Book Contemporary Precalculus by Thomas W. Hungerford pdf - Web Education

Distance Formula

Midpoint Formula

Slope of nonvertical line Equation of line with Equation of line with slope m

through and _{slopem through and y-intercept b}

y ! mx " b y # y1!m1x # x12 y2#y1 x2#x1 1x1,y12 1x2,y22 1x1, y12

a

x1"x2 2 , y1"y2 2

b

Midpoint M of segment PQ ! 21x1#x222"1y1#y222 Length of segment PQ ! b a c b h

Change of Base Formula

logbv !ln v_{ln b}

r

Natural Logarithms

Logarithms to Base

b

Special Notation

For v, w 0 and any u, k: For v, and any u, k: ln v means

means

Geometry

The Pythagorean Theorem

Area of a Triangle

Circles

Area ! pr2 Circumference ! 2pr Diameter ! 2r A !1 2bh c2_!a2_"b2 logb1vk2 ! k1logbv2 ln1vk_{2 ! k1ln v2}

logba_wv

b

!logbv #logbw ln

a

v

w

b

!ln v # ln w

logb1vw2 ! logbv "logbw ln1vw2 ! ln v " ln w log10v log v logbv ! u means bu!v ln v ! u means eu_!v logev w 7 0 7

The Quadratic Formula

are the solutions of ax2_{"bx " c !0}.

x !#b ± 2b2#4ac

2a

Equations and Graphs

The solutions of the equation are the x-intercepts of the graph of y ! f1x2.

f1x2 ! 0

Exponents

1c $ 02 c#r_! 1 cr 1d $ 02

a

c_d

b

r! c r dr 1cd2r_!cr_dr 1cr₂s_!crs cr cs!cr#s cr_cs_!cr"s

Multiplication and Factoring

u3_"v3_{! 1u " v21u}2_{#uv " v}2₂ u3_#v3_{! 1u # v21u}2_{"uv " v}2₂ 1u ; v22_!u2_{; 2uv " v}2 u2_#v2_{! 1u " v21u # v2} M P Q (x1, y1) (x2, y2)

Slope = m < 0 f(x)= mx + b Slope = m > 0 f(x)= mx + b x y Linear Functions x y Identity Function f(x)= x x y Constant Function f(x)= b x y b x y Cube Function f(x)= x3 x y Square Function f(x)= x2 x y

Square Root Function f(x)= x

Greatest Integer Function f(x)= [x] Absolute Value Function f(x)= x

x y x y 1 1 f(x)= xn _{(n even) f(x)= x}n _{(n odd)} x y Reciprocal Functions Power Functions x y x y x y f(x)_{= 1x} f(x)= 1_x2 f(x)= bx _{(b> 1) f(x)= b}x _{(0< b < 1)} Exponential Functions 1 x y x y 1 1 Logarithmic Functions x y x y 1 f(x)= log x f(x)= ln x ContinuesS

Rectangular and Parametric Equations for Conic Sections

Circles Ellipse Parabola

Center (h, k), radius r Center (h, k) Vertex (h, k)

1x # h22_{!4p1y # k2} 1x # h22 a2 " 1y # k22 b2 !1 1x # h22_{" 1y # k2}2_!r2 k h k h k h

y ! rx ! r sin t " k cos t " h 10 % t % 2p2 y ! bx ! a sin t " k cos t " h 10 % t % 2p2 (t any real)

y ! 1t # h2 2 4p "k

x ! t

Parabola Hyperbola Hyperbola

Vertex (h, k) Center (h, k) Center (h, k)

1y # k22 a2 # 1x # h22 b2 !1 1x # h22 a2 # 1y # k22 b2 !1 1y # k22_{!4p1x # h2} k h k h h k (t any real) y ! t x ! 1t # k2 2 4p "h y ! b tan t " k 10%t %2p2 x ! a cos t"h y ! a cos t"k 10 % t % 2p2 x ! b tan t " h 2π −2π 1 f(t)= sin t −1 t y 2π −2π 1 −1 t y 2π −2π t y f(t)= cos t f(t)= tan t Trigonometric Functions

Contemporary Precalculus

A Graphing Approach

5e

A Graphing Approach

5e

THOMAS W. HUNGERFORD

Saint Louis University

DOUGLAS J. SHAW

University of Northern Iowa

Australia • Brazil • Canada • Mexico • Singapore • Spain United Kingdom • United States

© 2009, 2004 Thomson Brooks/Cole, a part of The Thomson Corpora-tion. Thomson, the Star logo, and Brooks/Cole are trademarks used herein under license.

ALL RIGHTS RESERVED. No part of this work covered by the copy-right hereon may be reproduced or used in any form or by any means— graphic, electronic, or mechanical, including photocopying, recording, taping, web distribution, information storage and retrieval systems, or in any other manner—without the written permission of the publisher. Printed in the United States of America

1 2 3 4 5 6 7 11 10 09 08

ExamView® _{and ExamView Pro}® _{are registered trademarks of}

FSCreations, Inc. Windows is a registered trademark of the Microsoft Corporation used herein under license. Macintosh and Power Macin-tosh are registered trademarks of Apple Computer, Inc. Used herein under license.

© 2009 Thomson Learning, Inc. All Rights Reserved. Thomson Learn-ing WebTutorTM_{is a trademark of Thomson Learning, Inc.}

Library of Congress Control Number: 2007939806

STUDENT EDITION: ISBN-13: 978-0-495-10833-7 ISBN-10: 0-495-10833-2

Thomson Higher Education

10 Davis Drive

Belmont, CA 94002-3098 USA

Thomas W. Hungerford, Douglas J. Shaw

Mathematics Editor: Gary Whalen

Development Editors: Leslie Lahr, Kari Hopperstead Assistant Editor: Natasha Coats

Editorial Assistant: Rebecca Dashiell Technology Project Manager: Lynh Pham Marketing Manager: Joe Rogove Marketing Assistant: Ashley Pickering

Marketing Communications Manager: Darlene Amidon-Brent Project Manager, Editorial Production: Hal Humphrey Art Director: Vernon Boes

Print Buyer: Karen Hunt

Production Service: Laura Horowitz / Hearthside Publishing Services

Text Designer: Terri Wright / Terri Wright Design Photo Researcher: Terri Wright / Terri Wright Design Copy Editor: Grace Lefrancois

Illustrator: Hearthside Publishing Services Cover Designer: Terri Wright / Terri Wright Design

Cover Image: (front: Grand Canyon Skywalk, Hualapai Reservation, Arizona) ART FOXALL / UPI / Landov; (back) AP / Wide World

Photos

Cover Printer: R. R. Donnelley / Willard Compositor: ICC Macmillan Inc. Printer: R. R. Donnelley / Willard

For more information about our products, contact us at: Thomson Learning Academic Resource Center

1-800-423-0563

For permission to use material from this text or product, submit a request online at

http://www.thomsonrights.com.

Any additional questions about permissions can be submitted by e-mail to thomsonrights@thomson.com.

