Sadlier-oxford _ Algebra 1

Copyright © 2009 by William H. Sadlier, Inc. All rights reserved.


Chapter

Basic Concepts of AlgebraChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1-1 Rational and Irrational Numbers . . . . . . . . . 2

1-2 The Set of Real Numbers . . . . . . . . . . . . . . 4

1-3 Add and Subtract Real Numbers . . . . . . . . . 6

1-4 Multiply and Divide Real Numbers. . . . . . . . 8

Check Your Progress I (Lessons 1–4)

1-5 Integer Exponents . . . . . . . . . . . . . . . . . . . 10

1-6 The Order of Operations . . . . . . . . . . . . . . 12

1-7 Scientific Notation . . . . . . . . . . . . . . . . . . . 14

1-8 Algebraic Expressions . . . . . . . . . . . . . . . . 16

Check Your Progress II (Lessons 5–8)

1-9 Properties of Real Numbers. . . . . . . . . . . . 20

1-10 Sets and Operations . . . . . . . . . . . . . . . . . 22

1-11 Operations with Matrices: Addition and Subtraction . . . . . . . . . . . . . . 26

1-12 Operations with Matrices: Multiplication . . . . . . . . . . . . . . . . . . . . . . . 28

Check Your Progress III (Lessons 9–12)

1-13 Technology:Evaluate Numerical and Algebraic Expressions . . . . . . . . . . . . . . . . 30

1-14 Technology:Operations with Matrices . . . . . . . . . . . . . . 32

1-15 Problem-Solving Strategy:Make a Drawing. . . . . . . . . . . . . . . . . . . . . 34

Enrichment: Modular Arithmetic. . . . . . . . . . . . . . . . . . . 36

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Chapter

Linear EquationsChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2-1 Open Sentences and Solution Sets. . . . . . 40

2-2 Solve Addition and Subtraction Equations. . . . . . . . . . . . . . . . . . . . . . . . . . 42


2-3 Solve Multiplication and Division Equations. . . . . . . . . . . . . . . . . . . . . . . . . . 46

2-4 Solve Equations with Two Operations . . . . 50

2-5 Solve Multistep Equations . . . . . . . . . . . . . 54


2-6 Solve Absolute-Value Equations . . . . . . . . 58

2-7 Formulas and Literal Equations . . . . . . . . . 60


2-8 Technology:Solve Linear and Literal Equations . . . . . . 62

2-9 Problem-Solving Strategy:Solve a Simpler Problem . . . . . . . . . . . . . . 64

Enrichment: Diophantine Equations . . . . . . . . . . . . . . . 66

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Introduction to Algebra 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii–xviii

iv

8209-1_SB9_iv-xi_TOC:Layout 1 11/6/08 2:41 PM Page iv



v

Chapter

Linear InequalitiesChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3-1 Write and Graph Inequalities . . . . . . . . . . . 70

3-2 Solve Inequalities Using Addition orSubtraction . . . . . . . . . . . . . . . . . . . . . . . . 72


3-3 Solve Inequalities Using Multiplication or Division . . . . . . . . . . . . . . 74

3-4 Solve Multistep Inequalities . . . . . . . . . . . . 76


3-5 Solve Compound Inequalities . . . . . . . . . . 80

3-6 Solve Absolute-Value Inequalities . . . . . . . 84


3-7 Technology:Solve Linear Inequalities . . . . . . . . . . . . . . 86

3-8 Problem-Solving Strategy:Reason Logically . . . . . . . . . . . . . . . . . . . . 88

Enrichment: The Triangle Inequality Theorem . . . . . . . . 90

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Chapter

Relations and FunctionsChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4-1 Introduction to Relations . . . . . . . . . . . . . . 94

4-2 Introduction to Functions . . . . . . . . . . . . . . 96

4-3 Write Function Rules . . . . . . . . . . . . . . . . 100


4-4 Arithmetic Sequences . . . . . . . . . . . . . . . 102

4-5 Geometric Sequences . . . . . . . . . . . . . . . 106


4-6 Problem Solving:Review of Strategies. . . . . . . . . . . . . . . . . 110

Enrichment: Step Functions . . . . . . . . . . . . . . . . . . . . . 112

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . 114

8209-1_SB9_iv-xi_TOC:Layout 1 11/6/08 2:41 PM Page v



Chapter

Linear FunctionsChapter Opener. . . . . . . . . . . . . . . . . . . . . . . . . . 115

5-1 Identify Linear Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . 116

5-2 Direct Variation. . . . . . . . . . . . . . . . . . . . . 120

5-3 Equations in Slope-Intercept Form . . . . . 122


5-4 Equations in Point-Slope Form . . . . . . . . 126

5-5 Change the Form of a Linear Equation . . . . . . . . . . . . . . . . . . . . 128


5-6 Parallel and Perpendicular Lines . . . . . . . 132

5-7 Graph a Linear Inequality in the Coordinate Plane . . . . . . . . . . . . . . . . . . . 136

5-8 Absolute-Value Functions . . . . . . . . . . . . 138


5-9 Technology:Graph Linear Functions and Inequalities. . . . . . . . . . . . . . . . . . . . . . . . 140

