Ebook College algebra with trigonometry (9th edition): Part 1

Ebook College algebra with trigonometry (9th edition): Part 1 includes the following content: Chapter R: Basic algebraic operations, chapter 1 equations and inequalities, chapter 2 graphs, chapter 3 functions, chapter 4 polynomial and rational functions, chapter 5 exponential and logarithmic functions, chapter 6 trigonometric functions. Please refer to the documentation for more details.

College Algebra with Trigonometry

NINTH EDITION

College Algebra with Trigonometry

COLLEGE ALGEBRA WITH TRIGONOMETRY, NINTH EDITION Published by McGraw-Hill, a business unit of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas, New York, NY 10020 Copyright © 2011 by The McGraw-Hill Companies, Inc All rights reserved Previous editions © 2008, 2001, and 1999 No part of this publication may be reproduced or distributed in any form or

by any means, or stored in a database or retrieval system, without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning.

Some ancillaries, including electronic and print components, may not be available to customers outside the United States This book is printed on acid-free paper

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Chap-Library of Congress Cataloging-in-Publication Data

Barnett, Raymond A.

College algebra with trigonometry/Raymond A Barnett [et al.] — 9th ed.

p cm — (Barnett, Ziegler & Byleen’s precalculus series) Includes index.

ISBN 978-0-07-351950-0 — ISBN 0-07-351950-2 (hard copy : alk paper) 1 Algebra–Textbooks.

2 Trigonometry–Textbooks I Title QA154.3.B368 2011

512.13–dc22

2009019473 www.mhhe.com

The Barnett, Ziegler and Sobecki Precalculus Series

College Algebra, Ninth Edition

This book is the same as Precalculus without the three chapters on trigonometry.

ISBN 0-07-351949-9, ISBN 978-0-07-351-949-4

Precalculus, Seventh Edition

This book is the same as College Algebra with three chapters of trigonometry added.

The trigonometric functions are introduced by a unit circle approach

ISBN 0-07-351951-0, ISBN 978-0-07-351-951-7

College Algebra with Trigonometry, Ninth Edition

This book differs from Precalculus in that College Algebra with Trigonometry uses right

triangle trigonometry to introduce the trigonometric functions

ISBN 0-07-735010-3, ISBN 978-0-07-735010-9

College Algebra: Graphs and Models, Third Edition

This book is the same as Precalculus: Graphs and Models without the three chapters on

trigonometry This text assumes the use of a graphing calculator

ISBN 0-07-305195-0, ISBN 978-0-07-305195-6

Precalculus: Graphs and Models, Third Edition

This book is the same as College Algebra: Graphs and Models with three additional

chap-ters on trigonometry The trigonometric functions are introduced by a unit circle approach.This text assumes the use of a graphing calculator

ISBN 0-07-305196-9, ISBN 978-0-07-305-196-3

v

Raymond A Barnett, a native of and educated in California, received his B.A in

math-ematical statistics from the University of California at Berkeley and his M.A in matics from the University of Southern California He has been a member of the MerrittCollege Mathematics Department and was chairman of the department for four years Asso-ciated with four different publishers, Raymond Barnett has authored or co-authored 18 text-books in mathematics, most of which are still in use In addition to international Englisheditions, a number of the books have been translated into Spanish Co-authors includeMichael Ziegler, Marquette University; Thomas Kearns, Northern Kentucky University;Charles Burke, City College of San Francisco; John Fujii, Merritt College; Karl Byleen,Marquette University; and Dave Sobecki, Miami University Hamilton

mathe-Michael R Ziegler received his B.S from Shippensburg State College and his M.S and

Ph.D from the University of Delaware After completing postdoctoral work at the sity of Kentucky, he was appointed to the faculty of Marquette University where he heldthe rank of Professor in the Department of Mathematics, Statistics, and Computer Science

Univer-Dr Ziegler published more than a dozen research articles in complex analysis and co-authoredmore than a dozen undergraduate mathematics textbooks with Raymond Barnett and KarlByleen before passing away unexpectedly in 2008

Karl E Byleen received his B.S., M.A., and Ph.D degrees in mathematics from the

Uni-versity of Nebraska He is currently an Associate Professor in the Department of matics, Statistics, and Computer Science of Marquette University He has published a dozenresearch articles on the algebraic theory of semigroups and co-authored more than a dozenundergraduate mathematics textbooks with Raymond Barnett and Michael Ziegler

Mathe-Dave Sobecki earned a B.A in math education from Bowling Green State University, then

went on to earn an M.A and a Ph.D in mathematics from Bowling Green He is an ciate professor in the Department of Mathematics at Miami University in Hamilton, Ohio

asso-He has written or co-authored five journal articles, eleven books and five interactive CD-ROMs Dave lives in Fairfield, Ohio with his wife (Cat) and dogs (Macleod and Tessa).His passions include Ohio State football, Cleveland Indians baseball, heavy metal music, travel,and home improvement projects

About the Authors

Dedicated to the memory of Michael R Ziegler,

trusted author, colleague, and friend.

Preface xiv Features xvii Application Index xxviii

CHAPTER R Basic Algebraic Operations 1 CHAPTER 1 Equations and Inequalities 43 CHAPTER 2 Graphs 109

CHAPTER 3 Functions 161 CHAPTER 4 Polynomial and Rational Functions 259 CHAPTER 5 Exponential and Logarithmic Functions 327 CHAPTER 6 Trigonometric Functions 385

CHAPTER 7 Trigonometric Identities and Conditional Equations 461 CHAPTER 8 Additional Topics in Trigonometry 509

CHAPTER 9 Additional Topics in Analytic Geometry 571 CHAPTER 10 Systems of Equations and Matrices 625 CHAPTER 11 Sequences, Induction, and Probability 705

Appendix A Cumulative Review Exercises A1 Appendix B Special Topics A17

Appendix C Geometric Formulas A37 Student Answers SA1 Instructor Answers IA1 Subject Index I1