Whose presence in my life has greatly enriched it: Grace Parks Hungerford

Irene Parks Mills Florence M. Parks Ellen McGillicuddy

❖ ❖ ❖

For my daughter Francebelle Shaw, whose response to “Stop being so cute, I seriously can’t take it”

vii

Contents

Preface xi

To the Instructor xiv Ancillaries xvii To the Student xix

chapter

1

Basics 1

1.1

The Real Number System 2

1.1.A

Special Topics: Decimal Representation of Real Numbers 16

1.2

Solving Equations Algebraically 19

1.2.A

Special Topics: Absolute Value Equations 32

1.2.B

Special Topics: Variation 33

1.3

The Coordinate Plane 39

1.4

Lines 53

CHAPTER 1 REVIEW 69 ■ DISCOVERY PROJECT 1 75

chapter

2

Graphs and Technology 77

2.1

Graphs 78

2.2

Solving Equations Graphically and Numerically 92

2.3

Applications of Equations 101

2.4

Optimization Applications 114

2.5

Linear Models 120

CHAPTER 2 REVIEW 131 ■ DISCOVERY PROJECT 2 137

chapter

3

Functions and Graphs 141

3.1

Functions 142

3.2

Functional Notation 151

3.3

Graphs of Functions 161

3.3.A

Special Topics: Parametric Graphing 175

3.4

Graphs and Transformations 179

3.4.A

Special Topics: Symmetry 189

3.5

Operations on Functions 195

3.6

Rates of Change 204

3.7

Inverse Functions 217

CHAPTER 3 REVIEW 228 DISCOVERY PROJECT 3 236

(3,−1) (2, 4) x y 26 8 −6 −3 3 3 5 !5 !3

chapter

4

Polynomial and Rational Functions 239

4.1

Quadratic Functions 240

4.2

Polynomial Functions 250

4.2.A

Special Topics: Synthetic Division 259

4.3

Real Roots of Polynomials 262

4.4

Graphs of Polynomial Functions 270

4.4.A

Special Topics: Polynomial Models 283

4.5

Rational Functions 288

4.5.A

Special Topics: Other Rational Functions 304

4.6

Polynomial and Rational Inequalities 308

4.6.A

Special Topics: Absolute Value Inequalities 317

4.7

Complex Numbers 321

4.8

Theory of Equations 328

CHAPTER 4 REVIEW 334 ■ DISCOVERY PROJECT 4 340

chapter

5

Exponential and Logarithmic Functions 341

5.1

Radicals and Rational Exponents 342

5.1.A

Special Topics: Radical Equations 350

5.2

Exponential Functions 357

5.2.A

Special Topics: Compound Interest and the Number e 369

5.3

Common and Natural Logarithmic Functions 375

5.4

Properties of Logarithms 385

5.4.A

Special Topics: Logarithmic Functions to Other Bases 392

5.5

Algebraic Solutions of Exponential and Logarithmic Equations 399

5.6

Exponential, Logarithmic, and Other Models 409

CHAPTER 5 REVIEW 419 ■ DISCOVERY PROJECT 5 425

chapter

6

Trigonometric Functions 427

6.1

Angles and Their Measurement 428

6.1.A

Special Topics: Arc Length and Angular Speed 435

6.2

The Sine, Cosine, and Tangent Functions 442

6.2

Alternate: The Sine, Cosine, and Tangent Functions 452

6.3

Algebra and Identities 457

6.4

Basic Graphs 466

6.5

Periodic Graphs and Simple Harmonic Motion 477

6.5.A

Special Topics: Other Trigonometric Graphs 490

6.6

Other Trigonometric Functions 496

CHAPTER 6 REVIEW 505 ■ DISCOVERY PROJECT 6 512

x y 1 3 (0, 0) (3, 1) x y _{y= x} 1 1 g(x) f(x) AM signal

chapter

7

Trigonometric Identities and Equations 513

7.1

Basic Identities and Proofs 514

7.2

Addition and Subtraction Identities 523

7.2.A

Special Topics: Lines and Angles 532

7.3

Other Identities 535

7.4

Inverse Trigonometric Functions 545

7.5

Trigonometric Equations 555

CHAPTER 7 REVIEW 567 ■ DISCOVERY PROJECT 7 571

chapter

8

Triangle Trigonometry 573

8.1

Trigonometric Functions of Angles 574

8.1

Alternate: Trigonometric Functions of Angles 584

8.2

Applications of Right Triangle Trigonometry 588

8.3

The Law of Cosines 597

8.4

The Law of Sines 606

8.4.A

Special Topics: The Area of a Triangle 617

CHAPTER 8 REVIEW 620 ■ DISCOVERY PROJECT 8 624

chapter

9

Applications of Trigonometry 625

9.1

The Complex Plane and Polar Form for Complex Numbers 626

9.2

DeMoivre’s Theorem and nth Roots of Complex Numbers 632

9.3

Vectors in the Plane 639

9.4

The Dot Product 653

CHAPTER 9 REVIEW 662 ■ DISCOVERY PROJECT 9 666

chapter

10

Analytic Geometry 669

10.1

Circles and Ellipses 671

10.2

Hyperbolas 686

10.3

Parabolas 700

10.3.A

Special Topics: Parametric Equations for Conic Sections 713

10.4

Rotations and Second-Degree Equations 718

10.4.A

Special Topics: Rotation of Axes 722

10.5

Plane Curves and Parametric Equations 727

10.6

Polar Coordinates 743

10.7

Polar Equations of Conics 753

CHAPTER 10 REVIEW 763 ■ DISCOVERY PROJECT 10 768

2 −4 −6 6 h 10 46° 34° 36° (c, d) (a, b) (a – b, c – d) x u u – v v –v w y −10 −8 8 8

chapter

11

Systems of Equations 771

11.1

Systems of Linear Equations in Two Variables 772

11.1.A

Special Topics: Systems of Nonlinear Equations 784

11.2

Large Systems of Linear Equations 792

11.3

Matrix Methods for Square Systems 806

CHAPTER 11 REVIEW 819 ■ DISCOVERY PROJECT 11 823

chapter

12

Discrete Algebra 825

12.1

Sequences and Sums 826

12.2

Arithmetic Sequences 837

12.3

Geometric Sequences 844

12.3.A

Special Topics: Infinite Series 852

12.4

The Binomial Theorem 857

12.5

Mathematical Induction 864

CHAPTER 12 REVIEW 873 ■ DISCOVERY PROJECT 12 877

chapter

13

Limits and Continuity 879

13.1

Limits of Functions 880

13.2

Properties of Limits 890

13.2.A

Special Topics: The Formal Definition of Limit 897

13.3

Continuity 903

13.4

Limits Involving Infinity 914

CHAPTER 13 REVIEW 924 ■ DISCOVERY PROJECT 13 927

APPENDIX 1: Algebra Review 931 APPENDIX 2: Geometry Review 953 APPENDIX 3: Programs 964 Selected Answers 967 Index of Applications I-1 Subject Index I-4

y x 1 −1 −1 −2 2 1

xi

Preface

Content Changes in the Fifth Edition

The major changes in this edition include the following.

Chapter Tests In addition to the Chapter Review exercises, each chapter now has a chapter test (in two parts for longer chapters), with answers for both odd and even questions in the answer section.

Section Objectives The goals for student learning are summarized at the beginning of each section.

Discovery Projects A new Discovery Project has been added to Chapter 1. Exercises Every exercise in the previous edition was examined and a

signifi-cant number of them were updated, revised, or replaced. All in all, more than 20% of the exercises in this edition are new.

