beginning algebra connecting concepts through applications pdf

R.1 Operations with Integers 2Natural Numbers, Whole Numbers, and Integers • Number Lines • Relations Between Numbers • Absolute Value • Opposite of a Number • Operations with Integers

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Concept Investigations—Helping you to discover connections

In your own words, describe what the coef¿ cient (number in front) of x does to the

graph Remember to read graphs from left to right

2 Now graph the following equations that have negative coef¿ cients

a y 5 2 x

b y 5 22 x

c y 5 25 x

d y 5 28 x

In your own words, describe what a negative coef¿ cient of x does to the graph.

3 Graph the following equations with coefficients that are between zero and one.

Use your graphing calculator to examine the following

Start by setting up your calculator by doing the following steps

● Clear all equations from the Y5 screen (Press Y= , )

● Change the window to a standard window (Press ZOOM , (ZStandard).)Now your calculator is ready to graph equations The Y5 screen is where equations will be put into the calculator to graph them or evaluate them at input values Several simple equations will be graphed to investigate how the graph of an equation for a line

reacts to changes in the equation (Note that your calculator uses y as the dependent (output) variable and x as the independent (input) variable.)

1 Graph the following equations that have positive coef¿ cients on a standard window

Enter each equation in its own row (Y1, Y2, Y3, )

(Note: To enter an x, you use the X,T,⍜,n button next to the ALPHA button.)

Remember that a variable that

is by itself (x) has a coefficient of 1.

What’s That Mean?

Using Your TI Graphing Calculator

In entering fractions in the calculator, it is often best to use parentheses.

y 5 1/5 x

On many graphing calculators, parentheses are needed in almost all situations In some calculators, when

5 x

To be sure the calculator does what you intend, using parentheses is a good idea.

The TI-83 does not need parentheses in some situations, but in other situations, they are required To keep confusion down, one option is to use parentheses around every fraction Extra parentheses do not usually create

a problem, but not having them where they are needed can cause miscalculations

Concept Investigations are great learning tools to help you to

explore and generalize patterns and relationships such as the graphical and algebraic representations of the functions you’ll study

Calculator steps and tips are listed here Don’t forget to go

to the “Using the Graphing Calculator” Appendix when

you need further help

Describing math in your own words will help you understand

and remember the concepts

This particular Concept Investigation shows you

how algebraic concepts can be studied via pattern recognition This may be a review topic for you

Explanatory margin boxes

like the “What’s That Mean”

one here appear as needed

to ensure you understand key concepts or terms

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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it www.FreeEngineeringbooksPdf.com

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BEGINNING ALGEBRA

Connecting Concepts through Applications

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Beginning Algebra: Connecting Concepts

through Applications

Mark Clark & Cynthia Anfinson

Printed in the United States of America

1 2 3 4 5 6 7 15 14 13 12 11

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Rosemary, and my parents for their love and support

MC

To my husband Fred and son Sean, thank you for your love and support

CA

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MARK CLARKgraduated from California State University, Long Beach, with Bachelor’s and Master’s degrees in Mathematics He is a full-time Associate Professor at Palomar College and has taught there for the past

13 years He is committed to teaching his students through applications and using technology to help them both to understand the mathematics in context and to communicate their results clearly

CYNTHIA (CINDY) ANFINSON graduated from UC San Diego’sRevelle College, with a Bachelor of Arts degree in Mathematics Shewent to graduate school at Cornell University under the Army Science and Technology Graduate Fellowship and graduated from Cornell in 1989 with a Master of Science degree in Applied Mathematics She is currently

an Associate Professor of Mathematics at Palomar College and has been teaching there since 1995 Cindy Anﬁ nson was a ﬁ nalist for Palomar College’s 2002 Distinguished Faculty Award

About The CoverThis cover image tells a story Our goal with this cover was to represent

how people interact and connect with technology in their daily lives

We selected this cover as it illustrates the fundamental idea of the

Clark/Anﬁ nson series—connecting concepts to applications and rote

mathematics to the real world since the skills and concepts in this series have their foundation in applications from the world around us

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LINEAR EQUATIONS AND INEQUALITIES WITH ONE VARIABLE 133

LINEAR EQUATIONS WITH TWO VARIABLES 205 SYSTEMS OF LINEAR EQUATIONS 311

EXPONENTS AND POLYNOMIALS 415 FACTORING AND QUADRATIC EQUATIONS 479 RATIONAL EXPRESSIONS AND EQUATIONS 563 RADICAL EXPRESSIONS AND EQUATIONS 647 MODELING DATA 741

ANSWERS TO PRACTICE PROBLEMS A-1 ANSWERS TO SELECTED EXERCISES B-1

R 1 2 3 4 5 6 7 8 9 A B

Brief Contents

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R.1 Operations with Integers 2Natural Numbers, Whole Numbers, and Integers • Number Lines • Relations Between Numbers • Absolute Value • Opposite of a Number • Operations with Integers

• Order of Operations

R.2 Operations with Fractions 18Prime Numbers and Prime Factorization • Simplifying Fractions and Equivalent Fractions • Fractions on Number Lines • Addition and Subtraction of Fractions

• Multiplication and Division of Fractions • Order of Operations

R.3 Operations with Decimals and Percents 30Place Value • Relationships Between Fractions and Decimals • Graphing Decimals on a Number Line • Rounding Decimals • Addition and Subtraction of Decimals • Multiplication and Division

of Decimals • Order of Operations with Decimals • What Is a Percent? • Converting Between Percents, Decimals, and Fractions • Problem Solving with Percents

Rational Numbers • Irrational Numbers • The Real Number System • Exact and Approximate Answers

Chapter R Summary 51Chapter R Review Exercises 54Chapter R Test 57

Chapter R Projects 58

1.1 Exponents, Order of Operations, and Properties of Real Numbers 62Exponents • Scientiﬁ c Notation • Order of Operations • Properties of Real Numbers

1.2 Algebra and Working with Variables 75Constants and Variables • Evaluating Expressions • Unit Conversions • Deﬁ ning Variables

• Translating Sentences into Expressions • Generating Expressions from Input-Output Tables

Contents

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C o n t e n t s vii

1.3 Simplifying Expressions 94Like Terms • Addition and Subtraction Properties • Multiplication and Distributive Properties • Simplifying Expressions

1.4 Graphs and the Rectangular Coordinate System 105Data Tables • Bar Graphs • Scatterplots • Rectangular Coordinate SystemChapter 1 Summary 123

Chapter 1 Review Exercises 127Chapter 1 Test 130

Chapter 1 Projects 132

with One Variable

Variables

2.1 Addition and Subtraction Properties of Equality 134Recognizing Equations and Their Solutions • Addition and Subtraction Properties of Equality • Solving Literal Equations

2.2 Multiplication and Division Properties of Equality 148Multiplication and Division Properties of Equality • Solving Multiple-Step Equations • Generating Equations from Applications • More on Solving Literal Equations

2.3 Solving Equations with Variables on Both Sides 164Solving Equations with Variables on Both Sides • Solving Equations That Contain Fractions • Equations That Are Identities or Have No Solution • Translating Sentences into Equations and Solving

2.4 Solving and Graphing Linear Inequalities on a Number Line 175Introduction to Inequalities • Solving Inequalities • Interval Notation and Number Lines • Compound Inequalities

Chapter 2 Summary 191Chapter 2 Review Exercises 196Chapter 2 Test 199

Chapter 2 Projects 200Cumulative Review Chapters 1–2 202

3.1 Graphing Equations with Two Variables 206Using Tables to Represent Ordered Pairs and Data • Graphing Equations by Plotting Points • Graphing Nonlinear Equations by Plotting Points • Vertical and Horizontal Lines

3.2 Finding and Interpreting Slope 217Interpreting Graphs • Determining a Rate of Change • Calculating Slope • Interpreting Slope

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4 Systems of Linear Equations

3.3 Slope-Intercept Form of Lines 238Finding and Interpreting Intercepts from Graphs • Finding and Interpreting Intercepts from Equations • Slope-Intercept Form of a Line

3.4 Linear Equations and Their Graphs 253Graphing from Slope-Intercept Form • Graphing from General Form • Recognizing a Linear Equation • Parallel and Perpendicular Lines

3.5 Finding Equations of Lines 269Finding Equations of Lines Using Slope-Intercept Form • Finding Equations of Lines from Applications • Finding Equations of Lines Using Point-Slope Form • Finding Equations of Parallel and Perpendicular Lines

3.6 The Basics of Functions 285Relations • Functions • Vertical Line Test • Function Notation • Evaluating FunctionsChapter 3 Summary 297

4.1 Identifying Systems of Linear Equations 312Introduction to Systems of Equations • Solutions to Systems of Equations • Solving Systems Graphically • Types of Systems

4.2 Solving Systems Using the Substitution Method 333Substitution Method • Inconsistent and Consistent Systems • Practical Applications of Systems

of Linear Equations

4.3 Solving Systems Using the Elimination Method 350Using the Elimination Method • More Practical Applications of Systems of Linear Equations • Substitution or Elimination?