Brief Contents

Contents

Preface xiv

Features xvii

Applications Index xxviii

CHAPTER R Basic Algebraic

Operations 1R-1 Algebra and Real Numbers 2

R-2 Exponents and Radicals 11

R-3 Polynomials: Basic Operations and Factoring 21

R-4 Rational Expressions: Basic Operations 32

Chapter R Review 39

Chapter R Review Exercises 40

CHAPTER 1 Equations and

Inequalities 431-1 Linear Equations and Applications 44

1-2 Linear Inequalities 56

1-3 Absolute Value in Equations and Inequalities 65

1-4 Complex Numbers 74

1-5 Quadratic Equations and Applications 84

1-6 Additional Equation-Solving Techniques 97

Chapter 1 Review 104

Chapter 1 Review Exercises 106

Chapter 1 Group Activity: Solving a Cubic

Equation 108

CHAPTER 2 Graphs 109

2-1 Cartesian Coordinate Systems 110

2-2 Distance in the Plane 122

2-3 Equation of a Line 132

2-4 Linear Equations and Models 147

Chapter 2 Review 157

Chapter 2 Review Exercises 158

Chapter 2 Group Activity: Average Speed 160

Chapter 3 Review Exercises 252

Chapter 3 Group Activity:

Mathematical Modeling: Choosing a

Cell Phone Plan 257

CHAPTER 4 Polynomial and Rational

Functions 2594-1 Polynomial Functions, Division, and Models 260

4-2 Real Zeros and Polynomial Inequalities 278

4-3 Complex Zeros and Rational Zeros of Polynomials 288

4-4 Rational Functions and Inequalities 2984-5 Variation and Modeling 315

Chapter 4 Review 321Chapter 4 Review Exercises 323Chapter 4 Group Activity:

Interpolating Polynomials 326

CHAPTER 5 Exponential and Logarithmic

Functions 3275-1 Exponential Functions 3285-2 Exponential Models 3405-3 Logarithmic Functions 3545-4 Logarithmic Models 3655-5 Exponential and Logarithmic Equations 372Chapter 5 Review 379

Chapter 5 Review Exercises 380Chapter 5 Group Activity: Comparing Regression Models 383

CHAPTER 6 Trigonometric Functions 3856-1 Angles and Their Measure 386

6-2 Right Triangle Trigonometry 3956-3 Trigonometric Functions: A Unit Circle Approach 404

6-4 Properties of Trigonometric Functions 4146-5 More General Trigonometric Functionsand Models 428

6-6 Inverse Trigonometric Functions 441Chapter 6 Review 453

Chapter 6 Review Exercises 456Chapter 6 Group Activity: A Predator–Prey Analysis Involving Mountain Lions and Deer 460

CHAPTER 7 Trigonometric Identities and

Conditional Equations 4617-1 Basic Identities and Their Use 4627-2 Sum, Difference, and Cofunction Identities 4717-3 Double-Angle and Half-Angle Identities 4807-4 Product–Sum and Sum–Product Identities 488

7-5 Trigonometric Equations 493

Chapter 7 Review 504Chapter 7 Review Exercises 505

Chapter 7 Group Activity: From M sin Bt ⫹ N cos Bt to A sin (Bt ⫹ C)—A Harmonic Analysis

Tool 507

CHAPTER 8 Additional Topics in

Trigonometry 5098-1 Law of Sines 510

8-2 Law of Cosines 519

8-3 Vectors in the Plane 527

8-4 Polar Coordinates and Graphs 540

8-5 Complex Numbers and De Moivre’s Theorem 553

Chapter 8 Review 563Chapter 8 Review Exercises 567Chapter 8 Group Activity: Polar Equations of ConicSections 570

CHAPTER 9 Additional Topics in Analytic

Geometry 5719-1 Conic Sections; Parabola 572

9-2 Ellipse 581

9-3 Hyperbola 591

9-4 Translation and Rotation of Axes 604

Chapter 9 Review 620Chapter 9 Review Exercises 623Chapter 9 Group Activity: Focal Chords 624

CHAPTER 10 Systems of Equations

and Matrices 62510-1 Systems of Linear Equations 626

10-2 Solving Systems of Linear Equations Using

Gauss-Jordan Elimination 643

10-3 Matrix Operations 65910-4 Solving Systems of Linear Equations Using MatrixInverse Methods 672

10-5 Determinants and Cramer’s Rule 689

Additional Topics Available Online:

(Visit www.mhhe.com/barnett)10-6 Systems of Nonlinear Equations10-7 System of Linear In Equalities in Two Variables10-8 Linear Programming

Chapter 10 Review 698Chapter 10 Review Exercises 700Chapter 10 Group Activity: Modeling with Systems

of Linear Equations 703

CHAPTER 11 Sequences, Induction,

and Probability 70511-1 Sequences and Series 706

11-2 Mathematical Induction 71311-3 Arithmetic and Geometric Sequences 72211-4 Multiplication Principle, Permutations, and Combinations 733

11-5 Sample Spaces and Probability 74511-6 The Binomial Formula 760Chapter 11 Review 766Chapter 11 Review Exercises 768Chapter 11 Group Activity: Sequences Specified

Enhancing a Tradition of Success

The ninth edition of College Algebra with Trigonometry represents a substantial step

for-ward in student accessibility Every aspect of the revision of this classic text focuses on

making the text more accessible to students, while retaining the precise presentation of themathematics for which the Barnett name is renowned Extensive work has been done toenhance the clarity of the exposition, improving to the overall presentation of the content.This in turn has decreased the length of the text

Specifically, we concentrated on the areas of writing, exercises, worked examples, design,and technology Based on numerous reviews, advice from expert consultants, and direct cor-respondence with the many users of previous editions, this edition is more relevant and acces-sible than ever before

Writing Without sacrificing breadth or depth or coverage, we have rewritten explanations

to make them clearer and more direct As in previous editions, the text emphasizes tational skills, essential ideas, and problem solving rather than theory

compu-Exercises Over twenty percent of the exercises in the ninth edition are new These cises encompass both a variety of skill levels as well as increased content coverage, ensur-ing a gradual increase in difficulty level throughout In addition, brand new writing exer-cises have been included at the beginning of each exercise set in order to encourage a morethorough understanding of key concepts for students

exer-Examples Color annotations accompany many examples, encouraging the learning processfor students by explaining the solution steps in words Each example is then followed by asimilar matched problem for the student to solve Answers to the matched problems are located

at the end of each section for easy reference This active involvement in learning while readinghelps students develop a more thorough understanding of concepts and processes

Technology Instructors who use technology to teach college algebra with trigonometry,whether it be exploring mathematics with a graphing calculator or assigning homework andquizzes online, will find the ninth edition to be much improved