A number of smaller changes and additions—ranging from a few lines to a page or so—have been made throughout the book to improve coverage and clarity.

Organizational Changes

Several changes in the order and arrangement of topics have been made to provide more flexibility for the instructor.

Parametric Graphing The brief introduction in Chapter 3 is now in a section by itself and graph reading is covered in Section 3.3 (Graphs of Functions). Parametric equations for conic sections are now in a separate (optional) sec-tion of Chapter 10, rather than being included as parts of other secsec-tions. Arc Length and Angular Speed These topics are now in a separate (optional)

section instead of being included in Section 6.1.

Calculator Investigations These formerly appeared at the end of selected sec-tions at the beginning of the book and are now incorporated into the exercise sets for these sections.

This book provides the mathematical background needed for calculus for students who have had two or three years of high school mathematics. Topics that are essential for success in calculus are thoroughly covered, including functional notation, graph reading, the natural exponential and logarithmic functions, average rates of change, trigonometric functions of a real variable, and limits.

The mathematics is presented in an informal manner that stresses meaningful motivation, careful explanations, and numerous examples, with an ongoing focus on real-world problem solving. Technology is integrated into the presentation and students are expected to use it to participate actively in exploring topics from algebraic, graphical, and numerical perspectives.

Ongoing Features

All the helpful pedagogical features of earlier editions are retained here, such as: A Flexible Approach to Trigonometry that allows instructors to use the order of topics they prefer: Trig functions of a real variable can be covered before or after triangle trigonometry or both approaches can be combined (as explained in the chart on page xiv);

Graphing Explorations, in which students discover and develop useful mathe-matics on their own;

Cautions that alert students to common misconceptions and mistakes;

Technology Tips that provide assistance in carrying out various procedures on specific calculators;

Exercises that proceed from routine drill to those requiring some thought, including graph interpretation and applied problems, as well as Thinkers that challenge students to “think outside the box” (most of these are not difficult— just different);

Chapter Reviews that include a list of important concepts (referenced by section and page number), a summary of important facts and formulas, and a set of review questions;

Algebra Review Appendix for students who need a review of basic algebra; Geometry Review Appendix that summarizes frequently used facts from plane

geometry, with examples and exercises;

Program Appendix that provides a small number of programs that are useful for updating older calculator models or for more easily carrying out procedures discussed in the text.

Acknowledgments

First of all, the senior author is happy to welcome Doug Shaw as a coauthor. His insights and enthusiasm have had a very positive impact on this revision and will, I trust, continue to do so in the future.

We are particularly grateful to three people who have been associated with several editions of this book:

Leslie Lahr, our Developmental Editor, who has been involved in almost all aspects of the book, and from whose sage advice we have greatly benefited; Phil Embree of William Woods University, one of our accuracy checkers and the author of several Discovery Projects, whose helpful suggestions have definitely improved the final product;

Laura Horowitz of Hearthside Publication Services, whose calm coordination of all aspects of production has kept us sane and happy.

Thanks are also due to Heidi A. Howard, who pitched in on a variety of tasks whenever needed.

It is a pleasure to acknowledge the invaluable assis-tance of the Brooks/Cole staff:

Gary Whalen, Acquisitions Editor Joe Rogove, Marketing Manager Natasha Coats, Assistant Editor Lynh Pham, Editorial Assistant

Hal Humphrey, Senior Content Project Manager Vernon Boes, Senior Art Director

We also want to thank the outside production staff: Grace Lefrancois, Copy Editor

Jade Myers, Art Rendering Terri Wright of Terri Wright Design

Special thanks go to

John Samons, Florida Community College,

who did most of the accuracy checking of the manuscript, and to

Jerret Dumouchel, Florida Community College Arkady Hanjiev, North Hills College,

who prepared the Solution Manuals.

The students who have assisted in manuscript preparation also have our thanks: Varun Khanna (St. Louis Univer-sity), and Jordan Meyer, Ren Waddell and Kevan Irvine (University of Northern Iowa).

Finally, we want to thank the reviewers whose con-structive comments have played a crucial role in revising this text.

Jared Abwawo, Tacoma Community College Tony Akhlaghi, Bellevue Community College Donna Bernardy, Lane Community College Patrick DeFazio, Onondaga Community College Linda Horner, Columbia State Community College Douglas Nelson, Central Oregon Community College Stephen Nicoloff, Paradise Valley Community College Bogdan Nita, Montclair State University

Dennis Reissig, Suffolk County Community College Arnavaz Taraporevala, New York City College of

Technology

Thanks also go the reviewers of previous editions: Deborah Adams, Jacksonville University Kelly Bach, University of Kansas

David Blankenbaker, University of New Mexico Kathleen A. Cantone, Onondaga Community College Bettyann Daley, University of Delaware

Margaret Donlan, University of Delaware Patricia Dueck, Arizona State University Betsy Farber, Bucks County Community College Alex Feldman, Boise State University

Robert Fliess, West Liberty State College Betty Givan, Eastern Kentucky University William Grimes, Central Missouri State University Frances Gulick, University of Maryland

John Hamm, University of New Mexico Lonnie Hass, North Dakota State University Larry Howe, Rowan University

Conrad D. Krueger, San Antonio College

Ann Lawrance, Wake Technical Community College Charles Laws, Cleveland State Community College Anatoly S. Libgober, University of Illinois at Chicago Martha Lisle, Prince George’s Community College Matthew Liu, University of Wisconsin-Stevens Point Sergey Lvin, University of Maine

George Matthews, Onondaga Community College Nancy Matthews, University of Oklahoma Ruth Meyering, Grand Valley State University William Miller, Central Michigan University Philip Montgomery, University of Kansas Roger Nelsen, Lewis and Clark College Jack Porter, University of Kansas

Robert Rogers, University of New Mexico Barbara Sausen, Fresno City College

Hugo Sun, California State University at Fresno Stuart Thomas, University of Oregon

Trung G. Tran, Tacoma Community College Bettie Truitt, Black Hawk College

Jan Vandever, South Dakota State University Judith Wolbert, Kettering University Cathleen M. Zucco-Teveloff, Trinity College

The last word, as always, goes to our wives, Mary Alice Hungerford and Laurel Shaw, who have, as always, provided understanding and support when it was most needed.

Thomas W. Hungerford Douglas J. Shaw

This book contains more than enough material for two semesters. By using the chart on the facing page (and the similar ones at the beginning of each chapter that show the interdependence of sections within the chapter), you can easily design a course to fit the needs of your students and the constraints of time. When planning your syllabus, two things are worth noting.

Special Topics Sections with this label are usually related to the immediately preceding section and are not prerequisites for other sections of the text. Judg-ments vary as to which of these topics are essential and which can be omitted. So feel free to include as many or as few Special Topics sections as you wish. Trigonometry We believe that a precalculus course should emphasize trigono-metric functions of a real variable from the beginning. We also think that the advent of technology has made the cotangent, secant, and cosecant functions less necessary than was the case when calculations were done by hand. So they are not introduced until the end of Chapter 6. This is the approach given in the first column of the chart below.

However, we are well aware that some instructors feel strongly that triangle trigonometry should be introduced first, or that both approaches should be presented together, or that all six trig functions should be introduced at the same time, etc. The following chart (and its footnotes) show how to arrange things to suit your personal preferences.