4.4 Solving Linear Inequalities in Two Variables Graphically 365Linear Inequalities in Two Variables • Graphing Vertical and Horizontal Inequalities

4.5 Systems of Linear Inequalities 383Chapter 4 Summary 395

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5 Exponents and Polynomials

Equations

5.1 Rules for Exponents 416Product Rule for Exponents • Quotient Rule for Exponents • Power Rule for Exponents

• Powers of Products and Quotients

5.2 Negative Exponents and Scientiﬁ c Notation 426Negative Exponents • Using Scientiﬁ c Notation in Calculations

5.3 Adding and Subtracting Polynomials 438The Terminology of Polynomials • Adding and Subtracting Polynomials

5.4 Multiplying Polynomials 447Multiplying Polynomials • FOIL: A Handy Acronym • Special Products

5.5 Dividing Polynomials 458Dividing a Polynomial by a Monomial • Dividing a Polynomial by a Polynomial Using Long DivisionChapter 5 Summary 468

6.1 What It Means to Factor 480Factoring Out the Greatest Common Factor • Factoring by Grouping • Factoring Completely

6.2 Factoring Trinomials 492Factoring Trinomials of the Form x 2 + bx + c by Inspection • Factoring Trinomials of the

Form ax 2 + bx + c • More Techniques to Factor Completely

6.3 Factoring Special Forms 502Diﬀ erence of Squares • Perfect Square Trinomials • Summary of Factoring Techniques (Factoring Tool Kit)

6.4 Solving Quadratic Equations by Factoring 510Recognizing a Quadratic Equation • Zero-Product Property • Solving Quadratic Equations

by Factoring

6.5 Graphing Quadratic Equations 520Graphing Quadratic Equations by Plotting Points • The Relationship Between the Leading Coeﬃ cient a and the Graph • Locating the Vertex • Using the Axis of Symmetry

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7 Rational Expressions and

Equations

7.1 The Basics of Rational Expressions and Equations 564Evaluating Rational Expressions and Equations • Excluded Values • Simplifying Rational Expressions

7.2 Multiplication and Division of Rational Expressions 574Multiplying Rational Expressions • Expanding Unit Conversions • Dividing Rational Expressions • Basics of Complex Fractions

7.3 Addition and Subtraction of Rational Expressions 584Adding and Subtracting Rational Expressions with Common Denominators • Finding the Least Common Denominator (LCD) • Adding and Subtracting Rational Expressions with Unlike Denominators • Simplifying Complex Fractions

7.4 Solving Rational Equations 601Solving Rational Equations • Setting Up and Solving Shared Work Problems

7.5 Proportions, Similar Triangles, and Variation 616Ratios, Rates, and Proportions • Similar Triangles • VariationChapter 7 Summary 634

6.6 Graphing Quadratic Equations Including Intercepts 537Finding Intercepts • Putting It All Together to Sketch a GraphChapter 6 Summary 545

8.1 From Squaring a Number to Roots and Radicals 648Finding Square Roots • Evaluating Radical Expressions • Evaluating Radical Equations

• Simplifying Radical Expressions That Contain Variables • Finding Cube Roots

8.2 Basic Operations with Radical Expressions 659Simplifying More Complicated Radical Expressions • Adding and Subtracting Radical Expressions

8.3 Multiplying and Dividing Radical Expressions 670Multiplying Radical Expressions • Dividing Radical Expressions • Rationalizing the Denominator

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A Answers to Practice Problems A-1

Answers to Selected Exercises B-1

B Index I-1

8.4 Solving Radical Equations 684Checking Solutions to Radical Equations • Solving Radical Equations • Solving Applications Involving Radical Equations • Solving More Radical Equations

8.5 Solving Quadratic Equations by Using the Square Root Property 695Solving Quadratic Equations by Using the Square Root Property • Using the Pythagorean Theorem

8.6 Solving Quadratic Equations by Completing the Square and by the Quadratic Formula 707

Solving Quadratic Equations by Completing the Square • Solving Quadratic Equations by Using the Quadratic Formula

9.1 Modeling Linear Data 742Finding a Linear Model for Real Data • Finding a Linear Model • Using a Linear Model

to Make Estimates • Determining Model Breakdown

9.2 Working with Quadratic Models 763Determining Whether or Not the Graph of a Data Set Is Shaped Like a Parabola • Using Quadratic Models to Make Estimates • Determining When Model Breakdown Happens

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Company logo on T-shirts, 293

Craft fair sales, 331

Personalized candy bars, 252, 267 Photography, 720, 729

Plant nursery, 214, 638, 787 Plumbing, 131, 331 Real estate sales, 89, 118, 162 Restaurants, 42

Retail, 147, 199 Retirement gift, 570 Sales incentives, 258 Shipping, 87, 162, 163, 313 Spice mix, 412, 735 Sports equipment, 214 Stained-glass windows, 198 Surfboards, 571

Tea blend, 347 Theater ticket sales, 90, 91, 127 Tire store, 346

Toy manufacturing, 161, 277, 393, 553, 615 Trail mix, 403

Trailer hitch installation, 98 Trucking, 3, 625

Typesetting, 642 Union membership, 746, 761 Web design, 720

Window shutter production, 614

Education

Age and grade in school, 306 Alphabet Car toys, 175 Art supplies, 444 Cake-decorating class, 356 Chemistry lab, 54 Class enrollment, 189, 203, 474 College applications, 599 College credits, 78, 139 College library, 4 College printing costs, 78 Community college cost, 118 Community college enrollment, 762, 763 Course numbering system, 189, 203 Debate team, 571

Elementary school fundraising, 89 Gasoline for school district, 474 Gender of students, 39, 40, 49 Gifts for students, 128

Grading exams, 593 Graduation ceremony, 16 High school student cars, 295 High school track team, 331 Kindergarten class, 4, 103 Lab assignment, 615, 737 Mathletics team, 614 Museum fi eld trip, 565, 602 Piano teacher, 403

Preschool projects, 103 PTA gifts for teachers, 413 School bookstore, 188 School buses, 133, 161, 189 School supplies, 41, 171, 355 Science and engineering degrees, 106 Student heights, 287

Textbooks, 41, 295 Tuition costs, 61, 78, 84, 88, 91, 157, 283,

287, 295, 632 Tutoring, 89, 308, 734

Electronics

CD burning, 163, 267 Cell phone accessories, 348 Cell phone bill, 44, 84, 182, 188 Cell phone prices, 287

Computer hard drive, 185 Computer speed, 559 Data pits on compact discs, 436 Flat-screen TV, 199, 702 iPad, 39

LED fl at panel HDTV, 40 Mobile-phone cart, 346 Netbooks, 778 Network installation and maintenance, 777 Text messaging charges, 182, 188, 199 Transistor gate, 559

Video games, 777

Entertainment

Anniversary party, 93 Awards banquet, 613 Baby shower, 443, 614 Banquet at country club, 571 Banquet at hotel, 614 Baseball tickets, 296 Birthday party, 93 Cable TV, 49 Carnival, 256 Catering for conference, 93 Charity fund-raiser, 331, 362, 536 Cheerleader T-shirts, 97

College sports, 49, 84 Company Christmas party, 296 Football stadium, 629 Football tickets, 295, 393Applications Index

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Rock concert tickets, 229, 232, 236, 242, 251

Scout troop meeting, 376, 379

Capital equipment, 162

Car expense per mile, 84

CD (certifi cate of deposit), 363, 389, 413,

558, 735 Commission on sales, 89, 162, 206, 214,

324, 330, 346, 409, 412 Commission on stock options, 162, 163

Consumer debt, 121

Cost per item, 519, 536

Credit card debt, 252, 259, 267, 268, 304, 364

Hourly pay, 149, 306, 330

Internet payments transaction fee, 162

Investment accounts, 92, 128

Linear revenue model, 741

Loan origination fees, 145, 146, 161, 196

Personal income over fi ve years, 118

Profi t, 136, 141, 146, 147, 189, 197, 198,

199, 253, 267, 516, 519, 658, 693, 720,

729, 770, 777, 778, 787 Quarterly earnings, 108

Revenue, 253, 267, 348, 436, 554, 561, 720,

739, 764, 771, 777, 778

Shredding documents, 615 Simple interest, 164, 356, 357, 358,

362, 379 Stock account, 92, 363, 364, 392, 403,

405, 406 Stock prices, 295 Stock value, 432 Tax preparation, 189, 600 Total mortgage dollars in U.S., 162 Unit cost of production, 529 Value of used backhoe loader, 158 Weekly pay, 76, 81, 89, 91, 131, 135, 145,

188, 199, 208, 210, 237, 256, 293, 324,

346, 363, 375, 379, 625, 626, 632, 633, 737

Geography

Alaska population, 305 American Indian and Alaska Native population, 252

Arkansas population, 188 Atlantic Ocean storms, 117 Average daily low temperatures, 3, 4 Average high monthly temperatures, 119 City population growth, 217, 218, 228, 229,

231, 235, 241, 242, 250, 251, 299, 303, 304 Country populations, 294

Daily high temperatures, 16, 764, 766, 768,

782, 783, 787, 792 Daily low temperature, 49 Distance seen on surface of earth, 658, 738 Distance to horizon, 685

Elevation, 10, 17, 44, 49 Florida population, 189 Grand Canyon, 309 Hawaiian and Pacifi c Islander population, 252

Immigrants in U.S population, 107, 759, 760 Land plots, 174, 199

Los Angeles population, 16 Mexico City population, 789 Montana population, 786 Monthly low temperatures, 776, 777, 778 Nebraska population, 757

Nevada population, 759, 760 New Hampshire population, 757 New York population, 188 Norfolk, Virginia population, 16 North Carolina population, 780, 781 Ocean depths, 17

Ocean tide, 90 State population growth, 410 Temperatures in Antarctica, 109 Water in reservoir, 243 West Virginia population, 788

Cooking oil consumption, 456 Cookware purchase, 43 Dining area, 519 Electricity bill, 44, 49, 90 Fencing, 630

Flooring cost, 86 Food shopping, 29, 34, 35, 347, 362 Grocery coupons, 42

Grocery price comparisons, 577, 582, 617,

629, 642 Hallway rug, 519 Handyman, 331 High-fructose corn syrup consumption, 456 Home improvement project, 216

Lawn, 24, 85, 173 Living expenses, 375, 379, 383, 384,

392, 404 Mow curb, 643 Mowing the lawn, 599 Natural gas price, 252, 267 Nutrition, 42, 43

Oven/microwave combination, 331 Paint price comparisons, 639 Painting, 25, 29, 92, 612, 615, 642 Pie baking costs, 296

Plants for garden, 214, 303, 362 Plumber’s services, 131, 331 Price of oranges, 252 Recipes, 631 Renter’s expenses, 129 Roofi ng, 630

Spice prices, 617 Swimming pool, 612, 615 Tankless water heater, 185 Unit prices, 48, 50, 736, 737 Washer/dryer combination, 331 Water use, 216

Wireless-only households, 790

Medicine and Health

Asthma costs, 108 Autism diagnoses, 308 Body Mass Index, 646 Breast cancer fund-raising, 232, 245, 251 Calories burned by exercises, 231, 235, 251,