Refined “Technology Connections” boxes included at appropriate points in the text trate how problems previously introduced in an algebraic context may be solved using a graph-ing calculator Exercise sets include calculator-based exercises marked with a calculator icon.Note, however, that the use of graphing technology is completely optional with this text Weunderstand that at many colleges a single text must serve the purposes of teachers with widelydivergent views on the proper use of graphing and scientific calculators in college algebra withtrigonometry, and this text remains flexible regarding the degree of calculator integration.Additionally, McGraw-Hill’s MathZone offers a complete online homework system formathematics and statistics Instructors can assign textbook-specific content as well as cus-tomize the level of feedback students receive, including the ability to have students show

illus-their work for any given exercise Assignable content for the ninth edition of College Algebra

with Trigonometry includes an array of videos and other multimedia along with algorithmic

exercises, providing study tools for students with many different learning styles

Preface

Reflecting trends in the way college algebra with trigonometry is taught, the ninth tion emphasizes functions modeled in the real world more strongly than previous editions.

edi-In some cases, data are provided and the student is asked to produce an approximate responding function using regression on a graphing calculator However, as with previouseditions, the use of a graphing calculator remains completely optional and any such exam-ples or exercises can be easily omitted without loss of continuity

cor-Key Features

The revised full-color design gives the book a more contemporary feel and will appeal to

students who are accustomed to high production values in books, magazines, and nonprintmedia The rich color palette, streamlined calculator explorations, and use of color to sig-nify important steps in problem material work in conjunction to create a more visuallyappealing experience for students

An emphasis on mathematical modeling is evident in section titles such as “Linear

Equations and Models” and “Exponential Models.” These titles reflect a focus on the tionship between functions and real-world phenomena, especially in examples and exercises.Modeling problems vary from those where only the function model is given (e.g., when the

rela-model is a physical law such as F ⫽ ma), through problems where a table of data and the

function are provided, to cases where the student is asked to approximate a function fromdata using the regression function of a calculator or computer

Matched problems following worked examples encourage students to practice

prob-lem solving immediately after reading through a solution Answers to the matched probprob-lemsare located at the end of each section for easy reference

Interspersed throughout each section, Explore-Discuss boxes foster conceptual

under-standing by asking students to think about a relationship or process before a result is stated.Verbalization of mathematical concepts, results, and processes is strongly encouraged in theseexplanations and activities Many Explore-Discuss boxes are appropriate for group work.Refined Technology Connections boxes employ graphing calculators to show graph-

ical and numerical alternatives to pencil-and-paper symbolic methods for problem ing—but the algebraic methods are not omitted Screen shots are from the TI-84 Pluscalculator, but the Technology Connections will interest users of any automated graphingutility

solv-Think boxes (color dashed boxes) are used to enclose steps that, with some

experi-ence, many students will be able to perform mentally

Balanced exercise sets give instructors maximum flexibility in assigning homework A

wide variety of easy, moderate, and difficult level exercises presented in a range of lem types help to ensure a gradual increase in difficulty level throughout each exercise set.The division of exercise sets into A (routine, easy mechanics), B (more difficult mechan-ics), and C (difficult mechanics and some theory) is explicitly presented only in the Anno-tated Instructor’s Edition This is due to our attempt to avoid fueling students’ anxiety aboutchallenging exercises

prob-This book gives the student substantial experience in modeling and solving applied problems Over 500 application exercises help convince even the most skeptical student

that mathematics is relevant to life outside the classroom

An Applications Index is included following the Guided Tour to help locate

particu-lar applications

Most exercise sets include calculator-based exercises that are clearly marked with a

calculator icon These exercises may use real or realistic data, making them ally heavy, or they may employ the calculator to explore mathematics in a way that would

computation-be impractical with paper and pencil

As many students will use this book to prepare for a calculus course, examples and

exercises that are especially pertinent to calculus are marked with an icon

A Group Activity is located at the end of each chapter and involves many of the

con-cepts discussed in that chapter These activities require students to discuss and write aboutmathematical concepts in a complex, real-world context

Preface xv

Changes to this Edition

A more modernized, casual, and student-friendly writing style has been infused throughout the chapters without radically

changing the tone of the text overall This directly works toward a goal of increasing motivation for students to activelyengage with their textbooks, resulting in higher degrees of retention

A significant revision to the exercise sets in the new edition has produced a variety of important changes for both

stu-dents and instructors As a result, over twenty percent of the exercises are new These exercises encompass both a variety

of skill levels as well as increased content coverage, ensuring a gradual increase in difficulty level throughout In addition,brand new writing exercises have been included at the beginning of each exercise set in order to encourage a more thor-ough understanding of key concepts for students Specific changes include:

• The addition of hundreds of new writing exercises to the beginning of each exercise set These exercises encourage dents to think about the key concepts of the sections before attempting the computational and application exercises, ensur-ing a more thorough understanding of the material

stu-• An update to the data in many application exercises to reflect more current statistics in topics that are both familiar andhighly relevant to today’s students

• A significant increase the amount of moderate skill level problems throughout the text in response to the growing needexpressed by instructors

The number of colored annotations that guide students through worked examples has been increased throughout the

text to add clarity and guidance for students who are learning critical concepts

New instructional videos on graphing calculator operations posted on MathZone help students master the most

essen-tial calculator skills used in the college algebra course The videos are closed-captioned for the hearing impaired, subtitled

in Spanish, and meet the Americans with Disabilities Act Standards for Accessible Design Though these are an entirelyoptional ancillary, instructors may use them as resources in a learning center, for online courses, and to provide extra help

to students who require extra practice

Chapter R, “Basic Algebraic Operations,” has been extensively rewritten based upon feedback from reviewers to provide

a streamlined review of basic algebra in four sections rather than six Exponents and radicals are now covered in a singlesection (R-2), and the section covering operations on polynomials (R-3) now includes factoring

Chapter 10, “Systems of Equations and Matrices,” has been reorganized to focus on systems of linear equations, rather

than on systems of inequalities or nonlinear systems A section on determinants and Cramer’s rule (10-5) has been added.Three additional sections on systems of nonlinear equations, systems of linear inequalities, and linear programming are alsoavailable online

Design: A Refined Look with Your Students in Mind

The McGraw-Hill Mathematics Team has gathered a great deal of information about how to create a student-friendly book in recent years by going directly to the source—your students As a result, two significant changes have been made

text-to the design of the ninth edition based upon this feedback First, example headings have been pulled directly out intext-tothe margins, making them easy for students to find Additionally, we have modified the design of one of our existing fea-tures—the caution box—to create a more powerful tool for your students Described by students as one of the most use-ful features in a math text, these boxes now demand attention with bold red headings pulled out into the margin, alert-ing students to avoid making a common mistake These fundamental changes have been made entirely with the success

of your students in mind and we are confident that they will improve your students’ overall reaction to and enjoyment ofthe course