To the Instructor

OPTIONS FOR COVERING TRIGONOMETRY Mixed

[Trig functions of a real variable and basic triangle trig

Real Variable First Triangles First introduced as soon as possible]

Real Variable First Triangles First

Chapter 6* Chapter 8 (using Alternate Sections 6.1–6.4* Section 6.1

8.1 in place of 8.1) 8.1–8.2 Alternate 8.1

6.5–6.6 8.2

6.2–6.6* Chapter 6 (using Alternate

6.2 in place of 6.2)*

Chapters 7, 8, 9 in Chapters 7 & 9 in either Chapter 7, Sections 8.3–8.4 Chapter 7, Sections 8.3–8.4 any order† _order† _{and Chapter 9 in any order}† _{and Chapter 9 in any order}†

*For early introduction of cotangent, secant, and cosecant, incorporate Section 6.6 into Sections 6.2–6.4 by covering Part I of 6.6 at the end of 6.2;

Part II of 6.6 at the end of 6.3; Part III of 6.6 at the end of 6.4.

†_{Within each of these chapters, sections can be covered in several different orders. See the interdependence-of-sections chart at the beginning of the chapter.}

Interdependence of Chapters

1 Basics 2 Graphs and Technology 11 Systems of Equations* 3 Functions and Graphs 4 Polynomial and Rational Functions 5 Exponential and Logarithmic Functions 6 Trigonometric Functions 13 Limits and Continuity 7 Trigonometric Identities and Equations 9 Applications of Trigonometry 10 Analytic Geometry (Parametric & Polar Graphing)† 8 Triangle Trigonometry 12 Discrete Algebra

*Section 2.1 (graphs) is a prerequisite for Special Topics 11.1.A (Systems of Nonlinear Equations).

†_{Sections 10.1–10.4 (Conic Sections) depend only on Chapter 2. Trigonometry is an additional prerequisite for Special Topics 10.3.A and 10.4.A}

and Sections 10.5–10.7. 10 Analytic Geometry (Conic Sections)† xv

This text assumes the use of technology (a graphing calculator or computer with appropriate software). Discussions of calculators in the text apply (with obvious modifications) to computer software.

To address the fact that many students are unaware of the power of their graph-ing calculators, there are Technology Tips in the margins throughout the text. They provide general information and advice, as well as listing the proper menus or keys needed to care out procedures on specific calculators.* Unless noted otherwise,

Tips for also apply to

TI-84" TI-82, TI-83, TI-83"

TI-86 TI-85

TI-89 TI-92

Casio 9850 Casio 9750, Casio 9860, Casio 9970 HP-39gs HP-38, HP-39, HP-39"

*In addition there is a Program Appendix that provides users of older calculators with several helpful programs that are built in to newer calculators.

xvii

Ancillaries

Supplements for Instructors

COMPLETE SOLUTIONS MANUAL

ISBN-10: 0-495-55398-0 | ISBN-13: 978-0-495-55398-4

The complete solutions manual provides worked out solutions to all of the prob-lems in the text.

TEST BANK

ISBN-10: 0-495-55400-6 | ISBN-13: 978-0-495-55400-4

The Test Bank includes 8 tests per chapter as well as 3 final exams, each combin-ing multiple-choice, free-response, and fill-in-the-blank questions.

EXAMVIEW

®

_{WITH ALGORITHMIC EQUATIONS}

ISBN-10: 0-495-55417-0 | ISBN-13: 978-0-495-55417-2

Create, deliver, and customize tests and study guides (both print and online) in minutes with this easy-to-use assessment and tutorial system, which includes the Test Bank questions in electronic format. ExamView offers both a Quick Test Wizard and an Online Test Wizard that guide you step-by-step through the process of creating tests—you can even see the test you are creating on the screen exactly as it will print or display online.

TEXT-SPECIFIC DVD

S

ISBN-10: 0-495-55401-4 | ISBN-13: 978-0-495-55401-1

This set of video segments, available upon adoption of the text, offers a 10- to 20-minute problem-solving lesson for each section of each chapter.

JOININ™ STUDENT RESPONSE SYSTEM

ISBN-10: 0-495-55415-4 | ISBN-13: 978-0-495-55415-8

JoinIn™ content tailored to this text (on Microsoft®_PowerPoint®_{slides) allows}

you to pose book-specific questions and display answers seamlessly within the Microsoft PowerPoint slides of your own lecture. Use the slides with your favorite response system software.

ENHANCED WEBASSIGN—AN ASSIGNABLE HOMEWORK AND

TUTORIAL TOOL THAT SAVES YOU TIME

www.webassign.net/brookscole

Proven and reliable, Enhanced WebAssign allows you to easily assign, collect, grade, and homework assignments via the web. You save time, and students get interactive assistance plus relevant, immediate feedback. Features include:

•

Thousands of algorithmically generated homework problems based on up to 1,500 end-of-section exercises

•

Read It links to PDFs of relevant text sections

•

Watch It links to videos that provide further instruction on problems

•

Master It tutorials for step-by-step support

•

Chat About It links to live, online tutoring

•

Windows®_{and Apple}® _{Macintosh compatible; works with most web}

browsers (Firefox, Internet Explorer, Mozilla, Safari); does not require proprietary plug-ins

Supplements for Students

STUDENT SOLUTIONS MANUAL

ISBN-10: 0-495-55399-9 | ISBN-13: 978-0-495-55399-1

The student solutions manual provides worked out solutions to the odd-numbered problems in the text.

xix

To the Student

This text assumes the use of technology (a graphing calculator or computer with appropriate software). Discussions of calculators in the text apply (with obvious modifications) to computer software.

To help you get the most from your graphing calculator, there are Technology Tips in the margins throughout the text. They provide general information and advice, as well as listing the proper menus or keys needed to care out procedures on specific calculators.* Unless noted otherwise,

Tips for also apply to

TI-84" TI-82, TI-83, TI-83"

TI-86 TI-85

TI-89 TI-92

Casio 9850 Casio 9750, Casio 9860, Casio 9970 HP-39gs HP-38, HP-39, HP-39"

Getting the Most Out of This Course

With all this talk about calculators, don’t lose sight of this crucial fact: Technology is only a tool for doing mathematics.

You can’t build a house if you only use a hammer. A hammer is great for pound-ing nails, but useless for sawpound-ing boards. Similarly, a calculator is great for com-putations and graphing, but it is not the right tool for every mathematical task. To succeed in this course, you must develop and use your algebraic and geometric skills, your reasoning power and common sense, and you must be willing to work. The key to success is to use all of the resources at your disposal: your instructor, your fellow students, your calculator (and its instruction manual), and this book. Here are some tips for making the most of these resources.

Ask Questions Remember the words of Hillel: The bashful do not learn.

There is no such thing as a “dumb question” (assuming, of course, that you have attended class and read the text). Your instructor will welcome questions that arise from a serious effort on your part.

Read the Book Not just the homework exercises, but the rest of the text as well. There is no way your instructor can possibly cover the essential topics, clarify ambiguities, explain the fine points, and answer all your questions during class time. You simply will not develop the level of understanding you need to succeed in this course and in calculus unless you read the text fully and carefully.

*In addition there is a Program Appendix that provides users of older calculators with several helpful programs that are built in to newer calculators.

