, 2 = 3x 2 + (- 4xy) + 5x + (- y) + (-7) and - 7. Observe that the sign of a term is the sign that In this form, we can see that the immediately precedes it. terms (addends) are 3x2, -4xy, 5x, The terms 3x 2, -4xy, 5x, and -y are variable terms. The term - 7 is a constant - y, and -7. term. Each variable term has a numerical coefficient and a variable part. The numerical coefficient for the term 3x 2 is 3; the numerical coefficient for the term -4xy is - 4; the numerical coefficient for the term 5x is 5; and the numerical coefficient for the term -y is -1. When the numerical coefficient is 1 or - 1 (as in x and -x), the 1 is usually not written. To evaluate a variable expression, replace the variables by their given values and then use the Order of Operations Agreement to simplify the result. EXAMPLE 10 Evaluate a Variable Expression x3 - y3 a. Evaluate when x = 2 and y = - 3. x 2 + xy + y 2 b. Evaluate (x + 2y)2 - 4z when x = 3, y = - 2, and z = - 4. Solution x3 - y3 a. x 2 + xy + y 2 23 - ( -3)3 8 - ( - 27) 35 = = = 5 22 + 2( - 3) + ( -3)2 4 - 6 + 9 7 b. (x + 2y)2 - 4z [3 + 2(- 2)]2 - 4(- 4) = [3 + ( -4)]2 - 4( -4) = ( - 1)2 - 4(- 4) = 1 - 4( -4) = 1 + 16 = 17 Try Exercise 90, page 15 Simplifying Variable Expressions Addition, multiplication, subtraction, and division are the operations of arithmetic. Addition of the two real numbers a and b is designated by a + b. If a + b = c, then c is the sum and the real numbers a and b are called terms. Multiplication of the real numbers a and b is designated by ab or a # b. If ab = c, then c is the product and the real numbers a and b are called factors of c. The number - b is referred to as the additive inverse of b. Subtraction of the real numbers a and b is designated by a - b and is defined as the sum of a and the additive inverse of b. That is, a - b = a + ( - b) If a - b = c, then c is called the difference of a and b. 12 CHAPTER P PRELIMINARY CONCEPTS The multiplicative inverse or reciprocal of the nonzero number b is 1>b. The division of a and b, designated by a , b with b Z 0, is defined as the product of a and the reciprocal of b. That is, a , b = aa b 1 provided that b Z 0 b If a , b = c, then c is called the quotient of a and b. a The notation a , b is often represented by the fractional notation a>b or. The real b number a is the numerator, and the nonzero real number b is the denominator of the fraction. Properties of Real Numbers Let a, b, and c be real numbers. Addition Properties Multiplication Properties Closure a + b is a unique real ab is a unique real number. number. Commutative a + b = b + a ab = ba Associative (a + b) + c = a + (b + c) (ab)c = a(bc) Identity There exists a unique real There exists a unique real number 0 such that number 1 such that a + 0 = 0 + a = a. a # 1 = 1 # a = a. Inverse For each real number a, For each nonzero real there is a unique real number a, there is a number - a such that unique real number 1>a such that a # = # a = 1. a + ( - a) = ( -a) + a = 0. 1 1 a a Distributive a(b + c) = ab + ac EXAMPLE 11 Identify Properties of Real Numbers Identify the property of real numbers illustrated in each statement. a b 11 is a real number. 1 a. (2a)b = 2(ab) b. 5 c. 4(x + 3) = 4x + 12 d. (a + 5b) + 7c = (5b + a) + 7c 1# e. a 2ba = 1 # a f. 1 # a = a 2 Solution a. Associative property of multiplication b. Closure property of multiplication c. Distributive property d. Commutative property of addition P.1 THE REAL NUMBER SYSTEM 13 e. Inverse property of multiplication f. Identity property of multiplication Try Exercise 102, page 16 We can identify which properties of real numbers have been used to rewrite an expression by closely comparing the original and final expressions and noting any changes. For instance, to simplify (6x)2, both the commutative property and associative property of multiplication are used. Note (6x)2 = 2(6x) Commutative property of multiplication Normally, we will not show, as we = (2 # 6)x Associative property of