Tegrity Campus, a service that makes class time available all the time by automatically capturing every lecture in a

searchable format for students to review when they study and complete assignments, is an additional supplementarymaterial available with the new edition With a simple one-click start and stop process, you capture all computer screensand corresponding audio Students can then replay any part of any class with easy-to-use browser-based viewing on a

PC or Mac With Tegrity Campus, students quickly recall key moments by using Tegrity Campus’s unique search ture This search helps students efficiently find what they need, when they need it across an entire semester of classrecordings

Exploration and Discussion

Would you like to incorporate more discovery learning in

your course? Interspersed at appropriate places in every

section, Explore-Discuss boxes encourage students to

think critically about mathematics and to explore key

concepts in more detail Verbalization of

mathe-matical concepts, results, and processes is

encouraged in these Explore-Discuss boxes, as

well as in some matched problems, and in

prob-lems marked with color numerals in almost

every exercise set Explore-Discuss material can

be used in class or in an out-of-class activity

Examples and Matched Problems

Integrated throughout the text, completely worked

exam-ples and practice problems are used to introduce concepts

and demonstrate problem-solving techniques—algebraic,

graphical, and numerical Each example is followed by a

similar Matched Problem for the student to work

through while reading the material Answers to

the matched problems are located at the end of

each section for easy reference This active

involvement in the learning process helps

students develop a more thorough understanding

of algebraic concepts and processes

Z Midpoint of a Line Segment

The midpoint of a line segment is the point that is equidistant from each of the endp

A formula for finding the midpoint is given in Theorem 2 The proof is discussed i exercises.

Find the distance between the points (3, 5) and (2, 8).*

Let (x1, y1 )  (ⴚ3, 5) and (x2, y2 )  (ⴚ2, ⴚ8) Then,

Notice that if we choose (x1, y1 )  (2, 8) and (x 2, y2 )  (3, 5), then

so it doesn’t matter which point we designate as P1or P2

MATCHED PROBLEM 2 Find the distance between the points (6, 3) and (7, 5).

To graph the equation y ⫽ ⫺x3⫹ 2x, we use point-by-point plotting to obtain the

graph in Figure 5.

(A) Do you think this is the correct graph of

the equation? If so, why? If not, why?

(B) Add points on the graph for x ⫽ ⫺2,

⫺0.5, 0.5, and 2.

(C) Now, what do you think the graph looks

like? Sketch your version of the graph, adding more points as necessary.

(D) Write a short statement explaining any

conclusions you might draw from parts A,

B, and C.

ZZZ EXPLORE-DISCUSS 1

Z Figure 5

x y

⫺5

5

⫺5 5

Applications

One of the primary objectives of this book is to give the

student substantial experience in modeling and solving

real-world problems Over 500 application exercises help

convince even the most skeptical student that

mathemat-ics is relevant to everyday life An Applications

Index is included following the features to help

locate particular applications

54 L  8.8  5.1 log D for D (astronomy)

55 for t (circuitry)

56 for n (annuity)

The following combinations of exponential functions define four

of six hyperbolic functions, a useful class of functions in calculus

and higher mathematics Solve Problems 57–60 for x in terms of y The results are used to define inverse hyperbolic functions,

another useful class of functions in calculus and higher mathematics.

57 58

59 60

In Problems 61–68, use a graphing calculator to approximate to two decimal places any solutions of the equation in the interval

0  x  1 None of these equations can be solved exactly using

any step-by-step algebraic process.

70.COMPOUND INTEREST How many years, to the nearest year, will it take money to quadruple if it is invested at 6% compounded

ye x  e x2

ln (x  1)  ln (3x  1)  ln x

ln x  ln (2x  1)  ln (x  2) log (6x  5)  log 3  log 2  log x log x  log 5  log 2  log (x  3) log (x  3)  log (6  4x)

log x log 8  1 log 5 log x  2

20 5

119

18.9

No solution 3.14

Technology Connections

Technology Connections boxes integrated at

appropriate points in the text illustrate how

cepts previously introduced in an algebraic

con-text may be approached using a graphing

calculator Students always learn the algebraic

methods first so that they develop a solid grasp

of these methods and do not become

dependent The exercise sets contain

calculator-based exercises that are clearly marked with a

calculator icon The use of technology is

completely optional with this text All technology

features and exercises may be omitted without sacrificing

content coverage

Technology Connections

Figure 1 shows the details of constructing the logarithmic model of Example 5 on a graphing calculator.

0 0 100

Foundation for Calculus

As many students will use this book to prepare for a

calculus course, examples and exercises that are

especially pertinent to calculus are marked with

an icon

Group Activities

A Group Activity is located at the end of each chapter

and involves many of the concepts discussed in that

chap-ter These activities strongly encourage the

verbalization of mathematical concepts, results,

and processes All of these special activities are

highlighted to emphasize their importance CHAPTER 5

We have used polynomial, exponential, and logarithmic which equation provides the best fit for a given set of data? There mation about the type of data to help make a choice For example, length And we expect most populations to grow exponentially, at equations involves developing a measure of how closely an equa-

regres-ple Consider the data set in Figure 1, where L1 represents the x

data set is shown in Figure 2 Suppose we arbitrarily choose the

equation y1⫽ 0.6x ⫹ 1 to model these data (Fig 3).

Each of these differences is called a residual Note that three of

the residuals are positive and one is negative (three of the points

measure of the fit provided by a given model is the sum of the

(whether positive or negative or zero) makes a nonnegative tribution to the SSR.

con-(A) A linear regression model for the data in Figure 1 is given by

Compute the SSR for the data and y2 , and compare it to the

one we computed for y1

It turns out that among all possible linear polynomials, the

linear regression model minimizes the sum of the squares of the

called the least-squares line A similar statement can be made for

polynomials of any fixed degree That is, the quadratic regression bic regression model minimizes the SSR over all cubic polynomi- tial or logarithmic regression models Nevertheless, the SSR can models.

(B) Find the exponential and logarithmic regression models for

the data in Figure 1, compute their SSRs, and compare with the linear model.

(C) National annual advertising expenditures for selected years

since 1950 are shown in Table 1 where x is years since 1950

gression model would fit this data best: a quadratic model, a

y2⫽ 0.35x ⫹ 3

⫹ (7 ⫺ 5.8) 2 ⫽ 9.8 SSR⫽ (4 ⫺ 2.2) 2 ⫹ (5 ⫺ 3.4) 2 ⫹ (3 ⫺ 4.6) 2

Z Figure 1

Z Figure 2

Z Figure 3 y 1 ⫽ 0.6x ⫹ 1.