Be an Interactive Reader You can’t read a math book the way you read a novel or history book. You need pencil, paper, and your calculator at hand to work out the statements you don’t understand and to make notes of things to ask your fellow students and/or your instructor.

Do the Graphing Explorations When you come to a box labeled “Graphing Exploration,” use your calculator as directed to complete the discussion. Typi-cally, this will involve graphing one or more equations and answering some questions about the graphs. Doing these explorations as they arise will improve your understanding and clarify issues that might otherwise cause difficulties. Do Your Homework Remember that

Mathematics is not a spectator sport.

You can’t expect to learn mathematics without doing mathematics, any more than you could learn to swim without getting wet. Like swimming or dancing or reading or any other skill, mathematics takes practice. Homework assignments are where you get the practice that is essential for passing this course and suc-ceeding in calculus.

BASICS

On a clear day, can you see forever?

I

f you are at the top of the Smith Tower in Seattle, how far can you see? In earlier centuries, the lookout on a sailing ship was posted atop the highest mast because he could see farther from there than from the deck. How much farther? These questions, and similar ones, can be answered (at least approximately) by using basic algebra and geometry. See Example 4 on page 9 and Exercise 94 on page 15.

1

Chapter

© Neil Rabinowitz/CORBIS B 60° 90° 30° A C

2

Name Definition/Description

Natural numbers 1, 2, 3, 4, 5, . . .

Natural numbers are also called counting numbers or

positive integers.

Integers . . . , #5, #4, #3, #2, #1, 0, 1, 2, 3, 4, 5, . . .

The integers consist of the natural numbers, their negatives, and zero.

Interdependence of

Sections

1.1 The Real Number System

1.1.A Special Topics: Decimal Representation

of Real Numbers

1.2 Solving Equations Algebraically

1.2.A Special Topics: Absolute Value Equations

1.2.B Special Topics: Variation

1.3 The Coordinate Plane

1.4 Lines

T

his chapter reviews the essential facts about real numbers, equations, the coordinate plane, and lines that are needed in this course and in cal-culus. The Algebra Review Appendix at the end of the book is a prereq-uisite for this material.

1.2 1.1

1.3

1.4

1.1

The Real Number System

■

Identify important types of real numbers.

■

Simplify mathematical expressions.

■

Represent sets of real numbers with interval notation.

■

Graph intervals on a number line.

■

Use scientific notation.

■

Understand and apply the properties of square roots.

■

Understand and apply the properties of absolute value.

■

Compute the distance between two points on the number line.

Most of this book deals with the real number system, so it may be helpful to review the types of real numbers.

Continued

More information about decimal expansions of real numbers is given in Special Topics 1.1.A.

The relationships among the types of numbers in the preceding table are sum-marized in Figure 1–1, in which each set of numbers is contained in the set to its right. So, for example, integers are also rational numbers and real numbers, but not irrational numbers.

Rational numbers A rational number is a number that can be expressed as a

fraction &r

s&, with r and s integers and s $ 0, such as

&1 2&, #9.83 ! & # 1 9 0 8 0 3 &, 47 ! &4 1 7 &, 8&3 5& ! & 4 5 3 &. Alternatively, rational numbers are numbers that can be expressed as terminating decimals, such as .25 ! &1

4&, or as nonterminating repeating decimals in which a single digit or block of digits eventually repeats forever, such as

&5 3& !1.66666 ' ' ' or &1 3 6 6 6 2 5 & !.2174174174 ' ' '.

Irrational numbers An irrational number is a number that cannot be expressed as a

fraction with an integer numerator and denominator, such as the number p, which is used to calculate the area of a circle.* Alternatively, irrational numbers are numbers that can be expressed as nonterminating, nonrepeating decimals (no block of digits repeats forever).

*The proof that p is irrational is beyond the scope of this book. In the past you may have used 22/7 or 3.1416 as p and a calculator may display p as 3.141592654. However, these numbers are just

approximations ofp. Natural

numbers Integers numbersRational

Irrational numbers Real numbers Figure 1–1

ARITHMETIC

To avoid ambiguity when dealing with expressions such as 6 " 3 ( 5, mathemati-cians have made the following agreement, which is also followed by your calculator.

In light of this convention, there is only one correct way to interpret 6 " 3 ( 5: 6 " 3 ( 5 ! 6 " 15 ! 21. [Multiplication first, addition last]

Order of

Operations

In an expression without parentheses, multiplication and division are_{performed first (from left to right). Addition and subtraction are performed}

last (from left to right).

On the other hand, if you want to “add 6 " 3 and then multiply by 5,” you must use parentheses:

(6 " 3)

'

5 ! 9

'

5 ! 45.

This is an illustration of the first of two basic rules for dealing with parentheses.

For example,

8 " [11 # (6 ( 3)] ! 8 " (11 # 18) ! 8 " (#7) ! 1. Inside parentheses first

We assume that you are familiar with the basic properties of real number arithmetic, particularly the following fact.

The distributive law doesn’t usually play a direct role in easy computations, such as 4(3 " 5). Most people don’t say 4

'

3 " 4

'

5 ! 12 " 20 ! 32. Instead, they mentally add the numbers in parentheses and say 4 times 8 is 32. But when symbols are involved, you can’t do that, and the distributive law is essential. For example,

4(3 " x) ! 4

'

3 " 4x ! 12 " 4x.

THE NUMBER LINE AND ORDER

The real numbers are often represented geometrically as points on a number line, as in Figure 1–2. We shall assume that there is exactly one point on the line for every real number (and vice versa) and use phrases such as “the point 3.6” or “a number on the line.” This mental identification of real numbers and points on the line is often helpful.

Figure 1–2

The statement c ) d, which is read “c is less than d,” and the statement d * c (read “d is greater than c”) mean exactly the same thing:

clies to the left of d on the number line. For example, Figure 1–2 shows that #5.78 ) #2.2 and 4 *p.

−2 −3 −4 −5 −6 −7 −8 −8.6 −9 −1 0 1 2 3 4 5 6 7 8 9 −5.78 _{− 72} −2.2 3 p 6.3 2 337

Rules for

Parentheses

1. Do all computations inside the parentheses before doing any computa-_{tions outside the parentheses.}

2. When dealing with parentheses within parentheses, begin with the innermost pair and work outward.

Distributive

Law

For all real numbers a, b, c,

The statement c % d, which is read “c is less than or equal to d,” means Either c is less than d or c is equal to d.

Only one part of an “either . . . or” statement needs to be true for the entire state-ment to be true. So the statestate-ment 5 % 10 is true because 5 ) 10, and the statestate-ment 5 % 5 is true because 5 ! 5. The statement d + c (read “d is greater than or equal to c”) means exactly the same thing as c % d.

The statement b ) c ) d means

b ) c and simultaneously c ) d.

For example, 3 ) x ) 7 means that x is a number that is strictly between 3 and 7 on the number line (greater than 3 and less than 7). Similarly, b % c ) d means

b % c and simultaneously c ) d, and so on.