303, 305, 306, 557, 632, 640 Calories eaten per day, 409 Causes of death at ages 15–24, 316, 317 Chlamydia testing, 218

Coronary heart disease mortality, 327 Dental assistant, 203

Diabetes diagnoses, 231, 235, 251 Diabetic population of U.S., 760 Drug dosage, 576, 577, 578, 582 Exercise, 44

Fitness clubs, 328 Flu virus, 473 Height of children, 186, 784 Infl uenza season, 106

IV fl uids, 90 Leisure time physical activity, 762, 763 Life expectancy for Americans, 754 Life expectancy for Russian males, 751, 754 Maximum heart rate, 411

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Red blood cell, 475

Resting heart rate, 189, 732

Scanning medical records, 615

Smoking among young people, 761, 762

Target heart rate, 407, 760

Weight in relation to age, 306

Weight-loss diet, 211, 237, 244, 278, 283,

287, 308, 411, 617

Miscellaneous

Age of men at fi rst marriage, 749, 753, 756

Age of women at fi rst marriage, 751, 753,

Conference tote bags, 614

Copying church bulletins, 615

Long jump world record, 202

Marathon world records, 117

Olympic gold medals, 287

Perimeter, 157, 159, 163, 164, 173, 174,

331, 447, 473, 474, 475, 669, 735

Prehistoric counting bone, 1, 58

Prepaid phone card, 296

Politics, Government, and Military

American Recovery and Reinvestment Act

of 2009, 431

Crimes reported to police, 116

Gross domestic product (GDP), 436

Immigrants to United States, 107

Income tax withholding, 103

Income taxes, 90, 103

IRS standard mileage rate, 84 New York spending per person, 571 Postal rates, 90, 91, 757

Road crew, 578 Sales tax, 43, 56 Texas spending per person, 571 Total yearly taxes, 287 Traffi c accident investigation, 652, 657,

690, 692 Unemployment rate, 130, 760, 761 U.S national debt, 121, 431, 437 U.S Productivity Index, 131 Victory Obelisk in Moscow, 631 Washington Monument, 631 Washington state spending per person, 640

Recreation and Hobbies

Backpacking, 92, 138, 446 Band competition, 640 Baseball, 479, 529, 600 Basketball, 558, 571, 632 Bicycling, 89, 92, 215, 216, 402 Boat speed, 340, 342, 347 Bowling, 214

Boxing weight classes, 200 Camps for young people, 189 Cross country team, 614 Cross-country skiing, 583 Day camp T-shirts, 153 Dogs, 202, 294 Gym memberships, 283 Hand-painted dolls, 92 Hitting a ball, 529, 535, 536, 549, 554 LEGOLAND California, 189 Model rockets, 103, 643, 717, 718, 720, 731 Painting a mural, 599

Public park, 54, 216, 615 Ribbon for trim, 24 Rock climbing, 557 Rowing, 640 Running, 89, 640 Soccer, 202, 362, 600, 734, 736 Summer camp, 412

Throwing a ball, 519, 553 Track team, 331, 390 Volleyball, 146, 163, 619 White water rafting, 207 Women’s Club, 737 Woodworking workshop for kids, 175 YMCA activities, 446

Science and Engineering

Acid solutions, 311, 364 Alcohol solutions, 358, 364 Astronomical unit (AU), 433, 436, 437 Atomic diameter, 436

Biology lab assistant, 89 Blueprint, 629, 737 Conversion of units, 121, 128, 275,

282, 630 Diameters of planets, 437 Distances of planets, 473, 475

DNA molecule, 473 Electric charge of a proton, 433, 436, 437

Electricity generation, 308 Falling objects, 652, 657, 658, 689, 692,

697, 704, 705, 717, 718, 720, 728, 730,

731, 736, 739, 789 Golden rectangles, 644 Great Pyramid of Giza, 642 Guy wire, 702

Hooke’s Law, 632 Illumination of a light, 633 Lightning, 205, 216, 277 Light-years, 132, 415 Mass of an electron, 437 Masses of planets, 434, 436, 437 Nanotube, 431, 473

Pitch of roof, 647, 706 Plant cell, 431 Pressure in balloon, 633 Rain content of snow, 87, 274 Saline solutions, 93, 311, 359, 362,

364, 403 Scaling photographs, 563, 622, 630 Solar car, 582, 583

Sucrose solution, 363 Temperature changes, 10, 17 Temperature scales, 275 Temperatures and time of day, 90, 216,

237, 282 Width of a human hair, 436

Travel

Airplane passenger capacity, 189, 197 Airplane speed, 340, 341, 347, 363, 402,

412, 573, 734 Bike tour of Europe, 174 Boat speed, 363, 397, 406, 558 Bridge tolls, 296

Bus tour, 185 Commuter fl ight, 529 Cruise, 92

Currency conversions, 79–80 Cyclist’s speed, 582

Driving distance, 77, 81, 89, 135, 145, 160,

196, 199, 237, 306, 557, 632 Driving speed, 572, 577, 582, 627, 628,

633, 736, 738 Driving time, 92, 627, 628 Gasoline prices, 42, 118, 287, 295, 456 Gasoline used per week, 90

Miles per gallon, 283 Paddleboat speed, 342 Public transportation, 153, 161 Rental car, 198

Ship speed, 340 Ski trip, 446, 564, 601 Spring break trip, 474 Taxicab fare, 379 Tire purchase, 43 Towing charges, 188 Weight limit on car roof, 146, 147

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Our goal in writing this book is to help college students learn how to solve problems generated from realistic tions, even as they build a strong mathematical basis in beginning algebra We think that focusing our book on concepts and applications makes the mathematics more useful and vivid to college students One tried and true application that we will make more vivid is problem solving In our textbook, problem solving will be introduced as an organic, integrated part of the course, rather than separated off from the traditional skill set

applica-We include applications throughout the text, rather than in a stand-alone chapter or section Our goal is to tie applications

to realistic situations, providing straightforward background explanations of the mathematics so that students have a earth rationale in their heads for how to set up a problem At the same time, traditional algebra concepts and skills are presented throughout the book and the problem sets The applications give students a chance to practice their skills as well as to see the importance of mathematics in the real world We think that the combination of basic mathematical skills, concepts, and applica-tions will allow students to think critically about mathematics and communicate their results better, too

down-to-Discovery-Based Approach

Worked Examples with Practice Problems

Examples range from basic skills and techniques to

realistic applications with manageable data sets to

intriguing applications that call for critical thinking

about phenomena and daily activities In-text practice

problems help students to gain a sense of a concept

through an example and then to practice what was

taught in the preceding discussion

Worked examples and exercises require students to answer by explaining the meaning of concepts such as

slope, the x-intercept, and the y-intercept in context

The benefi t of requiring verbal explanations is that the

student learns to communicate mathematical concepts

and applications in a precise way that will carry over to

other disciplines and coursework such as psychology,

chemistry, and business

How many solutions are there?

CONCEPT INVESTIGATION

Consider the following linear inequality:

y # 22 x 1 5

1 Check to see whether the following points are in the solution set by substituting the

values into the inequality.

(0, 5) (2, 21) (4, 4) (0, 2) (3, 0) (22, 3)

2 Plot the points from part 1 that are in the solution set on the following graph Plot

only the points that made the inequality true.

19380_05_ch04_sec 4.4.indd 367 12/10/10 3:18 PM

Preface

Alice’s cell phone company charges $40 per month for the ¿ rst 400 minutes of use and minutes allowed in a month.

a Write an inequality to show that Alice must keep her monthly cell phone bill to at

most $45 (This is within her budget.)

b Solve the inequality from part a Write the solution in a complete sentence.

SOLUTION

a Since m is the number of minutes over 400, the monthly cell phone charges for

Alice are given by the expression 40 1 0.40m The inequality that represents Alice keeping her monthly cell phone bill to at most $45 is

her 250 texts allowed per month.

a Write an inequality to show that Amy must keep her monthly texting bill to at most

$10 (This is within her budget.)

b Solve the inequality from part a Write the solution in a complete sentence.

I l N i d N b Li

Solving inequalities that arise from applications

Example 4

Trang 19

Integrated “Student Work”

Clearly identifi able examples of “student work” appear throughout the text in Examples and Exercises These boxes present examples of correct and incorrect student work Students are asked to identify and fi x common stu-dent errors

Margin Notes

The margin contains three kinds of notes written to help the student with specifi c types of information:

1 Skill Connections provide a just- in-time review of

skills covered in previous sections of the text or courses, reinforcing student skill sets

2 Connecting the Concepts reinforce a concept by

show-ing relationships across sections and courses

3 The specifi c vocabulary of mathematics and

appli-cations is helpfully defi ned and reinforced through

margin notes called What’s That Mean?

Reinforcement of Visual Learning through

Graphs and Tables

Graphs and tables are used throughout the book to

organize data, examine trends, and have students gain

an understanding of how to graph linear and quadratic

equations The graphical and numeric approach helps

support visual learners, incorporating realistic

situa-tions into the text and reinforcing the graphs and data

that students see in their daily lives

Hand-drawn Graphs

A hand-drawn style graph is often used when

illustrat-ing graphillustrat-ing examples and answers to exercises The

hand-drawn style helps students visualize what their

work should look like

Graph the following linear equations using the slope and y-intercept Graph at least

to rise 2 and run 3

from the y-intercept

Repeat the rise/run process twice to ¿ nd two additional points

on the line and then connect them with a straight line.

b From the equation y 5 22 x 1 3, we see that the slope is 22 5 22 _

1 5 riserun and

the y-intercept is

the point 0, 3 Start by plotting the

y-intercept, 0, 3 , and then use the slope to rise 22 and run 1 from the

y-intercept Since the

rise is negative, we

go down 2 instead

of up Repeating this process twice yields two more points on the graph.