5-10 Technology:Families of Lines . . . . . . . . . . . . . . . . . . . 142

5-11 Problem-Solving Strategy:Consider Extreme Cases . . . . . . . . . . . . . 144

Enrichment: Slope in Coordinate Geometry . . . . . . . . 146

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . 148

Chapter

Systems of LinearEquations and InequalitiesChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . 149

6-1 Solve Systems of Linear EquationsGraphically . . . . . . . . . . . . . . . . . . . . . . . . 150

6-2 Solve Systems of Linear Equations by Substitution . . . . . . . . . . . . . . . . . . . . . 154


6-3 Solve Systems of Linear Equations by Elimination . . . . . . . . . . . . . . . . . . . . . 156

6-4 Solve Equivalent Systems of Linear Equations . . . . . . . . . . . . . . . . . . . 158


6-5 Apply Systems of Linear Equations . . . . . 160

6-6 Graph Systems of Linear Inequalities . . . 162


6-7 Technology:Graph Systems of Equations . . . . . . . . . . 166

6-8 Technology:Graph Systems of Inequalities . . . . . . . . . 168

6-9 Problem-Solving Strategy:Work Backward . . . . . . . . . . . . . . . . . . . . 170

Enrichment: Use Cramer’s Rule to Solve Systems of Linear Equations . . . . . . . . . . . . . . . . . 172

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . 174

vi

8209-1_SB9_iv-xi_TOC:Layout 1 11/6/08 2:42 PM Page vi



Chapter

Operations withPolynomialsChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . 175

7-1 Introduction to Polynomials . . . . . . . . . . . 176

7-2 Add and Subtract Polynomials . . . . . . . . . 178


7-3 Multiply a Polynomial by a Monomial . . . . 182

7-4 Model Binomial Multiplication . . . . . . . . . 184

7-5 Multiply Binomials . . . . . . . . . . . . . . . . . . 186

7-6 Multiply Polynomials . . . . . . . . . . . . . . . . 188


7-7 Divide a Polynomial by a Monomial . . . . . 190

7-8 Divide Polynomials Using Long Division . . . . . . . . . . . . . . . . . . . . . . 192


7-9 Problem-Solving Strategy:Find a Pattern. . . . . . . . . . . . . . . . . . . . . . 194

Enrichment: Pascal’s Triangle and the Expansion of (x + y)n . . . . . . . . . . . . . . . . 196

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . 198

Chapter

Factoring PolynomialsChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . 199

8-1 Common Monomial Factors. . . . . . . . . . . 200

8-2 Factor Trinomials: ax2 + bx + c, a = 1 . . . . . . . . . . . . . . . . . . 202

8-3 Factor Trinomials: ax2 + bx + c, a ≠ 1 . . . . . . . . . . . . . . . . . . 206


8-4 Special Product and Factoring: (a ± b)2 = a2 ± 2ab + b2 . . . . . . . . . . . . . . 210

8-5 Special Product and Factoring: (a + b)(a – b) = a2 – b2 . . . . . . . . . . . . . . 212


8-6 Factor by Grouping . . . . . . . . . . . . . . . . . 214

8-7 Factor Completely . . . . . . . . . . . . . . . . . . 216


8-8 Technology:Factor Polynomials Using a Graph. . . . . . 218

8-9 Problem Solving:Review of Strategies . . . . . . . . . . . . . . . . 220

Enrichment: Factor Sums and Differences of Cubes . . . . . . . . . . . . . . . . . . . . . . . . . 222

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . 224

vii

8209-1_SB9_iv-xi_TOC:Layout 1 11/6/08 2:43 PM Page vii



viii

Chapter

Radical Expressions andEquationsChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . 225

9-1 Simplify Radical Expressions. . . . . . . . . . 226

9-2 Add and Subtract Radical Expressions . . . . . . . . . . . . . . . . . . . . . . . 228


9-3 Multiply and Divide Radical Expressions . . . . . . . . . . . . . . . . . . . . . . . 230

9-4 Solve Radical Equations . . . . . . . . . . . . . 234


9-5 The Pythagorean Theorem . . . . . . . . . . . 236

9-6 Distance in the Coordinate Plane . . . . . . 238


9-7 Problem-Solving Strategy:Account for All Possibilities . . . . . . . . . . . 240

Enrichment: Extending the Pythagorean Theorem to Three Dimensions . . . . . . . . . . . . . . . . 242

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . 244

Chapter

Quadratic Functions andEquationsChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . 245

10-1 Identify Quadratic Functions and Their Graphs . . . . . . . . . . . . . . . . . . . . . . 246

10-2 Graph Quadratic Functions: Parabola . . . . . . . . . . . . . . . . . . . . . . . . . 250

10-3 Solve Quadratic Equations by Factoring . . . . . . . . . . . . . . . . . . . . . . . . . 254


10-4 Solve Verbal Problems Involving Quadratic Equations . . . . . . . . . . . . . . . . 258

10-5 Solve Quadratic Equations by Completing the Square . . . . . . . . . . . . . . 260


10-6 The Quadratic Formula and the Discriminant. . . . . . . . . . . . . . . . . . . . 262