0 0 10

10

0 0 10

10

For f(x) ⫽ x2⫹ 4x ⫹ 5, find and simplify:

(A) f(x ⫹ h) (B) f(x ⫹ h) ⫺ f(x) (C) f(x ⫹ h) ⫺ f(x)

h , h ⫽ 0

SOLUTIONS (A) To find f(x ⫹ h), we replace x with x ⫹ h everywhere it appears in the equation that

defines f and simplify:

(B) Using the result of part A, we get

Student Aids

Annotation of examples and explanations, in

small colored type, is found throughout the text

to help students through critical stages Think

Boxes are dashed boxes used to enclose steps

that students may be encouraged to perform

mentally

Screen Boxes are used to highlight important

definitions, theorems, results, and step-by-step

processes

If a principal P is invested at an annual rate r compounded m times a year, then

the amount A in the account at the end of n compounding periods is given by

Note that the annual rate r must be expressed in decimal form, and that

where t is years. n  mt,

A  Pa1  mb r n

ZDEFINITION 1Increasing, Decreasing, and Constant Functions

Let I be an interval in the domain of function f Then,

1 f is increasing on I and the graph of f is rising on I if

ZTHEOREM 1Tests for Symmetry

Symmetry with An equivalent respect to the: equation results if:

y axis x is replaced with ⫺x

x axis y is replaced with ⫺y

Origin x and y are replaced with ⫺x and ⫺y

The domain of f is all x values except or The value of a fraction is 0 if and only if the numerator is zero:

Subtract 4 from both sides.

Divide both sides by ⴚ3.

Caution Boxes appear throughout the text to

indicate where student errors often occur ZZZ CAUTION ZZZ A very common error occurs about now—students tend to confuse algebraic

expres-sions involving fractions with algebraic equations involving fractions.

Consider these two problems:

(A) Solve: (B) Add:

The problems look very much alike but are actually very different To solve the tion in (A) we multiply both sides by 6 (the LCD) to clear the fractions This works

equa-so well for equations that students want to do the same thing for problems like (B) The only catch is that (B) is not an equation, and the multiplication property of equal- ity does not apply If we multiply (B) by 6, we simply obtain an expression 6 times

as large as the original! Compare these correct solutions:

⫽3x

6 ⫹2x

6 ⫹606

⫽33ⴢ xⴢ 2⫹22ⴢ xⴢ 3⫹66ⴢ 10ⴢ 1

x

2 ⫹x3⫹ 10

Cumulative Review Exercise Sets are

provided in Appendix A for additional

reinforcement of key concepts

Chapter Review sections are provided at the

end of each chapter and include a thorough

review of all the important terms and symbols

This recap is followed by a comprehensive set

of review exercises

Work through all the problems in this cumulative review and problems are there, and following each answer is a number in italics indicating the section in which that type of problem is discussed Where weaknesses show up, review appropriate sections in the text.

10 Given the points A  (3, 2) and B  (5, 6), find:

(A) Distance between A and B.

(B) Slope of the line through A and B.

(C) Slope of a line perpendicular to the line through A and B.

11 Find the equation of the circle with radius and center:

(A) (0, 0) (B) (3, 1)

12 Graph 2x  3y  6 and indicate its slope and intercepts.

13 Indicate whether each set defines a function Find the domain

and range of each function.

Problems 16–18 refer to the function f given by the graph:

16 Find the domain and range of f Express answers in interval

notation.

17 Is f an even function, an odd function, or neither? Explain.

18 Use the graph of f to sketch a graph of the following:

25 For what real values of x does the following expression

represent a real number?

x

5

5

5 5

x  3  17 x  3 (1-6)

No solution (1-1) (1-6)

x  1,5

(1-1)

x 1 , 3

Domain: [2, 3]; range: [1, 2] (3-2) Neither (3-3)

5-1 Exponential Functions

The equation f (x)  b x , b  0, b  1, defines an exponential

func-tion with base b The domain of f is (, ) and the range is

(0, ) The graph of f is a continuous curve that has no sharp

cor-ners; passes through (0, 1); lies above the x axis, which is a tal asymptote; increases as x increases if b  1; decreases as x increases if b 1; and intersects any horizontal line at most once.

horizon-The function f is one-to-one and has an inverse We often use the

following exponential function properties:

3 For x  0, a x  b x if and only if a b.

As x approaches , the expression [1 (1兾x)] x

approaches the

ir-rational number e ⬇ 2.718 281 828 459 The function f(x)  e xis

called the exponential function with base e The growth of money

in an account paying compound interest is described by

A  P(1 r兾m) n

, where P is the principal, r is the annual rate, m

is the number of compounding periods in 1 year, and A is the

amount in the account after n compounding periods.

If the account pays continuous compound interest, the

amount A in the account after t years is given by A  Pe rt

.

5-2 Exponential Models

Exponential functions are used to model various types of growth:

1 Population growth can be modeled by using the doubling time

growth model where A is the population at time t,

is the population at time t 0,and d is the doubling time—

time zero and k is a positive constant called the relative growth

used for many other types of quantities that exhibit exponential growth as well.

2 Radioactive decay can be modeled by using the half-life decay

model where A is the amount at time t,

is the amount at time and h is the half-life—the time it

takes for half the material to decay Another model of radioactive decay, , where is the amount at time

zero and k is a positive constant, uses the exponential function

that exhibit negative exponential growth as well.

3 Limited growth—the growth of a company or proficiency at

learning a skill, for example—can often be modeled by the equation where A and k are positive constants.

Logistic growth is another limited growth model that is useful

for modeling phenomena like the spread of an epidemic, or sales of a new product The logistic model is where c, k, and M are positive constants A good comparison of these different

Exponential regression can be used to fit a function of the form to a set of data points Logistic regression can be used to find a function of the form

5-3 Logarithmic Functions

The logarithmic function with base b is defined to be the inverse

of the exponential function with base b and is denoted by y log b x.