Graphing lines using the slope and y-intercept

Example 1

4 5

2 3

1

–1 –2 –3

(6, 5)

(0, 1) –3 –1 0 1 5

–4 –5

–5 –4

4 5

2 3 1

–1 –2 –3

(2, –1)

(0, 3)

–3 –1 0 1 5

–4 –5

–5 –6 –4

For Exercises 51 through 80, simplify each exponential

expression List the exponent rule or other rule that was

xy2 x 4 y 3

xy

19380_06_ch05_sec 5.1.indd 425 12/10/10 4:25 PM

Skill Connection

The leading sign of a term

A term will take the sign that is in

front of it The expression

5x 2 10

has two terms: 5x and 210 The

second term is negative 10 because

subtraction in front of the 10 can

be thought of as “plus a negative.”

Recall that to subtract two integers,

change the sign of the second

integer and add For example,

b The two terms are 4 x 2 and 16x Factoring each term yields

4 x 2 5 2 ? 2 ? x ? x 5 2 2 ? x 2

16x 5 2 ? 2 ? 2 ? 2 ? x 5 2 4 ? x Remember to select the lowest power of the common factors The result is the GCF is 2 2 ? x,or 4 x.

c The two terms are 2x and 27 Factoring each term yields

equation zero are called excluded values.

DEFINITION

Excluded Values Any number that makes the denominator of a rational

expression or equation equal zero is called an excluded value These values will

be excluded from the possible values of the variable so that division by zero will not occur.

Connecting the Concepts

What makes fractions undeﬁ ned?

Remember from Section R.1 that division by zero is unde¿ ned, but dividing into zero is zero.

This is why we exclude values for variables that make the denominator zero

basic algebra rules to make an

expression simpler We simplify

expressions.

Evaluate: Substitute any given

values for variables and simplify the resulting expression or equation.

Solve: Isolate the given variable

in an equation on one side of the equal sign using the properties of equality This will result in a value that the isolated variable is equal

to We solve equations.

Trang 20

P r e f a c e xvii

Exercise Sets

The exercise sets include a balance of both applications- and skill-based problems developed with a clear level of progression in terms of level of diffi culty Most exercise sets begin with a few warm-up problems before focusing on applications and additional skills practice A combination of graphical, numerical, and algebraic skill problems are included throughout the book to help students see mathematics from several different perspectives End-of-book answers are written in full sentences to underscore the emphasis on student communication skills

Making Connections

Application problems require a complete-sentence

answer, encouraging students to consider the

reasonable-ness of their solution As a result, model breakdown is

discussed and is used as a way of teaching critical

think-ing about the reasonableness of answers in some of the

examples and problems throughout each chapter

Learning with Technology

Appropriate Use of the Calculator

Most exercises do not require calculator usage, although

the book has been written to support the use of a scientifi c

calculator Calculator Details margin boxes will appear

as necessary to instruct students on the correct use of a

scientifi c calculator In certain Concept Investigations,

the calculator is used to help students with arithmetic

so that they may concentrate on looking for patterns

In selected applications, the calculator is used to do the

numerical computations so that students can work with

more realistic problem situations

Review Material

Chapter R reviews prealgebra topics most necessary for beginning algebra It includes signed numbers, fractions, decimals, and percents Chapter R empha-sizes traditional student weak spots such as fractions, absolute value, the idea of opposites, the order-of-operations agreement, and ends with coverage of the real number system

Extensive end-of-chapter material includes Chapter Summaries, Review Exercises, Chapter

Tests, Chapter Projects, and Cumulative Reviews

Chapter Summaries revisit the big ideas of the

chapter and reinforce them with new worked-out

examples Students can also review and practice

what they have learned with the Chapter Review

Exercises before taking the Chapter Test

Scientiﬁ c Notation

When using positive exponents, we may encounter very large numbers Large numbers

is called scienti¿ c notation

What happened to my calculator display?

1 Use your calculator to ¿ ll in the following table.

Exponential Expression Numerical Value

Input Calculator Display

8 13 5.497558139E11 Notice the E just before the 11.

8 13 5.49755813 9 11 Notice the space between the number and the

exponent.

8 13 5.497558139 3 10 11

A calculator display like those above means that the calculator has gone into ti¿ c notation mode This means that this number is so large that it cannot be displayed

scien-of the number as it can

When a calculator displays 5.497558139E 11 or 5.49755813 9 11 , it means

5.497558139 3 10 11 5 549,755,813,900 When a calculator displays 5.497558139E 11 , the number 11 to the right of the E represents the exponent on 10.

5.497558139E 11 5 5.497558139 3 10 11

19380_02_ch01_sec 1.1-1.2.indd 64 06/10/10 7:13 PM

Simplifying Fractions and Equivalent Fractions

There are many times when we want to express the idea of a fractional part of thing For example, we may work only half a day and, therefore, receive half of our daily pay The fraction 1

some-2 represents half In a fraction such as 12 , there are special names for the number on the top of the fraction and the number on the bottom.

DEFINITION

Numerator and Denominator In the fraction a b , the value of a is called the

numerator The value of b is called the denominator The – symbol is called

the fraction bar.

numerator

a

b fraction bar denominator

When we write a fraction such as 2 4 , it represents the shaded part of the circle.

2–

4 ⫽

When we write the fraction 1

2 , it represents the shaded part of the circle Because these two fractions represent the

same portion of the circle, they are equivalent fractions This

means that 24 5 1 2 The right side of this equation, 2 4 5 1 2 , is

said to be reduced to lowest terms.

Fraction

A fraction is often thought of as

a part of a whole People will say

of the cost,” meaning that the object costs only part of what it should A fraction can also refer

to the breaking up of a whole

“The land was partitioned into fractions” means that the whole (the land) was broken into smaller parts (the fractions).

Sometimes a fraction will represent more than 1 In this case,

the fraction is called an improper

fraction.

Numerator and denominator

The denominator of a fraction

tells us (denominates) how many equal-sized pieces the whole is divided into For example, in the fraction 24 , the whole is divided

up into four equal-sized pieces

The numerator of a fraction tells

us (enumerates) how many of the equal-sized pieces to consider

In the fraction 2

4 , we count 2 of the 4 equal-sized pieces.

1–

2 ⫽

1 Write in exponential form: 6 6 6 6 6 [1.1]

2 Write in exponential form: 8 8 8 8 [1.1]

3 Write in exponential form: 3

5 Write in expanded form: 5 6 [1.1]

6 Write in expanded form: 22 3 [1.1]

7 Write in expanded form: 2 _2

5 4 [1.1]

8 Write in expanded form: _1

11 3 [1.1]

9 Find the value without a calculator: 23 3 [1.1]

10 Find the value without a calculator: 26 2 [1.1]

11 Find the value without a calculator: 4 9 2 [1.1]

12 Find the value without a calculator: 2 _2

3 3 [1.1]

13 Write using standard notation: 2.07 3 10 5 [1.1]

14 Write using standard notation: 5.67 3 10 6 [1.1]

2 Is t 5 4.2 and D 5 294 a solution to D 5 70t, where

D is the distance traveled in miles when driving for

t hours?

3 Solve and check the answer: 25 1 x 5 217

4 Solve the literal equation for the variable c.

P 5 a 1 b 1 c

5 Use the equation P 5 R 2 C, where P represents

the pro¿t R represents the revenue and C is

11 De¿ ne your variables and translate the problem into an

equation Solve the equation and check the answer The perimeter of a rectangular building lot is 330 feet The width of the lot is 40 feet Find the length of the lot.

12 Solve the literal equation for the variable x y 5 m x 1 b.

13 Translate the sentence into an equation and solve

The sum of four times a number and 7 is negative seventeen.

14 Solve the equation and check the answer

10 x 2 15 5 7x 2 9

15 Solve the equation and check the answer:

4 x 1 3 5 4 x 2 1 1 7

Trang 21

Cumulative Review Exercises

Cumulative reviews appear after every two chapters,

and group together the major topics across chapters

Answers to all the exercises are available to students

in the answer appendix

Chapter Projects

To enhance critical thinking, end-of-chapter

projects can be assigned either individually or

as group work Instructors can choose which

projects best suit the focus of their class and give

their students the chance to show how well they

can tie together the concepts they have learned

in that chapter Some of these projects include

online research or activities that students must

perform to analyze data and make conclusions

Factoring Tool Kit

This tool kit summarizes factoring techniques

Chapter-by-Chapter Overview

Chapter R The text begins in Chapter R by reviewing some prealgebra concepts This chapter helps students to acquire a

work-ing knowledge of prealgebra and to reviews those topics most necessary for beginnwork-ing algebra, such as arithmetic with signed

numbers, fractions, absolute value, the idea of opposites, and the order-of-operations agreement This chapter also reviews the

traditional student weak spots—fractions, decimals, and percents This chapter culminates in the real number system The Rule of

Four is covered implicitly throughout the book (numeric-symbolic-graphical-verbal), beginning with Chapter R

Chapter 1 Titled The Building Blocks of Algebra, this chapter centers on order of operations, the properties of real numbers,

variables, and graphing Students will become aware that they are making an important transition with this chapter Variables

and variable expressions are introduced using a variety of approaches For example, students will learn how to translate

sen-tences into expressions, as well as how to set up an input-output table and look for underlying patterns to generate expressions

Students are also introduced to graphing, with a detailed explanation of scale and how that relates to graphing real-life data and

reading real-life graphs The importance of units and unit conversions is also covered, with the goal of helping students contend

with units in real-life problems

Chapter 2 Linear equations and inequalities with one variable are discussed from the perspective of the properties of equality

After students become familiar with these properties, solving linear equations in one variable where the variable appears on

both sides is presented The focus is on the underlying concepts of the properties of equality to help students understand the

algebra, as well as develop a useful skill set The chapter ends with solving linear inequalities of one variable and graphing the

solution set on a number line Interval notation is presented in conjunction with number lines

Chapter 3 Linear equations in two variables is the primary topic of this chapter Students initially graph equations in two

vari-ables by generating a table of values Some of these graphs are nonlinear, such as absolute value equations and simple quadratic

equations Slope is introduced as a rate of change Students learn how to compute slope using rise/run, as well as the slope

formula Emphasis is given to interpreting slope in the context of an application Graphing lines from slope-intercept form and

general form are both presented Finding intercepts is covered, as well as interpreting intercepts in the context of an application

The chapter covers fi nding equations of lines using slope-intercept form and point-slope form Parallel and perpendicular lines

are also covered The chapter ends with an optional section covering the basics of functions

Cumulative Review

1 Write in exponential form: 2 x ? 2 x ? 2 x ? 2 x ? 2 x

2 Find the value without a calculator: 2 _1

11 Helena is driving to another city to visit her friends

She plans to average a speed of about 65 miles per hour The distance Helena can travel in one day can be calculated by using the equation

D 5 65t

where D is the distance she travels in miles if she

15 An exercise physiologist estimates that a man burns

about 124 calories per mile while running Let

C 5 124m, where C is the total number of calories

a man burns while running m miles Create a table

of points that make sense in this situation, and graph them Connect the points with a smooth curve.