10-7 Solve Quadratic Equations with the Quadratic Formula . . . . . . . . . . . . . . . . . . 264

10-8 Solve Linear-Quadratic Systems . . . . . . . 266


10-9 Technology:Find the Zeros of Polynomial Functions . . . . . . . . . . . . . . . . . . . . . . . . . 270

10-10 Technology:Families of Quadratic Functions . . . . . . . 272

10-11 Problem-Solving Strategy:Adopt a Different Point of View. . . . . . . . . 274

Enrichment: Reflective Properties of Parabolic Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . 276

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . 278

8209-1_SB9_iv-xi_TOC:Layout 1 11/6/08 2:43 PM Page viii



Chapter

Ratio, Proportion, andTrigonometryChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . 279

11-1 Ratios and Rates . . . . . . . . . . . . . . . . . . . 280

11-2 Apply Proportion to Scale Models . . . . . . 284

11-3 Calculate Relative Error . . . . . . . . . . . . . . 286


11-4 Apply Percents to Algebraic Problems . . . . . . . . . . . . . . . . . . . . . . . . . 288

11-5 The Trigonometric Ratios. . . . . . . . . . . . . 290


11-6 Use Trigonometric Ratios to Solve Right Triangles . . . . . . . . . . . . . . . . . . . . . 292

11-7 Use Trigonometric Ratios to Solve Verbal Problems. . . . . . . . . . . . . . . . . . . . 294


11-8 Technology:Graph the Sine and Cosine Functions . . . . . . . . . . . . . . . . . . . . . . . . . 298

11-9 Problem-Solving Strategy:Guess and Test . . . . . . . . . . . . . . . . . . . . 300

Enrichment: The Law of Sines. . . . . . . . . . . . . . . . . . . 302

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . 304

Chapter

Rational Expressions and Equations Chapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . 305

12-1 Introduction to Rational Expressions . . . . 306

12-2 Simplify Rational Expressions . . . . . . . . . 308

12-3 Multiply Rational Expressions . . . . . . . . . 310

12-4 Divide Rational Expressions . . . . . . . . . . 312


12-5 Combine Rational Expressions with Like Denominators . . . . . . . . . . . . . . 314

12-6 Combine Rational Expressions with Unlike Denominators . . . . . . . . . . . . 316

12-7 Mixed Expressions and Complex Fractions . . . . . . . . . . . . . . . . . . 318


12-8 Solve Rational Equations Resulting in Linear Equations . . . . . . . . . . . . . . . . . 320

12-9 Solve Rational Equations Resulting in Quadratic Equations . . . . . . . . . . . . . . 322



Enrichment: Continued Fractions. . . . . . . . . . . . . . . . . 326

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . 328

ix

8209-1_SB9_iv-xi_TOC:Layout 1 11/6/08 2:44 PM Page ix



x

Chapter

Exponential and OtherNonlinear FunctionsChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . 329

13-1 Inverse Variation . . . . . . . . . . . . . . . . . . . 330

13-2 Graph Rational Functions . . . . . . . . . . . . 332

13-3 Graph Radical Functions . . . . . . . . . . . . . 336


13-4 Identify Exponential Functions and Their Graphs. . . . . . . . . . . . . . . . . . . 338

13-5 Exponential Growth and Decay . . . . . . . . 342


13-6 Technology:Graph Rational Functions . . . . . . . . . . . . 346

13-7 Technology:Graph Radical Functions . . . . . . . . . . . . . 348

13-8 Technology:Compare Exponential Growth and Decay . . . . . . . . . . . . . . . . . . . . . . . . 350

13-9 Problem-Solving Strategy:Organize Data . . . . . . . . . . . . . . . . . . . . . 352

Enrichment: Geometric Series. . . . . . . . . . . . . . . . . . . 354

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . 356

Chapter

Data Analysis andProbabilityChapter Opener . . . . . . . . . . . . . . . . . . . . . . . . . 357

14-1 Sampling Techniques. . . . . . . . . . . . . . . . 358

14-2 Measures of Central Tendency and Range . . . . . . . . . . . . . . . . . . . . . . . . 362

14-3 Stem-and-Leaf Plots . . . . . . . . . . . . . . . . 364

14-4 Histograms. . . . . . . . . . . . . . . . . . . . . . . . 366

14-5 Quartiles and Box-and-Whisker Plots . . . 370


14-6 Percentiles . . . . . . . . . . . . . . . . . . . . . . . . 372

14-7 Scatter Plots. . . . . . . . . . . . . . . . . . . . . . . 374

14-8 Empirical Probability . . . . . . . . . . . . . . . . 378

14-9 Theoretical Probability . . . . . . . . . . . . . . . 380


14-10 Independent and Dependent Events . . . . 382

14-11 Mutually Exclusive Events . . . . . . . . . . . . 386

14-12 Conditional Probability . . . . . . . . . . . . . . . 388

14-13 Permutations . . . . . . . . . . . . . . . . . . . . . . 390

14-14 Combinations. . . . . . . . . . . . . . . . . . . . . . 392


14-15 Technology:Simulate Events. . . . . . . . . . . . . . . . . . . . 394

14-16 Technology:Calculator Statistics . . . . . . . . . . . . . . . . . 396


Enrichment: Geometric Probability . . . . . . . . . . . . . . . 400

Test Prep . . . . . . . . . . . . . . . . . . . . . . . . . 402

8209-1_SB9_iv-xi_TOC:Layout 1 11/6/08 2:44 PM Page x



xiv

references direct you to previously learned skills that will be called on as you work through the current lesson.