Certain sets of numbers, defined in terms of the order relation, appear fre-quently enough to merit special notation. Let c and d be real numbers with c ) d. Then

All four of these sets are called intervals from c to d. The numbers c and d are the endpoints of the interval. [c, d] is called the closed interval from c to d (both endpoints included and square brackets), and (c, d) is called the open inter-val from c to d (neither endpoint included and round brackets). Some examples are shown in Figure 1–3.*

Figure 1–3 −2 −3 −4 −5 −6 [−7, 2]

Interval Picture on the number line

−7 −1 0 1 2 3 −2 (0,π) −1 0 1 2 3 4 −2 (1, 4] −1 0 1 2 3 4 5 6 1 0 −1 −2 −3 [−2, 3) −4 2 3 4 5

Interval

Notation

[c, d] denotes the set of all real numbers x such that c % x % d.

(c, d) denotes the set of all real numbers x such that c ) x ) d. [c, d) denotes the set of all real numbers x such that c % x ) d. (c, d] denotes the set of all real numbers x such that c ) x % d.

*In Figures 1–3 and 1–4, a round bracket such as ) or ( indicates that the endpoint is not included, whereas a square bracket such as ] or [ indicates that the endpoint is included.

If b is a real number, then the half-line extending to the right or left of b is also called an interval. Depending on whether or not b is included, there are four pos-sibilities.

Some examples are shown in Figure 1–4.

Figure 1–4

In a similar vein, (!", ") denotes the set of all real numbers.

NEGATIVE NUMBERS AND NEGATIVES OF NUMBERS

The positive numbers are those to the right of 0 on the number line, that is, All numbers c with c * 0.

The negative numbers are those to the left of 0, that is, All numbers c with c ) 0. The nonnegative numbers are the numbers c with c + 0.

The word “negative” has a second meaning in mathematics. The negative of a number c is the number #c. For example, the negative of 5 is #5, and the neg-ative of #3 is #(#3) ! 3. Thus the negneg-ative of a negneg-ative number is a positive number. Zero is its own negative, since #0 ! 0. In summary,

−2 −3 −4 −5 [2,∞)

Interval Picture on the number line

3 2 4 5 1 0 −1 −2 −3 −4 −5 −1 0 1 2 3 4 5 −2 −3 −4 −5 −1 0 1 2 3 4 5 −2 −3 −4 −5 −1 0 1 2 3 4 5 (−3.5, ∞) (−∞, 2.75] (−∞, −1)

Interval

Notation

[b, ,) denotes the set of all real numbers x such that x + b.

(b, ,) denotes the set of all real numbers x such that x * b. (#,, b] denotes the set of all real numbers x such that x % b. (#,, b) denotes the set of all real numbers x such that x ) b.

NOTE

The symbol , is read “infinity,” and we call the set [b, ,) “the interval from b to infinity.” The symbol , does not de-note a real number; it is simply part of the notation used to label the first two sets of numbers defined in the previous box. Analogous remarks apply to the symbol #,, which is read “negative infinity.”

Negatives

The negative of the number c is #c.

If c is a positive number, then #c is a negative number. If c is a negative number, then #c is a positive number.

TECHNOLOGY

TIP

To enter a negative number, such as #5, on most calculators, you must use the negation key: (#) 5. If you use the subtraction key on such calculators and enter #5, the display will read

ANS #5

which tells the calculator to subtract 5 from the previous answer.

SCIENTIFIC NOTATION

In mid-2006 the U.S. national debt was 8.4 trillion dollars. Since one trillion is 1012_{(that is, 1 followed by 12 zeros), the national debt is the number}

8.4 ( 1012_.

This is an example of scientific notation. A positive number is said to be in scien-tific notation when it is written in the form

a (10n _{where 1 % a ) 10 and n is an integer.}

You should be able to translate between scientific notation and ordinary notation and vice versa.

EXAMPLE 1

Express these numbers in ordinary notation: (a) 1.55 ( 109 _{(b) 2.3 ( 10}#8

SOLUTION

(a) 1.55 ( 109_{!1.55 ( 1,000,000,000 ! 1,550,000,000}

You can do this multiplication in your head by using the fact that multiplying by 109_{is equivalent to moving the decimal point 9 places to the right.}

(b) 2.3 ( 10#8_{! &} 1 2 0 .3 8 & !.000000023*

This computation can also be done mentally by using the fact that dividing by 108_{is equivalent to moving the decimal point 8 places to the left. ■}

EXAMPLE 2

Write each of these numbers in scientific notation:

(a) 356 (b) 1,564,000 (c) .072 (d) .00000087

SOLUTION In each case, move the decimal point to the left or right to obtain a number between 1 and 10; then write the original number in scientific notation as follows.

(a) Move the decimal point in 356 two places to the left to obtain the number 3.56, which is between 1 and 10. You can get the original number back by multiplying by 100: 356 ! 3.56 ( 100. So the scientific notation form is

356 ! 3.56 ( 102

Note that the original decimal point is moved 2 places to the left and 10 is raised to the power 2.

(b) 1,564,000 ! 1.564 ( 1,000,000 ! 1.564 ( 106_{[Decimal point is moved}

6 places to the left, and 10 is raised to the 6th power.] (c) .072 ! 7.2 ( &

1 1

00& ! 7.2 ( 10#2[Decimal point is moved 2 places to the

right,and 10 is raised to #2.]

*Negative exponents are explained in the first section of the Algebra Review Appendix.

TECHNOLOGY

TIP

To enter 1.55 ( 109_{on a calculator,} use the EE key (labeled EEX on HP and 10x_{on Casio). Key in}

1.55 EE 9 and the display reads

1.55 E 9. Similarly, 2.3 ( 10#8_{is entered} as 2.3 EE #8 and the display reads 2.3 E #8.

(d) .00000087 ! 8.7 ( & 10,00 1

0,000

& !8.7 ( 10#7_{[Decimal point is moved 7}

places to the right, and 10 is raised to the #7.] ■

Scientific notation is useful for computations with very large or very small numbers.

EXAMPLE 3

(.00000002)(4,300,000,000) ! (2 ( 10#8_{)(4.3 ( 10}9₎

!2(4.3)10#8"9_!(8.6)101_{!86. ■}

Calculators automatically switch to scientific notation whenever a number is too large or too small to be displayed in the standard way. If you try to enter a number with more digits than the calculator can handle, such as 45,000,000,333,222,111, a typical calculator will approximate it using scientific notation as 4.500000033 E 16, that is, as 45,000,000,330,000,000.

SQUARE ROOTS

A square root of a nonnegative real number d is any number whose square is d. For instance both 5 and #5 are square roots of 25 because

52_{!25 and (#5)}2_!25.

The nonnegative square root (in this case, 5) is given a special name and notation.

For example,

!25" ! 5 because 5 + 0 and 52_!25.

The radical symbol always denotes a nonnegative number. To express the nega-tive square root of 25 in terms of radicals, we write #5 ! #!25".

Although #!25" is a real number, the expression !#25" is not defined in the real numbers because there is no real number whose square is #25. In fact, since the square of every real number is nonnegative,

No negative number has a square root in the real numbers. Some square roots can be found (or verified) by hand, such as

!225" ! 15 and !1.21" ! 1.1.

Usually, however, a calculator is needed to obtain rational approximations of roots. For instance, we know that !87" is between 9 and 10 because 92_{!81 and}

102_{!100. A calculator shows that !87" # 9.327379.}†

*The symbol !1"is called a radical.

†_{# means “approximately equal.”}

If d is a nonnegative real number, the principal square root of d is the

nonnegativenumber whose square is d. It is denoted !d".* Thus, !d" + 0 and _$!d"%2!d.