16 Use the equation y 5 x 2 1 3 to create a table of nine or more points and graph them Connect the points with

a smooth curve Clearly label and scale the axes Is the graph linear or nonlinear?

17 Find the slope of the line that goes through points

4, 23 and 22, 9

18 Find the x-intercept and the y-intercept of the equation

23x 1 8y 5 224 Graph the line using the intercepts

Clearly label and scale the axes.

19 Find the slope, x-intercept, and y-intercept of the line

2 x 1 7y 5 14.

20 Graph the line y 5 2 _3

5 x 2 2 using the slope and

y-intercept Graph at least three points Clearly label

and scale the axes.

21 Graph the line Clearly label and scale the axes

22 Determine whether or not the two lines are parallel,

perpendicular or neither Do not graph the lines The

3

The binomial x 1 5 is represented by the length of the rectangle The top of the rectangle has two lengths: one of length x and one of length 5 Adding these two lengths yields

x 1 5 The width of the rectangle represents the binomial x 1 3 Recall that the area

of a rectangle is length width Finding the area of each of the four smaller rectangles

yields the following:

5

x

Writt en Projec t

One or more people

Using the Box Met hod to Multiply Binomials

Trang 22

Chapter 4 Here, the theme of linear equations is continued and leads into a discussion of systems of linear equations Students

solve systems graphically, using the substitution method, as well as the elimination method Applications are presented, mixed

in with all three methods Solving a linear inequality in two variables graphically is covered, as well as solving systems of linear inequalities in two variables

Chapter 5 This chapter develops the exponent rules, including negative exponents Scientifi c notation, briefl y introduced in

Chapter 1, makes its appearance again in Chapter 5 in a more complete form Polynomials are defi ned, as well as the degree

of a polynomial Polynomial operations are then presented, addition, subtraction, multiplication, and division The section

on division of polynomials is broken up into three subsections, so that instructors can select the topics they need to cover for their course The three subsections are dividing a polynomial by a monomial, dividing a polynomial by a polynomial with no remainder, and fi nally, dividing a polynomial by a polynomial resulting in a remainder

Chapter 6 This chapter introduces factoring and quadratic equations Factoring out the GCF, factoring by grouping,

factor-ing trinomials, factorfactor-ing differences of squares and perfect square trinomials, and factorfactor-ing completely are all presented The topic of factoring is summarized with a “Factoring Tool Kit” to help students know which technique to use and when to use

it Students will learn to factor polynomials and solve quadratic equations by factoring and the square root property Graphing

quadratic equations where the x-intercepts may be determined by factoring fi nishes off Chapter Six in a section that pulls it all

together

Chapter 7 This chapter introduces rational expressions and equations Evaluating and simplifying rational expressions is

pre-sented, as well as the idea of excluded values Next, operations on rational expressions are covered: multiplying, dividing, ing, and subtracting rational expressions Adding and subtracting rational expressions is broken up into subsections on adding and subtracting with like denominators, fi nding a common denominator, and adding and subtracting with unlike denominators Solving rational equations is then covered, presenting applications from shared work problems Proportions, similar triangles, and direct and inverse variation end the chapter

add-Chapter 8 In add-Chapter Eight, radical expressions are introduced Square roots are emphasized, but cube roots are introduced

Simplifying radical expressions is covered, as well as adding and subtracting radical expressions Multiplying and dividing ical expressions is presented, including an introduction to rationalizing the denominator Solving radical equations is covered, including equations with an extraneous solution Now that students understand radicals, quadratic equations are revisited, this time solving them using the square root property The Pythagorean Theorem is covered The chapter concludes with a section

rad-on solving quadratic equatirad-ons by completing the square, and by using the quadratic formula A proof of the quadratic formula

is included as an optional chapter project

Chapter 9 This last chapter in the text is a bridge chapter to Intermediate Algebra, Connecting Concepts through Applications

This chapter has students explore modeling using the concept of “eyeball best-fi t” lines and quadratics from real-world data The idea is that students can readily understand how models relate to the real world and how some models may break down if not constructed carefully Appropriately sized data sets are used in this chapter, to keep the focus on the algebra concepts being studied As fi nding quadratic models is beyond the skill level of beginning algebra students, models are given to the student to evaluate and interpret

Appendices:

Appendix A Answers to Practice Problems

Appendix B Answers to Selected Exercises

For the Instructor:

Annotated Instructor’s Edition

The Annotated Instructor’s Edition provides the

complete student text with answers next to each

respective exercise As shown here, it includes new

classroom examples with answers to use in the

lec-ture that parallel each example in the text These could

also be used in class as additional practice problems

for students Teaching tips are also embedded to help

with pacing as well as key ideas that instructors may

want to point out to their students

Greensboro-Las Vegas–

Paradise, NV

1 Which city has the smaller population? Explain your response.

2 Which city’s population is growing faster? Explain your response. ▼

Trang 23

Complete Solutions Manual (0534419380)

Eve Nolan

The Complete Solutions Manual provides worked-out solutions to all of the problems in the text.

Instructor’s Resource Binder (0538736755)

Maria H Andersen, Muskegon Community College

Each section of the main text is discussed in uniquely designed Teaching Guides containing instruction tips, examples,

activi-ties, worksheets, overheads, assessments, and solutions to all worksheets and activities

PowerLecture with ExamView ® (0840054696)

This CD-ROM provides the instructor with dynamic media tools for teaching Create, deliver, and customize tests (both print

and online) in minutes with ExamView® Computerized Testing featuring algorithmic equations Microsoft® PowerPoint®

lec-ture slides, fi gures, and additional new examples from the annotated instructor’s edition are also included on this CD-ROM

Solution Builder

This online instructor database offers complete worked solutions to all exercises in the text, allowing you to create

custom-ized, secure solutions printouts (in PDF format) matched exactly to the problems you assign in class

Visit www.cengage.com/solutionbuilder

Text-Speciﬁ c Videos (0538734000)

Rena Petrello, Moorpark College

These 10- to 20-minute problem-solving lessons cover nearly every learning objective from each chapter in the text Recipient

of the “Mark Dever Award for Excellence in Teaching,” Rena Petrello presents each lesson using her experience teaching

online mathematics courses It was through this online teaching experience that Rena discovered the lack of suitable content for

online instructors, which caused her to develop her own video lessons and ultimately this video project These videos have won

four awards: two Telly awards, one Communicator Award, and one Aurora Award (an international honor) Students will love

the additional guidance and support when they have missed a class or when they are preparing for an upcoming quiz or exam

The videos are available for purchase as a set of DVDs or online via cengagebrain.com

Enhanced WebAssign with eBook (0538738103)

Exclusively from Cengage Learning, Enhanced WebAssign®, used by over one million

students at more than 1,100 institutions, allows you to assign, collect, grade, and record

homework assignments via the Web This proven and reliable homework system includes

thousands of algorithmically-generated homework problems, links to relevant textbook sections, video examples,

problem-specifi c tutorials, and more

The authors worked very closely with their media team, writing problems with Enhanced WebAssign in mind and selecting

exercises that offer a balance of applications and skill development Additionally, the “Master It” questions break questions into

multiple parts, forcing students to confi rm their understanding as they work through each step of a problem These problems

support the conceptual applied approach of the textbook, and instructors can choose to assign these or use the tutorial version

In addition, diagnostic quizzing for each chapter identifi es concepts that students still need to master, and directs them to the

appropriate review material Students will appreciate the interactive eBook, which offers search, highlighting, and note-taking

functionality, as well as links to multimedia resources All of this is available to students with Enhanced WebAssign

For the Student:

Student Solutions Manual (0534465331)

Eve Nolan

Contains fully worked-out solutions to all of the odd-numbered end-of-section exercises as well as the complete worked-out

solutions to all of the exercises included at the end of each chapter in the text, giving students a way to check their answers and

ensure that they took the correct steps to arrive at an answer

Student Workbook (1111568901)

Maria H Andersen, Muskegon Community College

Get a head-start The Student Workbook contains all of the assessments, activities, and worksheets from the Instructor’s

Resource Binder for classroom discussions, in-class activities, and group work.