Skills update

Clear, concisemathematical

list thepoints you willcover in the lesson.

objectives

Many lessons usemore than one

to addressthe fact that noteveryone learns thesame way. No matterwhat method isused, the instructionis developed

.step-by-step

method

summarize importantfacts, definitions, and mathematicalproperties that canbe used as helpful reinforcements.

Key concepts

The use of helps emphasizekey steps in theinstruction.

color

Algebra 1 Student SourceBook

8209-1_FM_Intro.qxd:Layout 1 11/13/08 4:31 PM Page xiv



xv

The logo points you to the Web-basedcomponents of the program. Here you will findhelpful resources and engaging activities.

Online

Follow the referencein the logo tofind the exercise setsthat you will completein the Practice Book.

Go To

questions help underscore the concepts of thelesson as you writeabout them in yourown words.

Discuss and Write

preparesyou for the types ofexercises you willencounter in thePractice Book.

Try These

Additional model variations ofthe lesson objectives.

examples

boxes modelthe reasoning andmental computationinvolved in computingand solving problems.

Think

Algebra 1 Student SourceBook

8209-1_FM_Intro.qxd:Layout 1 11/13/08 4:35 PM Page xv



xvi

Your SourceBookinstruction is summarized andillustrated to help youwith the exercises inthe Practice Book.

Read the instructionin the

before youbegin the exercises.display

teaching

The logohelps you easilylocate the pages inthe SourceBookthat correspond tothe current PracticeBook lesson.

Use With

Algebra 1 Student Practice Book

8209-1_FM_Intro.qxd:Layout 1 11/14/08 3:54 PM Page xvi



xvii

In every lesson, you will have the opportunity to applyyourskills.

problem-solving

isjust one of many end-of-lesson featuresthat encourage youto use higher-orderthinking skills.Additional end-of-lesson featuresinclude CriticalThinking, SpiralReview, Challenge,Write About It,Mental Math, andothers.

Test Preparation

The logo reminds you thatthere are more Practice exercisesand activities on the Web site.

Online Your teacher may create customizedworksheets and tests using the

.Practice/Test Generator

Algebra 1 Student Practice Book

8209-1_FM_Intro.qxd:Layout 1 11/13/08 4:36 PM Page xvii



xviii

Accelerate Expand Extend

In each chapter, the Enrichment topic is appropriate to the material you have been taught, and it will enrich you in one of three ways: It willaccelerate instruction to expose you to concepts that await you in thenot-too-distant future; it will expand on a presented topic; or it will extenda topic already taught and will apply it to a closely related theme. Thecommon thread is that the topics can be briefly, yet reasonably, andcompletely presented to you in a friendly fashion.

Some topics present mathematics in a historical context, which adds a great deal of interest and insight to the topic, while others may be more of a challenge that is intended to motivate you. In all cases, we hope you will find these Enrichment topics enjoyable so that you see mathematicsas a subject that can be fun, as well as extremely practical and useful.

These Enrichment topics go a long way to ensuring that you will be able to see the beauty of mathematics. It is your teacher’s goal to have youappreciate mathematics. We feel these Enrichment topics will help you reach that goal.

In this textbook, we provide some real topicsof enrichment. You may wonder what we meanby this, since we are sure you have been told manytimes that you are being “enriched.” Well, we aregoing to present topics that are not part of the regularcurriculum but that are within the scope of your courseof study. Some of these topics may be encountered in lateryears, but others may be missed if we do not include themnow. In all cases, we hope you enjoy them. To really do that,however, you must read them with an extra degree of enthusiasm!

8209-1_FM_Intro.qxd:Layout 1 11/25/08 11:19 AM Page xviii



xviii






8209-1_FM_Intro.qxd:Layout 1 11/24/08 4:20 PM Page xviii



xviii






• • abcdefghijklmnopqrstuvwxyz

Solve the system of equations. 13. 22

3 21

x y

x y

� �

� �

14. 3

2

6

x yx y

� �

� � �

15. 2

9

34 14

x y

x y

� �

� � �

16. yx

y x

� �

� �

9 2

3 211

17. 3

12

2

32

12

yx

yx

� �

� � �

18. x y

yx

� �

� �

34

254

54

38

a bc d

� 2 13 2

�

� � �4 � (�3) � –1e b

f d � 2 1

1 2

�

� � �4 � (�1) � �3

a ec f

� 2 23 1

� 2 � 6 � �4 x � �3�1 � 3; y � �4

�1 � 4; (3, 4)

19. The sum of the digits of a two-digit

number is 8. When the digits are reversed, the

number is increased by 36. Find the number.

20. Natural Delights mixes almonds that cost

$3.50/lb with cashews that cost $6.00/lb to

create 20 lb of mixed nuts. The mix is resold

for $4.50/lb. How many lb of each nut are

used to make the new mix?