TECHNOLOGY

TIP

To compute !7_"2_{"51"3 on a} calculator, you must use a pair of parentheses:

!(7_"2_"51)""3. Otherwise the calculator will not compute the correct answer, which is:

!7_"2_{"51"3 !!49 " 5"1""3} !!100""3 !13. Try it!

We shall often use the following property of square roots: !cd" ! !c"!d" for any nonnegative real numbers c and d.

For example, !9 ' 16" ! !9"!16" ! 3 ' 4 ! 12. But be careful—there is no similar

property for sums, as the Caution in the margin demonstrates.

EXAMPLE 4

Suppose you are located h feet above the ground. Because of the curvature of the earth, the maximum distance you can see is approximately d miles, where

d !

!

1.5h "

"

"

(3.587

"

(10#

_"

8_)h2_.

How far can you see from the 500-foot-high Smith Tower in Seattle and from the 1454-foot-high Sears Tower in Chicago?

SOLUTION For the Smith Tower, substitute 500 for h in the formula, and use your calculator:

d !

!

1.5(50

"

"

0) " (3.587

"

"

(10#

_"

8₎₅₀₀2_{# 27.4 miles.}

For the Sears Tower, you can see almost 47 miles because

d !

!

1.5(14

"

"

54) "(3.587

"

"

(10#

_"

8₎₁₄₅₄2

_"

_{# 46.7 miles. ■}

ABSOLUTE VALUE

On an informal level, most students think of absolute value like this: The absolute value of a positive number is the number itself.

The absolute value of a negative number is found by “erasing the minus sign.” If !c! denotes the absolute value of c, then, for example, !5! ! 5 and !#4! ! 4. This informal approach is inadequate, however, for finding the absolute value of a number such as p # 6. It doesn’t make sense to “erase the minus sign” here. So we must develop a more precise definition. The statement !5! ! 5 suggests that the absolute value of a nonnegative number ought to be the number itself. For negative numbers, such as #4, note that !#4! ! 4 ! #(#4), that is, the absolute value of the negative number #4 is the negative of #4. These facts are the basis of the formal definition.

EXAMPLE 5

(a) !3.5! ! 3.5 and !#7/2! ! #(#7/2) ! 7/2.

(b) To find !p # 6!, note that p# 3.14, so p # 6 ) 0. Hence, !p # 6! is defined to be the negative ofp # 6, that is,

!p # 6! ! #(p # 6) ! #p " 6.

(c) !5 #!2"! ! 5 # !2" because 5 # !2" + 0. ■

Absolute

Value

The absolute value of a real number c is denoted !c! and is defined as follows.

If c + 0, then !c! ! c. If c ) 0, then !c! ! #c.

CAUTION

If c and d are positive real numbers, then !c " d" $ !c" " !d". For example, !9 " 16" ! !25" ! 5, but !9" " !16" ! 3 " 4 ! 7.

Here are the important facts about absolute value.

EXAMPLE 6

Here are examples of the last three properties in the box. 3. !3! ! 3 and !#3! ! 3, so !3! ! !#3!. 4. If c ! 6 and d ! #2, then !cd! ! !6(#2)! ! !#12! ! 12 and !c!

'

!d! ! !6!

'

!#2! ! 6

'

2 ! 12, so !cd! ! !c!

'

!d!. 5. If c ! #5 and d ! 4, then

&

& d c_&

&

_!

&

_&#

4 5

&

&

!

&

#&5 4&

&

! &

5

4& and &! ! d c! ! & ! &!# !4 5 ! ! & ! &5 4&, so

&

& d c

&

&

! & ! ! d c! ! &. ■

When c is a positive number, then

!

c

"

2_{!c, but when c is negative, this is}

false.For example, if c ! #3, then

!

_"

c2_!

_!

_"

_(#3)2_{!!9" ! 3 (not #3),}

so

!

c

_"

2_{$c.In this case, however, !c! ! !#3! ! 3, so}

_!

_"

_c2_{!!c!. The same thing}

is true for any negative number c. It is also true for positive numbers (since !c! ! c when c is positive). In other words,

Properties of

Absolute Value

Property Description

1. !c! + 0 The absolute value of a number is nonnegative.

2. If c $ 0, then !c! * 0. The absolute value of a nonzero number is positive.

3. !c! ! !#c! A number and its negative have the same absolute value.

4. !cd! ! !c!

'

!d! The absolute value of the product of two numbers is the product of their absolute values.

5.

&

&

d c_&

&

_{! &}

! !

d c!

!

& (d $ 0) The absolute value of the quotient of two numbers is the quotient of their absolute values.

Square Roots

of Squares

For every real number c,

!

c

_"

2_!!c!.

TECHNOLOGY

TIP

To find !9 # 3p! on a calculator, key in

Abs (9 # 3p).

TheAbs key is located in this

menu/submenu: TI: MATH/NUM Casio: OPTN/NUM HP-39gs: Keyboard

When dealing with long expressions inside absolute value bars, do the com-putations inside first, and then take the absolute value.

EXAMPLE 7

(a) !5(2 # 4) " 7! ! !5(#2) " 7! ! !#10 " 7! ! !#3! ! 3.

(b) 4 # !3 # 9! ! 4 # !#6! ! 4 # 6 ! #2. ■

The caution shows that !c " d! ) !c! " !d! when c ! #3 and d ! 5. In the general case, we have the following fact.

DISTANCE ON THE NUMBER LINE

Observe that the distance from #5 to 3 on the number line is 8 units:

Figure 1–5

This distance can be expressed in terms of absolute value by noting that !(#5) # 3! ! 8.

That is, the distance is the absolute value of the difference of the two numbers. Furthermore, the order in which you take the difference doesn’t matter; !3 # (#5)! is also 8. This reflects the geometric fact that the distance from #5 to 3 is the same as the distance from 3 to #5. The same thing is true in the general case. −2 −3 −4 −5 −6 0 1 2 3 4 5 8 units −1

The Triangle

Inequality

For any real numbers c and d,

!c " d! % !c! " !d!.

Distance on the

Number Line

The distance between c and d on the number line is the number

!c # d! ! !d # c!.

CAUTION

When c and d have opposite signs, !c"d! is not equal to !c!"!d!. For example, when c! #3 and d!5, then

!c " d! ! !#3 " 5! ! 2, but

EXAMPLE 8

The distance from 4.2 to 9 is !4.2 # 9! ! !#4.8! ! 4.8, and the distance from

6 to !2" is !6 # !2"!. ■

When d ! 0, the distance formula shows that !c # 0! ! !c!. Hence,

Algebraic problems can sometimes be solved by translating them into equiv-alent geometric problems. The key is to interpret statements involving absolute value as statements about distance on the number line.

EXAMPLE 9

Solve the equation !x " 5! ! 3 geometrically.

SOLUTION We rewrite it as !x # (#5)! ! 3. In this form it states that

The distance between x and #5 is 3 units.*

Figure 1–6 shows that #8 and #2 are the only two numbers whose distance to #5 is 3 units:

Figure 1–6

Thus x ! #8 and x ! #2 are the solutions of !x " 5! ! 3. ■

EXAMPLE 10

The solutions of !x # 1! + 2 are all numbers x such that

The distance between x and 1 is greater than or equal to 2.