Trang 24

Enhanced WebAssign with eBook (0538738103)

Exclusively from Cengage Learning, Enhanced WebAssign®, used by over one million of

your fellow students at more than 1,100 institutions, allows your instructor to assign, collect,

grade, and record homework assignments via the Web This proven and reliable homework system includes thousands of algorithmically-generated homework problems and offers many tools to help you, including links to relevant textbook sections, video examples, and problem-specifi c tutorials You also have access to an interactive eBook with search, highlighting, and note-taking functionality as well as links to multimedia resources

In addition, diagnostic quizzing for each chapter identifi es concepts that you may still need to master, and directs you to the appropriate review material All of this is available to you with Enhanced WebAssign

Accuracy and Development Process

Periodically during the authoring process, there were several phases of

devel-opment where our manuscript was sent out to be reviewed by fellow college

mathematics instructors either via a traditional “paper” review and/or via

in-person focus groups The feedback we received, very often including very

thoughtful markups of the manuscript, related to accuracy, pacing,

order-ing of topics, accessibility and readorder-ing level, integration of technology, and

completeness of coverage At each stage, we analyzed and incorporated this

feedback into the manuscript

The manuscript also benefi ted from student and instructor feedback from class tests Bearing in mind the developmental math student and our applica-

tions-fi rst approach, we took great pains to ensure the reading level was

appro-priate, working closely with a developmental reading and writing instructor to

confi rm this

While revising the manuscript, a 10-person advisory board—made up of instructors who had seen the manuscript in different iterations—was specifi cally tasked with consider-

ing content queries and design and art questions

The fi nal manuscript was delivered to Brooks/Cole-Cengage Learning considerably improved following this rigorous review, development, and revision process Once in production, the manuscript was cycled through a process by specialized team members: production editor, copyeditor, accuracy reviewer, designer, proofreader, art editor, artists, and compositor Each team member pays special attention to accuracy and completeness As each phase of the cycle was completed, the manuscript was sent to us to verify any suggested changes or corrections By the time this manuscript was published, it had been through the many phases of the production process and at least 10 pair of highly trained eyes verifi ed its accuracy and completeness Please be assured that accuracy was our primary goal and we take great pride in having partnered with many people to deliver

an accurate, engaging and meaningful teaching tool for your beginning algebra classes

Advisory Board Members

Scott Barnett, Henry Ford Community College

Maryann Firpo, Seattle Central Community College

Kevin Fox, Shasta College

Brian Karasek, South Mountain Community College

Krystyna Karminska, Thomas Nelson Community College

Photo credits clockwise from the top: istockphoto.com/DOUGBERRY; Image © seanelliottphotography 2010 Used under license from Shutterstock.com; © Ian Leonard/Alamy; Image © ZTS Used under license from Shutterstock.com; Courtesy of White Loop Ltd; © Istockphoto.com/Chris Schmidt; © Istockphoto.com/clu; © Istockphoto/Carmen Martínez Banús

Reviewers

Authoring a textbook series is a huge undertaking and we are very grateful to so many colleagues who assisted us throughout the many stages of development It was a painstaking process made possible by your willingness to collaborate with us and share your thoughtful feedback and comments We thank you most sincerely for your many efforts on our behalf You all gave

of your time and expertise most generously: reviewing multiple rounds of manuscript, preparing for and attending our detailed

development focus groups, class testing the manuscript in its many iterations, and class testing and reviewing the online work problems we created We couldn’t have done this without you and your students and the series and its accompanying ancillary program are a testimony to that

home-Amy Keith, University of Alaska, Fairbanks Brianne Lodholtz, Grand Rapids Community College Marilyn Platt, Gaston College

Janice Roy, Montcalm Community College James Vallade, Monroe County Community College

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Katherine M Adams, Eastern Michigan University

Kent Aeschliman, Oakland Community College

Tim Allen, Delta College

Alex Ambriosa, Hillsborough Community College, Brandon

Campus

Ken Anderson, Chemeketa Community College

Frank Appiah, Maysville Community & Technical College

Sonia Avetisian, Orange Coast College

Margaret Balachowski, Everett Community College

Farshad Barman, Portland Community College

Scott Barnett, Henry Ford Community College

Alison Becker Moses, Mercer County Community College

Michelle Beerman, Pasco-Hernando Community College

Nadia Benakli, New York City College of Technology

David Bendler, Arkansas State University

Joel Berman, Valencia Community College East

Rebecca Berthiaume, Edison Community College

Scott Berthiaume, Edison Community College

Nina Bohrod, Anoka-Ramsey Community College

Jennifer Borrello, Grand Rapids Community College

Ron Breitfelder, Manatee Community College

Mary Brown, Eastern Oregon University

Reynaldo Casiple, Florida Community College

James Chesla, Grand Rapids Community College

Shawn Chiappetta, University of Sioux Falls

Lisa Christman, University of Central Arkansas

Astrida Cirulis, Concordia University Chicago

Adam Cloutier, Henry Ford Community College

Elizabeth Cunningham, Santa Barbara City College

Amy Cupit, Copiah-Lincoln Community College

Robert Diaz, Fullerton College

William Dickinson, Grand Valley State University

Susan Dimick, Spokane Community College

Christopher Donnelly, Macomb Community College

Randall Dorman, Cochise College

Karen Edwards, Diablo Valley College

Janet Evert, Eerie Community College, South

Maryann Faller, Adirondack Community College

Robert Farinelli, Community College of Allegheny County

Maryann Firpo, Seattle Central Community College

Thomas Fitzkee, Francis Marion University

Nancy Forrest, Grand Rapids Community College

Kevin Fox, Shasta College

Marcia Frobish, Grand Valley State University

Brian Goetz, Kellogg Community College

John Golden, Grand Valley State University

Gail Gonyo, Adirondack Community College

Lori Grady, University of Wisconsin, Whitewater

James Gray, Tacoma Community College

John Greene, Henderson State University

Kathryn Gundersen, Three Rivers College

Joseph Haberfeld, Columbia College Chicago

Susan Hahn, Kean University

Shelle Hartzel, Lake Land College

Mahshid Hassani, Hillsborough Community College

Stephanie Haynes, Davis & Elkins College

Julie Hess, Grand Rapids Community College Elaine Hodz, Florida Community College, Kent Kalynda Holton, Tallahassee Community College Sharon Hudson, Gulf Coast Community College Daniel Jordan, Columbia College Chicago Laura Kalbaugh, Wake Tech Community College Brian Karasek, South Mountain Community College Krystyna Karminska, Thomas Nelson Community College Ryan Kasha, Valencia Community College

Fred Katirae, Montgomery College Amy Keith, University of Alaska, Fairbanks Tom Kelley, Henry Ford Community College Gary S Kersting, North Central Michigan College Dennis Kimzey, Rogue Community College

Mike Kirby, Tidewater Community College

Mark Krasij, University of Texas at Arlington Theodore Lai, Hudson County Community College Ivy Langford, Collin County Community College Richard Leedy, Polk State College

Mary Legner, Riverside City College Allyn Leon, Imperial Valley Community College Andrea Levy, Seattle Central Community College Brianne Lodholtz, Grand Rapids Community College Daniel Lopez, Brookdale Community College Cathy Lovingier, Linn Benton Community College David Maina, Columbia College Chicago

Vivian Martinez, Coastal Bend College Jane Mays, Grand Valley State University Tim Mezernich, Chemeketa Community College Pam Miller, Phoenix College

Christopher Milner, Clark College Jeff Morford, Henry Ford Community College Brian Moudry, Davis & Elkins College Ellen Musen, Brookdale Community College and Palm Beach

Marilyn Platt, Gaston College Sandra Poinsett, College of Southern Maryland Michael Price, University of Oregon

Jill Rafael, Sierra College Linda Reist, Macomb Community College David Reynolds, Treasure Valley Community College Jolene Rhodes, Valencia Community College

Patricia Rhodes, Treasure Valley Community College Janice Roy, Montcalm Community College

Sean Rule, Central Oregon Community College Lily Sainsbury, Brevard Community College, Cocoa Mark Schwartz, Southern Maine Community College Cindy Scofi eld, Polk Community College

Lenora Shepherd, Atlantic University Sailakshmi Srinivasan, Grand Valley State University Barbara Tozzi, Brookdale Community College Jeannette Tyson, Valencia Community College

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James Vallade, Monroe County Community College

Andria Villines, Bellevue College

Xiaomin Wang, Rochester Community and Technical College

Matt Williamson, Hillsborough Community College,

South Shore

Deborah Wolfson, Suffolk County Community College

Fred Worth, Henderson State University

Carol Zavarella, Hillsborough Community College, Ybor

Developmental Focus Group Participants

Khadija Ahmed, Monroe Community College

Elaine Alhand, Ivy Tech Community College

Ken Anderson, Chemeketa Community College

Katherine Jankoviak Anderson, Schoolcraft College

Margaret Balachowski, Everett Community College

Farshad Barman, Portland Community College

Lois Bearden, Schoolcraft College

Clairessa Bender, Henry Ford Community College

Kathy Burgis, Lansing Community College

Caroline Castel, Delta College

Mariana Coanda, Broward College

Greg Cripe, Spokane Falls Community College

Alex Cushnier, Henry Ford Community College

Susan Dimick, Spokane Community College

Julie Fisher, Austin Community College

Will Freeman, Portland Community College

Irie Glajar, Austin Community College

Deede Furman, Austin Community College

James Gray, Tacoma Community College

Mahshid Hassani, Hillsborough Community College

Farah Hojjaty, North Central Texas College

Catherine Holl-Cross, Brookdale Community College

Susan Hord, Austin Community College

Liz Hylton, Clatsop Community College

Eric Kean, Western Washington University

Beth Kelch, Delta College

Dennis Kimzey, Rogue Community College

Becca Kimzey

Mike Kirby, Tidewater Community College

Patricia Kopf, Kellogg Community College

Kay Kriewald, Laredo Community College

Riki Kucheck, Orange Coast College

Joanne Lauterbur, Baker College

Ivy Langford, Collin County Community College

Melanie Ledwig, Victoria College

Richard Leedy, Polk State College

Brianne Lodholtz, Grand Rapids Community College

Daniel Lopez, Brookdale Community College

Babette Lowe, Victoria College

Vinnie Maltese, Monroe Community College

Tim Mezernich, Chemeketa Community College

Michael McCoy, Schoolcraft College Charlotte Newsome, Tidewater Community College Maria Miles, Mount Hood Community College Karen Miffl in, Palomar College

Pam Miller, Phoenix College Christopher Milner, Clark College Carol McKilip, Southwestern Oregon Community College Vivian Martinez, Coastal State College

Anna Maria Mendiola, Laredo Community College Ellen Musen, Brookdale Community College and Palm Beach

State College

Joyce Nemeth, Broward College Katrina Nichols, Delta College Joanna Oberthur, Ivy Tech Community College Ceci Oldmixon, Victoria College

Mary B Oliver, Baker College Diana Pagel, Victoria College Joanne Peeples, El Paso Community College Sandra Poinsett, College of Southern Maryland Bob Quigley, Austin Community College Jayanthy Ramakrishnan, Lansing Community College Patricia Rhodes, Treasure Valley Community College Billie Shannon, Southwestern Oregon Community College Pam Tindel, Tyler Junior College

Barbara Tozzi, Brookdale Community College James Vallade, Monroe Community College Thomas Wells, Delta College