21. Of the methods you have learned in this chapter for solving systems of linear equations,

which method do you prefer to use? Explain.

More Enrichment Topics

Cop

yrig

ht ©

by

Will

iam

H. S

adlie

r, In

c. A

ll rig

hts

rese

rved

.

162 Chapter 6

Enrichment Lesson, pages 172–173.• • abcdefghijklmnopqrstuvwxyz

Enrichment:Use Cramer’s Rule to Solve Systems of Linear Equations

Name

Date

Solve using Cramer’s Rule: 5

7 2

3 13

xy

y x

� �

� �First, rewrite the system in standard form: 5 2

73

13

x yx y

� �

� � .

Find the following determinants. a bc d

� ad � bc

5 2

3 1� � 5(�1) � 2(3) � �11

e bf d

� de � bf

7 2

13 1� � 7(�1) � 2(13) 5 �33

a ec f

� af � ce

5 7

3 13 � 5(13) � 7(3) 5 44

Then apply Cramer’s Rule. x �

e bf d

� a bc d

� �33 � (�11) � 3y �

a ec f

� a bc d

� 44 � (�11) � �4

So (3, �4) is the solution of the system 5

7 2

3 13

xy

y x

� �

� �. Evaluate the determinant. 1.

2 1

1 3�

2.

1 4

0 3

� 3.

3 2

2 1�

4.

�

�

1 2

4 1

5. 3 1

2 4

� 6.

3 1

4 1

�

�

7.

1 1

2 1

8.

6 2

3 1

�

�

9. � �5

7

32

10.

� �86

43

11.

5 3

6 9� �

12.

7 0

2 5� �

(2 • 3) � (�1 • 1) 6 � (�1) 6 � 1; 7

Cop

yrig

ht ©

by

Will

iam

H. S

adlie

r, In

c. A

ll rig

hts

rese

rved

.

Chapter 6 161Chapter 6 173

More Enrichment Topics

Evaluate each determinant.1.

2.3.

Solve each system of equation using Cramer’s Rule.

� �

� �

3 4

2 1

8 1

3 0

�

�

3 2

4 5

�

pages 161–162 for exercise sets.

NETS 10228 • Sadlier (ALG1SB) • 8209-1 172-173 • tmd 07-30-08 • Edit b

•• abcdefghijklmnopqrstuvwxyz

4. 4x � y � �3

3x � 2y � 45. 2x � 3y � 9

5y � 2x � 116. y � 3x � 4

y � �3x � 4

⎧⎪

⎨⎪

⎩

⎧⎪

⎨⎪

⎩

⎧⎪

⎨⎪

⎩

7. Discuss and Write Explain how Cramer’s Rule is used to solve .

Describe the system.5x � 3y � 9

�5x � 3y � 12

⎧⎪

⎨⎪

⎩

To apply Cramer’s Rule, the system of equations must be in standard form.Use Cramer’s Rule to solve the system .• First, rewrite the system in standard form.

• Then apply Cramer’s Rule.

x � �

� �

�

So the solution of the system is x � and y � 10.�When using Cramer’s Rule to solve a system of linear equations that is a dependent

system, both numerator and denominator have their determinant equal to zero.Use Cramer’s Rule to show that the system

is dependent.

x ��

� �

So the system is dependent and has infinitely many solutions.

2x � y � �5

�6x � 2y � 5

⎧⎪

⎨⎪

⎩

2x � y � 5

2y � 6x � 5

⎧⎪

⎨⎪

⎩

00

�24 � 24�12 � 12

(4)(�6) � (�8)(3)(2)(�6) � (�4)(3)

4 3

8 6

2 3

4 6

� �

� �

2x � 3y � 4

�4x � 6y � �8

⎧⎪

⎨⎪

⎩

�5�2

�10 � 54 � 6

52

52

(�5 • 2) � (5)(�1)(2 • 2) � (�6)(�1)

� �

�

�

5 1

5 2

2 1

6 2

2x � y � 5

2y � 6x � 5

⎧⎪

⎨⎪

⎩

y � �

� �

� 10

(2 • 5) � (�6)(�5)(2 • 2) � (�6)(�1)

2 5

6 5

2 1

6 2

�

�

�

�

�20�2

10 � 304 � 6

y ��

� � 00

�16 � 16�12 � 12

(2)(�8) � (�4)(4)(2)(�6) � (�4)(3)

2 4

4 8

2 3

4 6

� �

� �

172 Chapter 6

TS 10228 • Sadlier (ALG1SB) • 8209-1 172-173 • tmd 07-30-08 • Edit bjj 10.10.08 • 3RD••abcdefghijklmnopqrstuvwxyz

Enrichment:Use Cramer’s Rule to Solve Systems of Linear Equations

Objective To solve systems of linear equations in two variables using Cramer’s Rule

You have solved systems of linear equations in two variables graphically and

by using substitution and elimination. Another method involves a technique

known as . This method was developed by the Swiss mathematician

Gabriel Cramer (1704–1752) in 1750. Cramer, who received his doctorate degree

at age 18, is known for his early acceleration in mathematics.