Figure 1–7 shows that the numbers 2 or more units away from 1 are the numbers

xsuch that

x % #1 or x +3.

So these numbers are the solutions of the inequality. ■

Figure 1–7 −2 −3 −4 −5 −1 0 1 2 3 4 5 2 2 −2 −3 −4 −5 −6 −7 −8 −9 −10 0 1 2 3 3 −1 3

Distance

to Zero

!c! is the distance between c and 0 on the number line.

*It’s necessary to rewrite the equation first because the distance formula involves the difference of two numbers, not their sum.

EXAMPLE 11

The solutions of !x # 7! ) 2.5 are all numbers x such that

The distance between x and 7 is less than 2.5.

Figure 1–8 shows that the solutions of the inequality, that is, the numbers within 2.5 units of 7, are the numbers x such that 4.5 ) x ) 9.5, that is, the interval

(4.5, 9.5). ■ Figure 1–8 10 9 11 6 5 7 8 3 2 4 2.5 9.5 4.5 2.5

Year Percentage change

1996 3.0 1997 2.3 1998 1.6 1999 2.2 2000 3.4 2001 2.8 2002 1.6 2003 2.3 2004 2.7 2005 2.5

*U.S. Bureau of Labor Statistics; data for 2005 is for the first half.

EXERCISES

1.1

1. Draw a number line and mark the location of each of these numbers: 0, #7, 8/3, 10, #1, #4.75, 1/2, #5, and 2.25.

2. Use your calculator to determine which of the following rational numbers is the best approximation of the irrational numberp.

&2₇2&, &3₁5₁₃&,5 &1₃0₃3_,,₁&,9₀9₂3 &2₇,5₉0₈8_,4,4₅&.2₈9_,0,7₀8₀7 If your calculator says that one of these numbers equals p, it’s lying. All you can conclude is that the number agrees withp for as many decimal places as your calculator can handle (usually 12–14).

In Exercises 3–8, b, c, and d are real numbers such that b ) 0, c * 0, and d ) 0. Determine whether the given number is posi-tive or negaposi-tive.

3. #b 4. #c 5. bcd

6. b # c 7. bc # bd 8. b2_{c # c}2_d

In Exercises 9 and 10, use a calculator and list the given num-bers in order from smallest to largest.

9. &1 3 8 7 9 &, &4 6 5 9 8 1 7 &, !47", 6.735, !27", &2 5 0 2 4 3 0 & 10. &3 1 8 7 5 7 &, !10", &1 6 8 3 7 &, p, !!85"", 2.9884

In Exercises 11–19, express the given statement in symbols.

11. #4 is greater than #8.

12. #17 is less than 6.

13. p is less than 100.

14. xis nonnegative.

15. zis greater than or equal to #4.

16. tis negative.

17. dis not greater than 7.

18. cis at most 3.

19. zis at least #17.

In Exercises 20–24, fill the blank with ), !, or * so that the resulting statement is true.

20. #6 #2 21. 5 #3

22. 3'4 .75 23. 3.1416 p

24. 1'3 .33

The consumer price index for urban consumers (CPI-U) meas-ures the cost of consumer goods and services such as food, housing, transportation, medical costs, etc. The table shows the yearly percentage increase in the CPI-U over a decade.*

In Exercises 25–29, let p denote the yearly percentage increase in the CPI-U. Find the number of years in this period which satisfied the given inequality.

25. p +2.8 26. p )2.6 27. p *2.3

28. p %3.0 29. p *3.4

In Exercises 30–36, fill the blank so as to produce two equiva-lent statements. For example, the arithmetic statement “a is negative” is equivalent to the geometric statement “the point a

lies to the left of the point 0.”

Arithmetic Statement Geometric Statement

30. a + b

31. alies c units to the right of b

32. alies between b and c

33. a # b *0

34. ais positive

35. alies to the left of b

36. a " b ) c (b * 0)

In Exercises 37–42, draw a picture on the number line of the given interval.

37. (0, 8] 38. (0, ,) 39. [#2, 1]

40. (#1, 1) 41. (#,, 0] 42. [#2, 7)

In Exercises 43–48, use interval notation to denote the set of all real numbers x that satisfy the given inequality.

43. 5 % x % 10 44. #2 % x % 7

45. #3 ) x ) 14 46. 7 ) x ) 77

47. x + #9 48. x +12

In Exercises 49–53, express the given numbers (based on 2006 estimates) in scientific notation.

49. Population of the world: 6,506,000,000

50. Population of the United States: 298,400,000

51. Average distance from Earth to Pluto: 5,910,000,000,000 meters

52. Radius of a hydrogen atom: .00000000001 meter

53. Width of a DNA double helix: .000000002 meter

In Exercises 54–57, express the given number in normal deci-mal notation.

54. Speed of light in a vacuum: 2.9979 ( 108_{miles per second} 55. Average distance from the earth to the sun: 1.50 ( 1011_meters 56. Electron charge: 1.602 ( 10#27_coulomb

57. Proton mass: 1.6726 ( 10#19_kilogram

58. One light-year is the distance light travels in a 365-day year. The speed of light is about 186,282.4 miles per second.

(a) How long is 1 light-year (in miles)? Express your an-swer in scientific notation.

(b) Light from the North Star takes 680 years to reach the earth. How many miles is the North Star from the earth?

59. The gross federal debt was about 8365 billion dollars in 2006, when the U.S. population was approximately 298.4 million people.

(a) Express the debt and the population in scientific notation. (b) At that time, what was each person’s share of the

fed-eral debt?

60. Apple reported that it had sold 28 million iPods through the end of 2005 and that 14 million iPods were sold in the first quarter of 2006. If the rate in the first quarter of 2006 con-tinues through the end of 2008, how many iPods will be sold? Express your answer in scientific notation.

In Exercises 61–68, simplify the expression without using a cal-culator. Your answer should not have any radicals in it.

61. !2"!8" 62. !12"!3" 63.

(

&3 5&

)(

&1 5 2 &

)

64.

(

&1 2&

)(

&1 6&

)(

& 1 1 2 &

)

65. !6" " !2" $!2" # !3"% 66. !12" $!3" # !27"% 67. !u_{" (u any real number)}4 _{68. !3x"!75x" (x + 0)}3

In Exercises 69–78, simplify, and write the given number with-out using absolute values.

69. !3 # 14! 70. !(#2)3! 71. 3 # !2 # 5!

72. #2 # !#2! 73. !(#13)2_{! 74. #!#5!}2 75. !p #!2"! 76. !!2" # 2! 77. !3 # p! " 3

78. !4 #!2"! # 5

In Exercises 79–84, fill the blank with ), !, or * so that the resulting statement is true.

79. !#2! !#5! 80. 5 !#2!

81. !3! #!4! 82. !#3! 0

83. #7 !#1! 84. #!#4! 0

In Exercises 85–92, find the distance between the given numbers.

85. #3 and 4 86. 7 and 107

87. #7 and 15/2 88. #3/4 and #10

89. p and 3 90. p and #3

91. !2" and !3" 92. p and!2"

93. Galileo discovered that the period of a pendulum depends only on the length of the pendulum and the acceleration of gravity. The period T of a pendulum (in seconds) is

T !2p

(

&

g l

&

)

,