Charles Wickman, Everett Community College Eric Wiesenauer, Delta College

Andy Villines, Bellevue Community College Beverly Vredevelt, Spokane Falls Community College Mary Young, Brookdale Community College

Class Testers

Tim Allen, Delta College Scott Barnett, Henry Ford Community College Jennifer Borrello, Grand Rapids Community College Damon Ellingston, Seattle Central Community College Maryann Firpo, Seattle Central Community College Marcia Frobish, Grand Valley State University Rathi Kanthimathi, Seattle Central Community College Daniel Lopez, Brookdale Community College

Ellen Musen, Brookdale Community College and Palm Beach

State College

Mary Stinnett, Umqua Community College Barbara Tozzi, Brookdale Community College Randy Trip, Jefferson Community College Thomas Wells, Delta College

Michael White, Jefferson Community College Eric Wiesenauer, Delta College

Paul Yu, Grand Valley State University

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To the Student

Welcome to a course on beginning algebra This text was designed to help you learn the algebra skills and concepts you need

in future math, science, and other courses Critical algebra skills are presented to give you a solid foundation in mathematics

and concepts are presented Concepts are presented so that you understand the “why” of what you are doing The applications

selected are based more in real-life situations, so you can learn how to interpret mathematical results

We hope that you will fi nd the real-life applications to be interesting and helpful in understanding the material An emphasis

is placed on thinking critically about the results you fi nd throughout the book and communicating these results clearly The

answers in the back of the book will help demonstrate this level of thinking and communication We believe improving these

skills will benefi t you in many of your classes and in life

There are several features of the book that are meant to help you review and prepare for exams Throughout the sections are

defi nition boxes and margin features that point out key skills or vocabulary that will be important to your success in mastering

the skills and concepts We encourage you to use the end-of-chapter reviews and chapter tests as an opportunity to try a

variety of problems from the chapter and test your understanding of the concepts and skills being covered There is a review of

prealgebra topics at the beginning of the text–Chapter R if you fi nd yourself in need of a refresher

We hope these tools will be a good reference if you fi nd yourself in need of a little more help Remember that your instructor

and fellow classmates are some of the best resources you have to help you be successful in this course and in using this text

Good luck in your Beginning Algebra course!

Sincerely,Mark ClarkCindy Anfi nson

Acknowledgments

We would like to thank Eve Nolan and Ellen Musen for their many helpful suggestions for improving the text We are also

grateful to Scott Barnett, who has done an excellent job with the accuracy checking for this text We also thank the editorial,

production, and marketing staffs of Brooks/Cole: Charlie Van Wagner, Carolyn Crockett, Rita Lombard, Stefanie Beeck,

Carrie Jones, Jennifer Cordoba, Cheryll Linthicum, Gordon Lee, Mandee Eckersley, Heleny Wong, and Darlene Macanan for

all of their help and support during the development and production of this edition Thanks also to Vernon Boes, Leslie Lahr,

and Lisa Torri for their work on the design and art program, and to Barbara Willette and Susan Reiland for their copyediting

and proofreading expertise We especially want to thank Don Gecewicz, who did an excellent job of ensuring the accuracy

and readability of this edition, mentoring us through this process Our gratitude also goes to Cheryll Linthicum and Jill Traut,

who had an amazing amount of patience with us throughout production We truly appreciate all the hard work and efforts of

the entire team

Mark ClarkCynthia Anfi nson

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throughout recorded history and even in prehistoric times The Ishango bone was found

in 1960 in Africa, in an area near the border between Uganda and the Congo The bone is that

of a baboon, and it is currently estimated to be more than 20,000 years old The tally marks on the bone were originally thought to be counting marks, but some scientists now believe that the marks indicate knowledge of multiplication and division by two This bone shows that prehistoric peoples had a fi rm understanding of counting.

Review of Prealgebra

R

and Percents

R.1 R.2 R.3

R.4

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LEARNING OBJECTIVES

Distinguish natural numbers, whole numbers, and integers

Use number lines

Find the absolute value of an integer

Find the opposite

Use the order-of-operations agreement for integers

Natural Numbers, Whole Numbers, and Integers

People have been doing mathematics continuously throughout recorded history There is strong evidence that in prehistory (before recorded history), people were using mathematics First, they invented numbers and counting Then people invented ways of combining numbers Arithmetic is the study of combining numbers through the operations

of addition, subtraction, multiplication, and division

Humans like to classify objects into various sets For example, in the animal dom, animals have been classifi ed into sets such as the reptiles, amphibians, birds, mam-mals, and fi sh In the plant world, some classifi cations are ferns, conifers, and fl owering plants Mathematicians also like to classify numbers into sets

king-The fi rst set of numbers that people used are the natural (or counting) numbers

Ancient peoples did not see numbers as ideas, as we do today They used numbers to count things up For example, “two sheep” is how ancient peoples would have used the value 2 It is a big step to make the mental shift from “two sheep” to “two things”

and then to “2” as an idea with no “things” attached to it Zero (0) is not in the set

of counting numbers, as the ancients would have had no reason to count a set of no sheep

Zero was a diffi cult concept for people to grasp, even though zero is taken for granted today Zero seems to have developed from two different needs One is the use of zero as

a placeholder, and the other is a symbol to represent that no objects are present As an example of zero as a placeholder, in our number system, 102 is a different number from

12 The value of zero holds the place of tens, and the zero is necessary to distinguish between the numbers 102 and 12 It appears that zero as a placeholder developed in both India and Mexico (in the Mayan civilization) independently

Zero also represents the number 0 or, as the ancients would have seen it, no objects are present This idea of zero as a symbol was published in three different Indian texts about 600 C.E When we add zero to the set of natural numbers, we call this new set the

whole numbers.

DEFINITIONS

Natural Numbers The natural numbers consist of the set {1, 2, 3, }.

Whole Numbers The whole numbers consist of the set {0, 1, 2, 3, }.

When we count the number of wheels on a wheeled vehicle (such as a bike or a car),

it is clear that the vehicle can have 1, 2, 3, 4, or more wheels Counting the number of

wheels is an example of using values from the set of natural numbers

Suppose we want to count up the number of people in a room The room can contain

0, 1, 2, people The value of 0 needs to be included, since the room could be empty

The number of people in the room takes on values from the set of whole numbers.

Sets

Mathematicians call a collection

of objects a set.

The curly brace symbols, { },

in mathematics indicate a set.

The symbols, , denote a

pattern These symbols mean

that the set continues in this way

indefi nitely.

A set with no elements is

called the empty set It is written

{ } or .

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Integers Whole Numbers Natural Numbers

The use of negative numbers fi rst appeared in ancient China (200 B.C.E to 220 C.E.)

as answers to certain equations Negative numbers also appeared when traders balanced

their accounts and had to show a loss As with the number 0, it took a while for the

concept of negative numbers to be widely accepted Today, we use negative numbers

in different fi elds, such as the measurement of extremely cold temperatures,

double-entry bookkeeping, and describing profi t-loss situations The set of positive and

nega-tive whole numbers is called the set of integers In weather forecasts, the temperature of

the day is often measured in integers The temperature on a cold winter’s day in Fargo,

North Dakota, could be 230°F The summer temperature in San Diego, California,

could be 75°F Pure oxygen freezes at about 2361°F

DEFINITION

Integers The integers consist of the set { , 23, 22, 21, 0, 1, 2, 3, }.

The diagram at the right will help you to visualize the number systems The natural numbers are contained in the set of whole numbers, and the set of whole numbers is

contained in the set of integers This diagram is a visual way to see that if a number is a

natural number, then it is also a whole number and an integer

1 Place each number in the smallest possible set it fi ts into.

2 List all the natural numbers between 25 and 4, including the endpoints 25 and 4.

3 List all the whole numbers between 27 and 3, including the endpoints 27 and 3.

4 List all the integers between 23 and 5, including the endpoints 23 and 5.

What kind of number is that?

Determine the set(s), natural numbers, whole numbers, or integers, to which all possible

numbers described by the statement could belong

a The number of dogs boarding at a kennel during a two-week period.

b The average daily low temperature, in degrees Fahrenheit, in Fairbanks, Alaska,

over the course of a year

c The number of tires on a truck used for long-haul trucking (long-distance

transportation of goods) in the United States

SOLUTION

a This value (the number of dogs) is a whole number The number of dogs boarding

at a kennel could be 0, 1, 2, Since the kennel could be empty on any given day (0 dogs), the value 0 is included as a possibility These values are also integers

b This is an example from the set of integers The average daily low in Fairbanks

was 213°F in January to 50°F in July The temperatures range from negative numbers to positive numbers

c This is an example from the natural numbers, whole numbers, or integers Trucks

can have four or more tires, so the number of tires could be 4, 5, 6,

Classifying numbers

Example 1

Integers Whole Numbers Natural Numbers

▼

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PRACTICE PROBLEM FOR EXAMPLE 1Determine the set(s), natural numbers, whole numbers, or integers, to which all possible numbers described by the statement could belong.

a The number of children attending kindergarten at Park View Elementary School.

b The average daily low temperature, in degrees Fahrenheit, in Minneapolis,

Minnesota, over the course of a year

c The number of people in the quiet study room in the library at Irvine Valley

Community College

Number Lines

Number lines are a useful way to visualize numbers A number line is like a ruler It

is straight with a consistent scale This means the tick marks on the ruler are the same

distance apart, and the number of units the marks are apart is called the scale Numbers

get larger (increase) as we move to the right on the number line Numbers get smaller (decrease) as we move to the left on the number line The positive numbers are to the right of 0, and the negative numbers are to the left of 0

The number line below has scale of 1 This is because the tick marks on the line are

The scale of a number line is the

distance between the consistent

and evenly spaced tick marks on

the number line.

Trang 32

Graph the numbers 23, 0, 1, and 4 on the following number line.