�Cramer’s Rule is a formula that solves a system of linear equations by using

to find the values of the variables. A determinant is a real number

obtained by evaluating a square array of numbers (a matrix) by a specific rule.

The determinant of a 2-by-2 matrix , denoted by , is

defined as � ad � bc. Notice that the symbol for a matrix is

enclosed in brackets, whereas the symbol for a determinant is a pair of

vertical lines similar to the absolute-value symbol.

Evaluate each determinate.

• � (3 • 6) � (4 • 2)

� 18 � 8

� 10

�The solution of the system is x � and y � .

Note that the denominator of the values of x and y is the determinant made up of the

coefficients of the equations, . The numerator of the value of x has the column of the

coefficients of x replaced by the column of constants. The numerator of the value of y is the same

as the denominator, and has the column of coefficients of y replaced by the column of constants.

Use Cramer’s Rule to solve the system .

x � �

� �

� 3

So the solution of the system is x � 3 and y � 2.

�30�10

�24 � 6�6 � 4

(8)(�3) � (6 • 1)

(2)(�3) � (4 • 1)

8 1

6 3

2 1

4 3

�

�

2x � y � 8

4x � 3y � 6

⎧⎪

⎨⎪

⎩

a b

c d

a b

c d

⎡

⎣⎢

⎤

⎦⎥

a b

c d

determinants

Cramer’s Rule

ax � by � e

cx � dy � f

⎧⎪

⎨⎪

⎩

a

c

a

c

e

f

b

d

e

f

b

d

b

d

a

c

a

c

b

d

3 2

4 6

y � �

� �

� 2

�20�10

(2 • 6) � (4 • 8)

(2)(�3) � (4 • 1)

2 8

4 6

2 1

4 3� 12 � 32�6 � 4

• � (�5 • 2) � (8)(�1)

� �10 � 8

� �2

� �5 1

8 2

Gabriel Cramer

8209-1_FM_Intro.qxd:Layout 1 11/24/08 5:09 PM Page xviii



Skills UpdateI. Place Value . . . . . . . . . . . . . . . . . . . . . . . . . 403

II. Estimation: Rounding and Compatible Numbers. . . . . . . . . . . . . . . . . . 403

III. Divisibility Rules . . . . . . . . . . . . . . . . . . . . . 404

IV. Prime and Composite Numbers. . . . . . . . . . 404

V. GCF and LCM . . . . . . . . . . . . . . . . . . . . . . . 404

VI. Multiply and Divide Decimals. . . . . . . . . . . . 405

VII. Mixed Numbers and Fractions. . . . . . . . . . . 406

VIII. Add and Subtract Fractions . . . . . . . . . . . . . 406

IX. Multiply and Divide Fractions. . . . . . . . . . . . 406

X. Decimals, Fractions, and Percents . . . . . . . 407

XI. The Coordinate Plane . . . . . . . . . . . . . . . . . 407

XII. Transformations on the Coordinate Plane. . . . . . . . . . . . . . . . . . . . . 408

XIII. Metric and Customary Systems ofMeasurements. . . . . . . . . . . . . . . . . . . . . . . 408

XIV. Basic Geometric Terms and AngleClassifications. . . . . . . . . . . . . . . . . . . . . . 409

XV. Polygons and Circles . . . . . . . . . . . . . . . . 410

XVI. Similarity and Congruence . . . . . . . . . . . . 411

XVII. Triangles and Quadrilaterals . . . . . . . . . . . 411

XVIII. Perimeter and Area of Polygons . . . . . . . . 412

XIX. Volume and Surface Area . . . . . . . . . . . . . 412

XX. Double Line and Double Bar Graphs . . . . 413

XXI. Line Plots and Circle Graphs . . . . . . . . . . 413

XXII. Probability and Odds . . . . . . . . . . . . . . . . 414

TI-Nspire™ Handbook . . . . . . . . . . . . . . . . . . . . 415

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

End-of-Book Materials

xi

8209-1_SB9_iv-xi_TOC:Layout 1 11/6/08 2:45 PM Page xi



Critical Thinking

The Groth family pays 10.86 cents per kilowatt-

hour for electricity in their home. Their total

monthly usage in June was 888 kilowatt-hours.

To save energy in July, the Groths want to cut

their monthly usage by . If they succeed, how

much less will the family pay for electricity for the

month of July than they would have paid for June?

In This Chapter You Will:● Identify types of rational numbers and

recognize irrational numbers● Find square roots of perfect squares

and approximate square roots of nonperfect squares

● Add, subtract, multiply, and divide signednumbers

● Classify and graph real numbers● Simplify numerical expressions using the

order of operations

● Perform operations on numbers in scientific notation

● Apply set operations of intersection andunion

● Use matrices to organize data and performoperations on matrices

● Apply the strategy: Make a Drawing● Look for new vocabulary words

in each lessonhighlighted

● A number is a mathematical conceptused to describe and assess quantity.

● Addition and subtraction are inverse operations, as are multiplication and division.

● The prime factorization of a numbershows the number as a product ofprime factors.

● A fraction is in simplest form when its numerator and denominator have

no common factor other than 1.