So y logb x if and only if x  b y , b 0, b  1 The domain of a

logarithmic function is (0, ) and the range is (, ) The graph

of a logarithmic function is a continuous curve that always passes

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sta-MathZone also helps ensure consistent assignment delivery across several sectionsthrough a course administration function and makes sharing courses with other instruc-tors easy In addition, instructors can also take advantage of a virtual whiteboard by set-ting up a Live Classroom for online office hours or a review session with students.For more information, visit the book’s website (www.mhhe.com/barnett) or contactyour local McGraw-Hill sales representative (www.mhhe.com/rep)

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Preface xxv

Tegrity Campus’s unique search feature This search helps students efficiently find what theyneed, when they need it across an entire semester of class recordings Help turn all yourstudents’ study time into learning moments immediately supported by your lecture

To learn more about Tegrity watch a 2 minute Flash demo at http://tegritycampus.mhhe.com

Instructor Solutions Manual Prepared by Fred Safier of City College of San Francisco,this supplement provides detailed solutions to exercises in the text The methods used tosolve the problems in the manual are the same as those used to solve the examples in thetextbook

Student Solutions Manual Prepared by Fred Safier of City College of San Francisco,

the Student’s Solutions Manual provides complete worked-out solutions to odd-numbered

exercises from the text The procedures followed in the solutions in the manual matchexactly those shown in worked examples in the text

Lecture and Exercise Videos The video series is based on exercises from the textbook

J D Herdlick of St Louis Community College-Meramec introduces essential definitions,theorems, formulas, and problem-solving procedures Professor Herdlick then works throughselected problems from the textbook, following the solution methodologies employed by theauthors The video series is available on DVD or online as part of MathZone The DVDsare closed-captioned for the hearing impaired, subtitled in Spanish, and meet the Americanswith Disabilities Act Standards for Accessible Design

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Computerized Test Bank (CTB) Online Available through the book’s website, this puterized test bank, utilizing Brownstone Diploma® algorithm-based testing software,enables users to create customized exams quickly This user-friendly program enablesinstructors to search for questions by topic, format, or difficulty level; to edit existing ques-tions or to add new ones; and to scramble questions and answer keys for multiple versions

com-of the same test Hundreds com-of text-specific open-ended and multiple-choice questions areincluded in the question bank Sample chapter tests and a sample final exam in MicrosoftWord®and PDF formats are also provided

xxvi Preface

Acknowledgments

In addition to the authors, many others are involved in the successful publication of a book

We wish to thank personally the following people who reviewed the text and offered able advice for improvements:

invalu-Marwan Abu-Sawwa, Florida Community College at JacksonvilleGerardo Aladro, Florida International University

Eugene Allevato, Woodbury UniversityJoy Becker, University of Wisconsin–StoutSusan Bradley, Angelina College

Ellen Brook, Cuyahoga Community College, Eastern CampusKelly Brooks, Pierce College

Denise Brown, Collin County Community CollegeCheryl Davids, Central Carolina Technical CollegeTimothy Delworth, Purdue University

Marcial Echenique, Broward Community CollegeGay Ellis, Missouri State University

Jackie English, Northern Oklahoma College EnidMike Everett, Santa Ana College

Nicki Feldman, Pulaski Technical CollegeJames Fightmaster, Virginia Western Community CollegePerry Gillespie, University of North Carolina at FayettevilleVanetta Grier-Felix, Seminole Community College

David Gurney, Southeastern Louisiana UniversityCeleste Hernandez, Richland College

Fredrick Hoffman, Florida Atlantic UniversitySyed Hossain, University of Nebraska at KearneyGlenn Jablonski, Triton College

Sarah Jackson, Pratt Community CollegeCharles Johnson, South Georgia CollegeLarry Johnson, Metropolitan State College of DenverCheryl Kane, University of Nebraska LincolnRaja Khoury, Collin County Community CollegeBetty Larson, South Dakota State UniversityOwen Mertens, Southwest Missouri State UniversityDana Nimic, Southeast Community College, Lincoln CampusLyn Noble, Florida Community College at JacksonvilleLuke Papademas, DeVry University, DeVry Chicago CampusDavid Phillips, Georgia State University

Margaret Rosen, Mercer County Community CollegePatty Schovanec, Texas Tech University

Eleanor Storey, Front Range Community College, Westminster CampusLinda Sundbye, Metropolitan State College of Denver

Cynthia Woodburn, Pittsburg State UniversityMartha Zimmerman, University of LouisvilleBob Martin, Tarrant County College

Susan Walker, Montana Tech of the University of MontanaLynn Cleaveland, University of Arkansas

Michael Wodzak, Viterbo UniversityRyan Kasha, Valencia Community CollegeFrank Juric, Brevard Community CollegeJerry Mayfield, North Lake CollegeAndrew Shiers, Dakota State UniversityRichard Avery, Dakota State University

Preface xxvii

Mike Everett, Santa Ana CollegeGreg Boyd, Murray State CollegeSarah Cook, Washburn UniversityNga Wai Liu, Bowling Green State UniversityDonald Bennett, Murray State UniversitySharon Suess, Asheville-Buncombe Technical Community CollegeDale Rohm, University of Wisconsin at Stevens Point

George Anastassiou, The University of MemphisBill White, University of South Carolina UpstateLinda Sundbye, Metropolitan State College of DenverKhaled Hussein, University of Wisconsin

Diane Cook, Okaloosa Walton CollegeCeleste Hernandez, Richland CollegeThomas Riedel, University of LouisvilleThomas English, College of the MainlandHayward Allan Edwards, West Virginia University at ParkersburgDebra Lehman, State Fair Community College

Nancy Ressler, Oakton Community CollegeMarwan Zabdawi, Gordonn CollegeIanna West, Nicholls State UniversityTzu-Yi Alan Yang, Columbus State Community CollegePatricia Jones, Methodist University

Kay Geving, Belmont UniversityLinda Horner, Columbia State Community CollegeMartha Zimmerman, University of LouisvilleFaye Childress, Central Piedmont Community CollegeBradley Thiessen, Saint Ambrose University

Pamela Lasher, Edinboro University of Pennsylvania

We also wish to thankCarrie Green for providing a careful and thorough check of all the mathematical calculations

in the book (a tedious but extremely important job)

Fred Safier for developing the supplemental manuals that are so important to the success of atext

Mitchel Levy for scrutinizing our exercises in the manuscript and making recommendationsthat helped us to build balanced exercise sets

Tony Palermino for providing excellent guidance in making the writing more direct andaccessible to students