PRACTICE PROBLEM FOR EXAMPLE 3

Graph the numbers 26, 8, 23, and 7 on the following number line

Relations Between Numbers

When we look at a number line, we can see that some numbers are larger than others

Mathematicians express this idea by saying that the integers are a number system that

has order Order means that we can organize a set of numbers from smallest to largest

The symbols given in the following table are mathematical notation used to express

order in numbers

a $ b a is greater than or equal to b 22 $ 24

a , b , c b is between a and c, not

The symbols #, ,, $, and Þ are called inequality symbols, and the fi rst six rows in

the table are called inequalities.

Fill in each blank column with , or Draw a number line to visualize the correct

relationship between the values This is especially helpful in comparing two negative

In the last row of the table, we

wrote a < b The < symbol means

“approximately equal to.” It lets the reader know that this solution

is not exact The solution has been rounded or chopped off in some way When we fi nd a decimal approximation to an exact answer,

we will use the approximately equal to symbol <.

Notation

A set of symbols that represents something can also be called

notation For example, musical

notation is formed by musical notes and other musical symbols.

▼

Trang 33

on the number line.

PRACTICE PROBLEM FOR EXAMPLE 4Fill in each blank with , or If needed, draw a number line to visualize the correct relationship between the values

1 What is the distance between 26 and 0?

4 Can a distance be positive?

5 Can a distance be negative?

6 Can a distance be zero?

What is the distance between a number and zero?

To discuss distance on a number line, we use the idea of absolute value Absolute

value is defi ned to be a distance measurement, so it must be a positive number or 0 (nonnegative) Distances cannot be negative Therefore, absolute value can never result

in a negative number

DEFINITION

Absolute Value The absolute value of a number is the distance between

that number and 0

Note The symbols are the notation for absolute value

Nonnegative

When a quantity takes on only

values that are 0 or positive

(that is, greater than or equal to

0), mathematicians will say the

quantity is nonnegative.

Trang 34

For example, 24 5 4, since the distance between 24 and 0 is 4 See the number line below.

Find the value of the absolute value expressions

25, 0, 2, 3, 12 Convert back to the original notation

25, 0 , 22 , 3 , 12 PRACTICE PROBLEM FOR EXAMPLE 6

Place the following numbers in increasing order (from smallest to largest):

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C H A P T E R R R e v i e w o f P r e a l g e b r a

8

Opposite of a Number

Look at the following number line The numbers 7 and 27 are the same distance from

0, so they have the same absolute value

Numbers that have the same absolute value but different signs are called opposites

Another way to read the number 27 (negative 7) is the opposite of 7

DEFINITION

Opposite of a Real Number The opposite of any number has the same

absolute value but differs in sign

Note Read 2a as the “opposite of a,” not as “negative a.”

First, rewrite the math expression as a sentence Then fi nd the value of the expression

a 2 8 b 2 21 c 2 0

Operations with Integers

We can use the number line to visualize adding and subtracting integers (signed numbers) Let’s discuss addition fi rst To add signed numbers, we move to the right on the number line when adding a positive number Suppose we want to add 28 1 7 We start on the number line at 28 and count over 7 units to the right (since 7 is a positive number) The solution is 21

Drawing a number line each time we want to add or subtract signed numbers can

be time-consuming We usually add and subtract signed numbers using the following steps

Rewriting an expression in sentence form

Example 7

Expression

In English, an expression can

mean a word or a phrase In

mathematics, an expression is

a mathematical phrase It is a

combination of numbers and

symbols An expression does not

contain an equal sign.

Signed Numbers

In algebra, a signed number refers

to a number with either a positive

or negative sign Positive numbers

are written with no sign Thus,

5 5 15 Negative numbers are

always written with the negative

sign, such as 25.

Trang 36

Steps to Add or Subtract Integers

1 To add integers with the same sign: Add the absolute values of the numbers

Attach the same sign of the numbers to the sum

2 To add integers with different signs: Take the absolute value of each

number Subtract the smaller absolute value from the larger Attach the sign

of the number that is larger in absolute value

3 To subtract integers: Change the sign of the second integer (reading left to

right) and add as explained above

Add the integers

SOLUTION

a These two integers have different signs Take the absolute value of each number

and subtract the smaller absolute value from the larger Subtract 15 from 29

29 2 15 5 14

The number that is larger in absolute value is –29, so the answer will be negative

Attach a negative sign to 14

229 1 15 5 214

The fi nal answer is –14.

b These two integers have different signs Take the absolute value of each number

and subtract the smaller absolute value from the larger Subtract 13 from 18

18 2 13 5 5

Since the number that is larger is 118, the answer is 15 or just 5.

Add the integers

Subtract the integers

SOLUTION

a The two integers are being subtracted Change the sign of the second number

(7 to 27) Add the two numbers using the rule for adding integers

3 2 7 5 3 1 27 5 24

The fi nal answer is 24.

b The two integers are being subtracted Change the sign of the second integer

(29 to 9) Add the two numbers using the rule for adding integers

▼

Trang 37

PRACTICE PROBLEM FOR EXAMPLE 9Subtract the integers.

a The temperature can be found by adding 25 1 8 These integers have different

signs The number with the larger absolute value is 8

8 2 5 5 3 Subtract 5 from 8, as the number with the larger absolute value is 8.

25 1 8 5 3 The result is positive, as the number with the larger absolute value

was positive 8.

The fi nal temperature is 3°F.

b The temperature can be found by subtracting 212 2 2 To subtract two integers,

change the sign of the second integer and add Therefore, we compute

212 1 22 5 214

The fi nal temperature is 214°F.

PRACTICE PROBLEM FOR EXAMPLE 10The highest point in the state of California is Mount Whitney, which has an elevation of 14,494 feet, and the lowest point in California is Death Valley, which is 282 feet below sea level

Source: www.netstate.com/states/geography.

a Write the elevation of Mount Whitney as an integer Note: A positive elevation will

be above sea level

b Write the elevation of Death Valley as an integer Note: A negative elevation will be

below sea level

c Find the difference in elevation between the highest and lowest points in the state

of California

Absolute value can be used to fi nd the distance between two points on a number line

DEFINITION

Distance Between Two Points on the Number Line The distance between

two points on a number line, a and b, can be found as b 2 a

Rising and falling temperatures

Example 10

120 100 80 60 40 20 0 –20 –40

50 40 30 20 10 0 –10 –20 –30 –40

Trang 38

Find the distance between 28 and 23.

SOLUTION

One way to fi nd the distance between 28 and 23 is to graph the two points on a number

line and count the spaces in between them

Since there are fi ve spaces in between 28 and 23, the distance between these two numbers is 5

Another way to fi nd the distance between these two points is to use the formula

b 2 a Let a 5 28 and b 5 23 and substitute into the formula.

b 2 a 5 23 2 28

5 23 + 8

5 5

5 5Note that if we switch the order of the points, the formula still gives the correct answer

Let a 5 23 and b 5 28 and substitute into the formula.

b 2 a 5 28 2 23

5 28 + 3

5 25

5 5PRACTICE PROBLEM FOR EXAMPLE 11

Find the distance between 29 and 22

The two other basic operations are multiplication and division Recall that cation of natural numbers is shorthand for a repeated addition So 3 4 5 4 1 4 1 4

multipli-5 3 1 3 1 3 1 3 So 3 4 is the same as adding 4 three times or adding 3 four times

Likewise, division of natural numbers is a shortcut for a repeated subtraction In

12 4 3 5 4, when we repeatedly subtract 3 from 12, we get

12 2 3 5 9

9 2 3 5 6

6 2 3 5 3

3 2 3 5 0The number of times we subtract 3 from 12 is 4 Therefore, 12 4 3 5 4

Writing multiplication and division problems as repeated additions and/or tions can be time-consuming That is why people memorize multiplication tables The

subtrac-following Concept Investigation will review the rules for multiplying signed numbers

Finding the distance between two points on a number line

Trang 39

2 What happens to the result as the number you multiply 4 by goes down by 1 in

5 When a positive number and a negative number are multiplied together, is the result

a positive or negative number?

6 Use a pattern to complete the following table.

7 When a negative number and a negative number are multiplied together, is the

result a positive or negative number?

Steps to Multiply or Divide Signed Numbers

1 To multiply or divide two integers with the same sign: Multiply or divide

the absolute values of the numbers The solution is always positive

2 To multiply or divide two integers with different signs: Multiply or divide

the absolute values of the two numbers The solution is always negative

Multiply the following integers

a 23 7 b 29 26

SOLUTION

a 23 7 5 221The answer is negative, since the two integers have different signs

b 29 26 5 54The answer is positive, since the two integers have the same sign

Multiplying integers

Example 12

▼

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S E C T I O N R 1 O p e r a t i o n s w i t h I n t e g e r s 13

Multiply the following integers

The answer is positive, since the two integers have the same sign.

Divide the following integers

a 20 4 24 b 21628 c 227 4 3

Keep in mind that division by 0 means that 0 is the divisor or is in the denominator

When 0 is the dividend, such as 0 4 16 or _16 , the answer is 0 It is important to note 0

that division by zero is undefi ned (sometimes, mathematicians say the result of such

an operation “does not exist” or “DNE”) We know that 16 4 2 5 8 or _162 5 8, since

8 2 5 16 Now suppose 16 4 0 5 c or 16 _0 5 c, where c is some real number This

would mean that 0 c 5 16, which is impossible Any number multiplied by 0 is 0

Therefore, we say that division by zero is undefi ned mathematically

Basic Operations The basic operations used in arithmetic are as follows.

1 Addition The two (or more) numbers being added are called addends or terms The result of an addition is called a sum

2 Subtraction The result of a subtraction is called a difference.

3 Multiplication The two (or more) numbers being multiplied are called factors The result of a multiplication is called a product.

4 Division The result of a division is called a quotient.

Dividing integers

Example 13

Connecting the Concepts

Is there more than one way

to write divide?

There are several ways to show division One way is with the 4 sign, such as 8 4 24 Another way is with a fraction bar:

8 4 24 5 _248 Another way is using long division:

8 4 24 means 24 8

Dividend, Divisor, and Quotient

The parts of a division problem have special names.

Dividend 4 divisor 5 quotient.

Written using a fraction bar,

we have dividend

divisor 5 quotientSometimes, when a fraction bar is used, the parts are called the numerator and denominator.

numerator _

denominator

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