Do You Remember? For Practice Exercises:

PRACTICE BOOK, pp. 1–38

● Skills Update Practice● Practice Activities● Audio Glossary● Vocabulary Activities

● Technology Videos● Enrichment Activities● Electronic SourceBook

For Chapter Support:

www.progressinmathematics.com

Virtual Manipulatives

15

Chapter 1 1

8209-1_SB9_001_CO_CH01:Layout 2 11/3/08 1:51 PM Page 1



2 Chapter 1

1-1Update your skills. See page 404 III, IV.

Rational and Irrational NumbersObjective To identify types of rational numbers • To find square roots of perfect squares• To approximate square roots of nonperfect squares • To recognize irrational numbers

During a 4-month period, an environmentalist recorded the followingchanges to a lake’s water level: 4 in., �2 in., 0.41 in., and in.

The numbers recorded above are rational numbers.

�A is the quotient of two integers, a and b, written , with b � 0.

All of the following types of numbers are rational numbers.

• Integers, which are whole numbers and their opposites, are represented by the set {. . . , �3, �2, �1, 0, 1, 2, 3, . . .}.

• Fractions and mixed numbers, such as , � , and 2 , can be positive or negative.

• Decimals can also be positive or negative. They are eitherterminating or repeating.

, such as 0.41, 3.0, �5.7, have a finite number of digits.

, like the examples that follow, have a sequence of one or more digits that repeat indefinitely.2.333 . . . , which can be written as 2.3,and �0.636363 . . . , or �0.63

�Multiplying a number by itself, or raising it to the second power, is finding the square of that number. Some examples are given below.

square of ( )2� • � square of �7 (�7)2 � �7 • �7 � 49

square of 0.5 (0.5)2 � 0.5 • 0.5 � 0.25 opposite of the square of 4 �42 � �(4 • 4) � �16

, or square numbers, are the squares of natural numbers. Some examples of perfect squares are shown in the table below.

The set of perfect squares is {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, . . .}.

�A of a number is one of two equal factors of that number. The positive square root of a number is called the . It is indicated with the symbol , called a . The expression under a radical sign is called the

. The of a number is indicated by writing a negative sign in front of the radical.

� � 5 � � �5

Use an ellipsis (three dots) or use an overbar toshow that one or more digits repeat in a decimal.

255 5•25 negative square rootprincipal square root

radicandradical sign

principal square root

Perfect squares

negative square root

square root

964

38

38

38

38

Natural Number 1 2 3 4 5 6 7 8 9 10

Square of the Number 12 22 32 42 52 62 72 82 92 102

Perfect Square 1 4 9 16 25 36 49 64 81 100

117

14

38

Repeating decimals

Terminating decimals

38

14

ab

rational number

NETS 10228 • Sadlier (ALG1SB) • 8209-1 002-003 • tmd 04.22.08 • Edit bjj 07.15.08 • 3RD

Remember:Natural, or Counting, Numbers:{1, 2, 3, 4, . . .}Whole Numbers: {0, 1, 2, 3, 4, . . .}

Every integer a can be written as .a1

Read as “the principalsquare root of 25.”

Read as “the negativesquare root of 25.”

8209-1_002-003:Layout 2 7/31/08 4:02 PM Page 2



Chapter 1 3Lesson 1-1 for exercise sets.

Practice & Activities

NETS 10228 • Sadlier (ALG1SB) • 8209-1 002-003 • tmd 04.22.08 • Edit bjj 07.15.08 • 3RD

�

� � �

� �

� �(3 • 5) � �15

( )( )3 5 3 5• •

3 3 5 5• • •225

2251

�

� �23

23•

23

2( )49

49

23

2Remember: The primefactorization of a numbershows the number as theproduct of prime factors.

ThinkThe perfect squares in orderare 1, 4, 9, 16, 25, . . . .19 is between 16 and 25.

Key ConceptIrrational NumbersIrrational numbers are numbers that

cannot be expressed in the form ,

where a and b are integers and b � 0.

ab

ThinkWhat number

squared equals ?49

Find each square root.

For each number, list all the terms that apply: whole number, integer, rational number, and irrational number.

1. �321.11 2. 45 3. 0.010203 . . . 4. 1.358

5. 6. 7. 8.

Find each square root. If the radicand is a nonperfect square, between which two consecutive integers would the square root fall?

9. 10. 11. 12.

13. A contractor is building a patio in the shape of a square. The patio will cover 945 square feet. Estimate the length of the side of the patio to the nearest integer.

14. Discuss and Write Which of these numbers is irrational: , , ? Explain.

227 144

3 6.13636

400 205 � 4981 � 77

102� 36

�To approximate the square root of a , a number that is not the square of a natural number, find two consecutive integers that the square root is between.

Between what two consecutive integers is ?

19 is between 16 and 25.

is between and .

� 4 and � 5 Find each square root.

So is between 4 and 5.

�Square roots of nonperfect squares are examples of numbers that are not rational. Numbers that are not rational are called irrational. The following are .

• positive or negative decimals that are nonterminating and nonrepeating or have a pattern in their digits but do not repeat exactly.4.13216582 . . . , �0.5050050005 . . .