Pat Steele for carefully editing and correcting the manuscript

Christina Lane for editorial guidance throughout the revision process

Sheila Frank for guiding the book smoothly through all publication details

All the people at McGraw-Hill who contributed their efforts to the production of this book.Producing this new edition with the help of all these extremely competent people has been

a most satisfying experience

Cell phone cost, 174

Cell phone subscribers, 382

Cost of high speed internet, 174Counting card hands, 742Counting code words, 736Counting serial numbers, 742–743Court design, 102

Crime statistics, 326Cryptography, 684–685, 688, 703

Data analysis, 160, 346Daylight hours, 503Delivery charges, 187, 658Demographics, 147Depreciation, 155–156, 159, 353, A–4Design, 104, 107, 590

Diamond prices, 152–153Dice roles, 748, 756, 759Diet, 702

Distance-rate-time problems, 50–51, 92Divorce, 277

Earthquakes, 366–368, 371, 376, 379, 382, A–7Earth science, 55, 64, 352, 394, 642

Ecology, 371Economics, 20, 42, 55, 64, 732, 769Economy stimulation, 729–730Electrical circuit, 42, 439, 440, A–11Electric current, 502, 506

Electricity, 320Employee training, 314Energy, 64

Engineering, 96, 132, 220–221, 321, 391, 394, 395, 403,

427, 428, 440, 452, 507, 518, 526, 527, 569, 580, 590,

624, 732, A–11Epidemics, 345–346Evaporation, 203, 234Explosive energy, 371Eye surgery, 503

Fabrication, 298Falling object, 220, 256Finance, 339, 642, 732, A–4Fire lookout, 518

Flight conditions, 156Flight navigation, 156Fluid flow, 203, 234Food chain, 732Forces, 569Forestry, 155, 160

Gaming, 351Gas mileage, 220

APPLICATION INDEX

xxviii

Genealogy, 732

Genetics, 321

Geometry, 31, 55, 103, 287, 321, 401, 403, 404, 428, 459,

479, 487, 503, 526, 527, 552, 658, 689, 733, A–7,A–11, A–12

Nutrition, 658, 671

Oceanography, 146–147Officer selection, 739Olympic games, 157Optics, 502–503Ozone levels, 113

Packaging, 31, 298Parachutes, 156Pendulum, 21Petty crime, 657Photography, 321, 352, 378, 395, 452, 733Physics, 122, 146, 320–321, 326, 403, 404, 427, 428, 440,

502, 712, 732, A–4, A–8Physiology, 314–315Player ranking, 672Politics, 107, 671Pollution, 234, 440Population growth, 340–341, 351–352, 378, 381, 732, A–7Predator-prey analysis, 460

Present value, 339, 382Price and demand, 93, 95, 121, 249–250, 256, A–4 – A–5Price and supply, 121, 250

Prize money, 726Production costs, 202, 670Production rates, 642Production scheduling, 638–639, 642–643, 658, 664, 688Profit and loss analysis, 213–214, 220, 221–222, 230–231,A–4

Projectile flight, 220Projectile motion, 211, A–27 – A–28, A–31Psychology, 56, 64, 321

Purchasing, 654–655Puzzle, 703, 732–733

Quality control, 770Quantity-rate-time problems, 51–52

Radian measure, 394Radioactive decay, 342–343Radioactive tracers, 351Rate of descent, 156Rate problems, 174Rate-time, A–4Rate-time problems, 55–56, 107, 641Regression, 346

Relativistic mass, 21Replacement time, 315Resolution of forces, 539Resource allocation, 658, 688, 702Restricting access, 459

Resultant force, 534–535, 539Retention, 315

Revenue, 242–243, 277, 698Rocket flight, 368–369

APPLICATION INDEX xxix

Safety research, 203

Sailboat racing, 553

Salary increment, 712

Sales and commissions, 187, 662–663

Search and rescue, 526

Seasonal business cycles, 459

Telephone charges, 187Telephone expenditures, 153–154Temperature, 122, 435–436, 459, A–12Temperature variation, 441

Thumbtack toss, 754Timber harvesting, 202–203Tournament seeding, 671–672Traffic flow, 703–704

Training, 353Transportation, 96, 769Underwater pressure, 151

Weather, 175Weather balloon, 234Weight of fish, 271Well depth, 103Wildlife management, 353, 382Wind power, 392

Work, 326Zeno’s paradox, 733xxx APPLICATION INDEX

College Algebra with Trigonometry

Basic Algebraic

Operations

ALGEBRA is “generalized arithmetic.” In arithmetic we add, subtract,

multiply, and divide specific numbers In algebra we use all that we

know about arithmetic, but, in addition, we work with symbols that

represent one or more numbers In this chapter we review some

im-portant basic algebraic operations usually studied in earlier courses

C

CHAPTER R OUTLINE

R-1 Algebra and Real Numbers

R-2 Exponents and Radicals

R-3 Polynomials: Basic Operationsand Factoring

R-4 Rational Expressions: BasicOperations

Chapter R Review

2 C H A P T E R R BASIC ALGEBRAIC OPERATIONS

Z The Set of Real Numbers

Z The Real Number Line

Z Addition and Multiplication of Real Numbers

Z Further Operations and Properties

The numbers are examples of real numbers Because the symbols

we use in algebra often stand for real numbers, we will discuss important properties of thereal number system

Informally, a real number is any number that has a decimal representation So the real

numbers are the numbers you have used for most of your life The set of real numbers,

denoted by R, is the collection of all real numbers The notation (read “ is an

element of R”) expresses the fact that is a real number The set Z { , 2, 1,

0, 1, 2, } of the natural numbers, along with their negatives and zero, is called the set

of integers We write (read “Z is a subset of R”) to express the fact that every

ele-ment of Z is an eleele-ment of R; that is, that every integer is a real number Table 1 describes

the set of real numbers and some of its important subsets Study Table 1 and note in

par-ticular that N Z Q R.

No real number is both rational and irrational, so the intersection (overlap) of the sets

Q and I is the empty set (or null set), denoted by .The empty set contains no elements,

(((

Table 1 The Set of Real Numbers

N Natural numbers Counting numbers (also called positive 1, 2, 3,

integers)

Z Integers Natural numbers, their negatives, and 0 , 2, 1, 0, 1, 2,

(also called whole numbers)

Q Rational numbers Numbers that can be represented as a 兾b,

where a and b are integers and

decimal representations are repeating or terminating

nonrepeating and nonterminating decimal 2.71828182 †

numbers

R Real numbers Rational numbers and irrational numbers

*The overbar indicates that the number (or block of numbers) repeats indefinitely.

†Note that the ellipsis does not indicate that a number (or block of numbers) repeats indefinitely.