multiplication did at the right, all the steps involved in the simplification of a = 12x variable expression. For instance, To simplify 3(4p + 5), use the distributive property. we will just write (6x)2 = 12x, 3(4p + 5) = 3(4p) + 3(5) Distributive property 3(4p + 5) = 12p + 15, and 3x 2 + 9x 2 = 12x 2. It is important = 12p + 15 to know, however, that every step in the simplification process Terms that have the same variable part are called like terms. The distributive property depends on one of the properties is also used to simplify an expression with like terms such as 3x 2 + 9x 2. of real numbers. 3x 2 + 9x 2 = (3 + 9)x 2 Distributive property = 12x 2 Note from this example that like terms are combined by adding the coefficients of the like terms. Question Are the terms 2x2 and 3x like terms? EXAMPLE 12 Simplify Variable Expressions Simplify. a. 5 + 3(2x - 6) b. 4x - 237 - 5(2x - 3)4 Solution a. 5 + 3(2x - 6) = 5 + 6x - 18 Use the distributive property. = 6x - 13 Add the constant terms. b. 4x - 237 - 5(2x - 3)4 = 4x - 237 - 10x + 154 Use the distributive property to remove the inner parentheses. = 4x - 23 - 10x + 224 Simplify. = 4x + 20x - 44 Use the distributive property to remove the brackets. = 24x - 44 Simplify. Try Exercise 120, page 16 # Answer No. The variable parts are not the same. The variable part of 2x2 is x x. The variable part of 3x is x. 14 CHAPTER P PRELIMINARY CONCEPTS An equation is a statement of equality between two numbers or two expressions. There are four basic properties of equality that relate to equations. Properties of Equality Let a, b, and c be real numbers. Reflexive a = a Symmetric If a = b, then b = a. Transitive If a = b and b = c, then a = c. Substitution If a = b, then a may be replaced by b in any expression that involves a. EXAMPLE 13 Identify Properties of Equality Identify the property of equality illustrated in each statement. a. If 3a + b = c, then c = 3a + b. b. 5(x + y) = 5(x + y) c. If 4a - 1 = 7b and 7b = 5c + 2, then 4a - 1 = 5c + 2. d. If a = 5 and b(a + c) = 72, then b(5 + c) = 72. Solution a. Symmetric b. Reflexive c. Transitive d. Substitution Try Exercise 106, page 16 EXERCISE SET P.1 In Exercises 1 and 2, determine whether each number is In Exercises 9 to 18, perform the operations given that an integer, a rational number, an irrational number, a A { 3, 2, 1, 0, 1, 2, 3}, B {2, 0, 2, 4, 6}, prime number, or a real number. C {0, 1, 2, 3, 4, 5, 6}, and D { 3, 1, 1, 3}. 1 9. A ´ B 10. C ´ D 1. - , 0, -44, p, 3.14, 5.05005000500005 Á , 181, 53 5 11. A ¨ C 12. C ¨ D 5 5 1 2. , , 31, -2 , 4.235653907493, 51, 0.888 Á 17 7 2 13. B ¨ D 14. B ´ (A ¨ C) In Exercises 3 to 8, list the four smallest elements of 15. D ¨ (B ´ C) 16. (A ¨ B) ´ (A ¨ C) each set. 3. 52x ƒ x positive integers6 4. 5 ƒ x ƒ ƒ x integers6 17. (B ´ C) ¨ (B ´ D) 18. (A ¨ C) ´ (B ¨ D) 5. 5 y ƒ y = 2x + 1, x natural numbers6 In Exercises 19 to 24, perform the operation, given A is 6. 5 y ƒ y = x2 - 1, x integers6 any set. 7. 5z ƒ z = ƒ x ƒ , x integers6 19. A ´ A 20. A ¨ A 8. 5z ƒ z = ƒ x ƒ - x, x negative integers6 21. A ¨ 22. A ´ P.1 THE REAL NUMBER SYSTEM 15 23. If A and B are two sets and A ´ B = A, what can be said In Exercises 63 to 74, use absolute value notation to about B? describe the given situation. 63. d(m, n) 64. d( p, 8) 24. If A and B are two sets and A ¨ B = B, what can be said about B? 65. The distance between x and 3 66. The distance between a and -2 In Exercises 25 to 36, graph each set. Write sets given in interval notation in set-builder notation, and write sets 67. The distance between x and -2 is 4. given in set-builder notation in interval notation. 25. ( -2, 3) 26. 31, 54 68. The distance between z and 5 is 1. 27. 3-5, -14 69. The distance between a and 4 is less than 5. 28. (-3, 3) 70. The distance between z and 5 is greater than 7. 29. 32, q ) 30. (- q , 4) 71. The distance between x and -2 is greater than 4. 31. 5x ƒ 3 6 x 6 56

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