• square roots of nonperfect squares,

• pi, symbolized by the Greek letter � (3.14 and are rational approximate values.)

2516

� 25042227

irrational numbers

Find two nearby perfectsquares that 19 is between.

2516

19

19

19

nonperfect square

8209-1_002-003:Layout 2 8/1/08 11:56 AM Page 3



4 Chapter 1

1-2

NETS 10228 • Sadlier (ALG1SB) • 8209-1 004-005 • bjj/cps 04.22.08 • Edit sp 07.16.08 • 3RD

Real Numbers

Rational NumbersIntegers

Whole NumbersNaturals Zero

IrrationalNumbers

Key ConceptCompleteness Property for Points on theNumber LineEvery real number corresponds to exactly one pointon a number line, and every point on the numberline corresponds to exactly one real number.

Update your skills. See page 406, VII.

The Set of Real Numbers Objective To classify real numbers • To graph real numbers on a number line • To compare andorder real numbers • To find the absolute value and additive inverse of a real number • To understandand apply the Closure Property

The set of consists of all rational numbers and irrational numbers. The at the right shows the relationships among natural numbers, whole numbers, integers, rational numbers, and irrational numbers.

To which sets of numbers does each one of these real numbers belong: 0.54, , , and ?

�To decide to which sets of numbers a real number belongs, you may need to rename the number in a different form.

• 0.54 � 0.5444 . . . a repeating decimal • � �9 an integer

rational number integer; rational number

• � 4 a natural number • a nonperfect square radicand

natural number; whole number; integer; irrational numberrational number

�There is a one-to-one correspondencebetween the set of real numbers and the points on a number line. This is illustrated by the

.

The origin of a number line is zero. Points to the right of zero correspond to positive numbers. Points to the left of zero correspond to negative numbers.Positive and negative numbers are often called signed numbers. The number zero is neither positive nor negative.

The point that corresponds to a real number is called the graph of the number. The number line below shows the graph of , �3 , �, 2, and �1.7.

�A number line can help you compare and order real numbers. The farther to the right a number is on the number line, the greater it is.

Use the number line above to compare and order the following numbers from least to greatest.

�1.7 � �3 � � 2 is between and �.

In order from least to greatest: �3 , �1.7, , 2, �

45 2 2

245

Completeness Property for Pointson the Number Line

�4 �3 �2 �1 0 1 2 3 4

� �3 �1.745 √ 2 2

2 45

6

� 81

4411

98� 81

Venn diagram

real numbers

4411

8209-1_004-005:Layout 2 7/31/08 4:03 PM Page 4



Chapter 1 5Lesson 1-2 for exercise sets.

Practice & Activities

Give an example to illustrate the type of number described.

1. a real number that is not rational

Graph the numbers on a number line. Then writethe numbers in order from least to greatest.

3. �2.2, , 0, � , �1.3

6. Discuss and Write Explain why {�1, 1} is closed under division.

529

NETS 10228 • Sadlier (ALG1SB) • 8209-1 004-005 • bjj/cps 04.22.08 • Edit sp 07.16.08 • 3RD

�4 �3 �2 �1 0

3.5 units 3.5 units

Opposites

1 2 3 4

�When you apply an operation (for example, addition) to any numbers in a set and the result is also a number of that set, the set is said to be closed under the operation. This is called the .

Finding a that shows that a set of numbers is not closed under a given operation is one way to test closure for that set under the given operation. A single counterexample proves that a statement is false.

counterexample

Closure Property

• The set of real numbers is closed under addition. True, whenever you add two real numbers, the sum is always a real number.

• The set of integers is closed under division. False, 3 � 5 � is not an integer.3

535

Key ConceptAdditive Inverse PropertyFor any real number a,

a � (�a) � 0 and �a � a � 0.

2. a rational number that is not an integer

Find the value of the expression.

4. �|9 • 75| 5. �|3.9 � 5.2 |

�(�30) • �(�12) 30 • 12

360

Find the oppositeof each factor.

|�6.9 | � |2.9 |6.9 � 2.9

4

Find the absolute valueof each number.

�The of a real number, n, written |n |, is the distance from 0 to n on a number line. Since distance is always positive, the absolute value of a nonzero number is always positive.

The absolute value of 3.5 is 3.5. Write: |3.5 | � 3.5The absolute value of �3.5 is 3.5. Write: |�3.5 | � 3.5

�Two real numbers are opposites (or ) if they are on opposite sides of 0 and they are the same distance from 0 on a number line. The sum of a real number and its additive inverse is 0.

The opposite (or additive inverse) of 3.5 is �3.5. Write: �(3.5) � �3.53.5 � (�3.5) � 0The opposite (or additive inverse) of �3.5 is 3.5. Write: �(�3.5) � 3.5�3.5 � [�(�3.5)] � 0The opposite (or additive inverse) of 0 is 0. Write: �(0) � 00 � 0 � 0

�You can find the value of an expression involving absolute values or additive inverses of real numbers.

Find the value of each expression.

absolute value

additive inverses

8209-1_004-005:Layout 2 7/31/08 4:03 PM Page 5



Documents

Sadlier-oxford _ Algebra 1