12, , 1 3 7, 1.414213 ,

5.272727

b 0; 4, 0, 1, 25, 

3 , 2 , 3.67, 0.333,*

so it is true that every element of the empty set is an element of any given set In otherwords, the empty set is a subset of every set.

Two sets are equal if they have exactly the same elements The order in which the

ele-ments of a set are listed does not matter For example,

{1, 2, 3, 4}  {3, 1, 4, 2}

A one-to-one correspondence exists between the set of real numbers and the set of points

on a line That is, each real number corresponds to exactly one point, and each point toexactly one real number A line with a real number associated with each point, and viceversa, as in Figure 1, is called a real number line, or simply a real line Each number asso-

ciated with a point is called the coordinate of the point The point with coordinate 0 is

called the origin The arrow on the right end of the line indicates a positive direction The

coordinates of all points to the right of the origin are called positive real numbers, and

those to the left of the origin are called negative real numbers The real number 0 is

nei-ther positive nor negative

S E C T I O N R – 1 Algebra and Real Numbers 3

Z Figure 1 A real number line.

7.64 Origin

␲

兹27 43

How do you add or multiply two real numbers that have nonrepeating and nonterminatingdecimal expansions? The answer to this difficult question relies on a solid understanding

of the arithmetic of rational numbers The rational numbers are numbers that can be

writ-ten in the form a 兾b, where a and b are integers and b  0 (see Table 1 on page 2) The

numbers 7兾5 and 2兾3 are rational, and any integer a is rational because it can be ten in the form a 兾1 Two rational numbers a兾b and c兾d are equal if ad  bc; for example,

writ-35兾10  7兾2 Recall how the sum and product of rational numbers are defined:

Z DEFINITION 1 Addition and Multiplication of Rationals

For rational numbers a 兾b and c兾d, where a, b, c, and d are integers and

Addition and multiplication of rational numbers are commutative; changing the order in

which two numbers are added or multiplied does not change the result

Addition is commutative.

Multiplication is commutative.

Addition and multiplication of rational numbers is also associative; changing the grouping

of three numbers that are added or multiplied does not change the result:

Addition is associative.

Multiplication is associative.

Furthermore, the operations of addition and multiplication are related in that multiplication

distributes over addition:

Left distributive law

Right distributive law

The rational number 0 is an additive identity; adding 0 to a number does not change

it The rational number 1 is a multiplicative identity; multiplying a number by 1 does not

change it Every rational number r has an additive inverse, denoted r; the additive inverse

of 4兾5 is 4兾5, and the additive inverse of 3兾2 is 3兾2 The sum of a number and its

addi-tive inverse is 0 Every nonzero rational number r has a multiplicative inverse, denoted

r1; the multiplicative inverse of 4兾5 is 5兾4, and the multiplicative inverse of 3兾2 is 2兾3.The product of a number and its multiplicative inverse is 1 The rational number 0 has nomultiplicative inverse

4 C H A P T E R R BASIC ALGEBRAIC OPERATIONS

EXAMPLE 1 Arithmetic of Rational Numbers

Perform the indicated operations

(A) (B) (C) (D) (179)1 (6  92)1

12 ⴝ103

S E C T I O N R – 1 Algebra and Real Numbers 5

MATCHED PROBLEM 1* Perform the indicated operations

(A) (B) (C) (D)

exam-The number 6 repeats indefinitely.

The block 142857 repeats indefinitely.

Terminating expansion

Conversely, any decimal expansion that is repeating or terminating represents a rationalnumber (see Problems 49 and 50 in Exercise R-1)

The number is irrational because it cannot be written in the form a 兾b, where a and

b are integers, (for an explanation, see Problem 89 in Section R-3) Similarly, isirrational But which is equal to 2, is a rational number In fact, if n is a positive integer,

then is irrational unless n belongs to the sequence of perfect squares 1, 4, 9, 16, 25,

(see Problem 90 in Section R-3)

We now return to our original question: how do you add or multiply two real bers that have nonrepeating and nonterminating decimal expansions? Although we willnot give a detailed answer to this question, the key idea is that every real number can

num-be approximated to any desired precision by rational numnum-bers For example, the tional number

irra-is approximated by the rational numbers

.Using the idea of approximation by rational numbers, we can extend the definitions ofrational number operations to include real number operations The following box summa-rizes the basic properties of real number operations

141,421100,000 1.41421

14,14210,000  1.4142

1,4141,000 1.414

6 C H A P T E R R BASIC ALGEBRAIC OPERATIONS

Z BASIC PROPERTIES OF THE SET OF REAL NUMBERS

Let R be the set of real numbers, and let x, y, and z be arbitrary elements of R.

Addition Properties

Associative:

Commutative:

Identity: 0 is the additive identity; that is, 0  x  x  0  x for all

x in R, and 0 is the only element in R with this property.

Inverse: For each x in R, is its unique additive inverse; that is,

and x is the only element in R relative to x with this property.

Multiplication Properties

Associative:

Commutative:

Identity: 1 is the multiplicative identity; that is, for all x in R,

(1)x  x(1)  x, and 1 is the only element in R with this

property

inverse; that is, and is the only element

in R relative to x with this property.

x  (x)  (x)  x  0, x

x  y  y  x (x  y)  z  x  (y  z)

x  y

EXAMPLE 2 Using Real Number Properties

Which real number property justifies the indicated statement?

(A)(B)(C)(D)(E) If then a  b  0, b  a.

(x  y)(a  b)  (x  y)a  (x  y)b (2x  3y)  5y  2x  (3y  5y)

a(b  c)  (b  c)a (7x)y  7(xy)

SOLUTIONS (A) Associative (ⴢ)

(B) Commutative (ⴢ)(C) Associative ()(D) Distributive

It is important to remember that

Division by 0 is never allowed.

S E C T I O N R – 1 Algebra and Real Numbers 7

MATCHED PROBLEM 2 Which real number property justifies the indicated statement?

(E) If then



Subtraction of real numbers can be defined in terms of addition and the additive inverse If

a and b are real numbers, then is defined to be Similarly, division can be

defined in terms of multiplication and the multiplicative inverse If a and b are real

num-bers and b 0, then a b (also denoted a 兾b) is defined to be a ⴢ b1

(a  b)  c  c  (a  b)

4 (2  x)  (4  2)  x

Z DEFINITION 2 Subtraction and Division of Real Numbers

For all real numbers a and b:

sub-Z THEOREM 1 Properties of Negatives

For all real numbers a and b: