algebra 1 interactive notebook ebook

Contributing AuthorDinah Zike

AlgebraConcepts and Applications

ConsultantDouglas Fisher, PhD

Director of Professional DevelopmentSan Diego State University

San Diego, CA

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act, no part of this book may be reproduced in any form, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-868488-9 Algebra: Concepts and Applications (Louisiana Student Edition)Noteables™: Interactive Study Notebook with Foldables™

1 2 3 4 5 6 7 8 9 10 024 09 08 07 06 05 04

Algebra: Concepts and Applications iii

Foldables. . . . . . . . . . . . . . . . . . . . . . . . . . 1Vocabulary Builder . . . . . . . . . . . . . . . . . . 21-1 Writing Expressions and

Equations. . . . . . . . . . . . . . . . . . . . . 41-2 Order of Operations . . . . . . . . . . . . 71-3 Comm. and Assoc. Properties . . . . . 91-4 Distributive Property . . . . . . . . . . . 111-5 A Plan for Problem Solving . . . . . . 131-6 Collecting Data . . . . . . . . . . . . . . . 151-7 Displaying and Interpreting Data . . 19Study Guide . . . . . . . . . . . . . . . . . . . . . . 23

Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 27Vocabulary Builder . . . . . . . . . . . . . . . . . 282-1 Integers on a Number Line . . . . . . 302-2 The Coordinate Plane . . . . . . . . . . 332-3 Adding Integers. . . . . . . . . . . . . . . 352-4 Subtracting Integers . . . . . . . . . . . 372-5 Multiplying Integers . . . . . . . . . . . 392-6 Dividing Integers . . . . . . . . . . . . . . 41Study Guide . . . . . . . . . . . . . . . . . . . . . . 43

Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 47Vocabulary Builder . . . . . . . . . . . . . . . . . 483-1 Rational Numbers . . . . . . . . . . . . . 503-2 Adding and Subtracting

Rational Numbers . . . . . . . . . . . . . 533-3 Mean, Median, Mode, Range . . . . 553-4 Equations. . . . . . . . . . . . . . . . . . . . 573-5 Solving Equations by Using Models . 593-6 Solving (�) and (�) Equations . . . . 613-7 Solving Equations Involving

Absolute Value . . . . . . . . . . . . . . . . 63Study Guide . . . . . . . . . . . . . . . . . . . . . . 65

Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 69Vocabulary Builder . . . . . . . . . . . . . . . . . 704-1 Multiplying Rational Numbers . . . . 714-2 Counting Outcomes . . . . . . . . . . . . . 74

4-3 Dividing Rational Numbers . . . . . . . 764-4 Solving (�) and (�) Equations. . . . . 794-5 Solving Multi-Step Equations. . . . . . 824-6 Variables on Both Sides. . . . . . . . . . 854-7 Grouping Symbols . . . . . . . . . . . . . 87Study Guide . . . . . . . . . . . . . . . . . . . . . . 89

Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 93Vocabulary Builder . . . . . . . . . . . . . . . . . 945-1 Solving Proportions . . . . . . . . . . . . 965-2 Scale Drawings and Models. . . . . . 995-3 The Percent Proportion . . . . . . . . 1015-4 The Percent Equation . . . . . . . . . 1045-5 Percent of Change . . . . . . . . . . . . 1075-6 Probability and Odds . . . . . . . . . . 1095-7 Compound Events . . . . . . . . . . . . 111Study Guide . . . . . . . . . . . . . . . . . . . . . 113

Foldables . . . . . . . . . . . . . . . . . . . . . . . . 117Vocabulary Builder . . . . . . . . . . . . . . . . 1186-1 Relations . . . . . . . . . . . . . . . . . . . 1206-2 Equations as Relations . . . . . . . . . 1236-3 Graphing Linear Relations . . . . . . 1266-4 Functions . . . . . . . . . . . . . . . . . . . 1296-5 Direct Variation . . . . . . . . . . . . . . 1326-6 Inverse Variation . . . . . . . . . . . . . 135Study Guide . . . . . . . . . . . . . . . . . . . . . 137

Foldables . . . . . . . . . . . . . . . . . . . . . . . . 141Vocabulary Builder . . . . . . . . . . . . . . . . 1427-1 Slope . . . . . . . . . . . . . . . . . . . . . . 1447-2 Point-Slope Form . . . . . . . . . . . . . 1467-3 Slope-Intercept Form . . . . . . . . . . 1487-4 Scatter Plots . . . . . . . . . . . . . . . . . 1517-5 Graphing Linear Equations . . . . . 1547-6 Families of Linear Graphs . . . . . . 1577-7 Parallel and Perpendicular

Lines. . . . . . . . . . . . . . . . . . . . . . . 160Study Guide . . . . . . . . . . . . . . . . . . . . . 163

Contents

iv Algebra: Concepts and Applications

Foldables . . . . . . . . . . . . . . . . . . . . . . . . 167Vocabulary Builder . . . . . . . . . . . . . . . . 1688-1 Powers and Exponents. . . . . . . . . 1708-2 Multiply and Divide Powers. . . . . 1728-3 Negative Exponents. . . . . . . . . . . 1748-4 Scientific Notation . . . . . . . . . . . . 1768-5 Square Roots . . . . . . . . . . . . . . . . 1788-6 Estimating Square Roots . . . . . . . 1808-7 The Pythagorean Theorem . . . . . 181Study Guide . . . . . . . . . . . . . . . . . . . . . 183

Foldables . . . . . . . . . . . . . . . . . . . . . . . . 187Vocabulary Builder . . . . . . . . . . . . . . . . 1889-1 Polynomials . . . . . . . . . . . . . . . . . 1899-2 Add and Subtract Polynomials . . 1929-3 Multiplying a Polynomial

by a Monomial. . . . . . . . . . . . . . . 1959-4 Multiplying Binomials . . . . . . . . . 1989-5 Special Products . . . . . . . . . . . . . . 200Study Guide . . . . . . . . . . . . . . . . . . . . . 202

Foldables . . . . . . . . . . . . . . . . . . . . . . . . 205Vocabulary Builder . . . . . . . . . . . . . . . . 20610-1 Factors . . . . . . . . . . . . . . . . . . . . . 20710-2 Factoring: Distributive Property . 20910-3 Factoring: x2 � bx � c . . . . . . . . . 21110-4 Factoring: ax2 � bx � c . . . . . . . . 21310-5 Special Factors . . . . . . . . . . . . . . . 215Study Guide . . . . . . . . . . . . . . . . . . . . . 217

Foldables . . . . . . . . . . . . . . . . . . . . . . . . 221Vocabulary Builder . . . . . . . . . . . . . . . . 22211-1 Graphing Quadratic Functions. . . 22411-2 Families of Quadratic Functions . 22711-3 Graphing Quadratic Equations . . 23011-4 Factoring Quadratic Equations . . 23311-5 Completing the Square . . . . . . . . 23511-6 The Quadratic Formula . . . . . . . . 23711-7 Exponential Functions . . . . . . . . . 239Study Guide . . . . . . . . . . . . . . . . . . . . . 241

Foldables . . . . . . . . . . . . . . . . . . . . . . . . 245Vocabulary Builder . . . . . . . . . . . . . . . . 246

12-1 Inequalities and Their Graphs . . . 24712-2 Solving (�) and (�) Inequalities . . 25012-3 Solving (�) and (�) Inequalities . . 25212-4 Solving Multi-Step Inequalities . . 25412-5 Solving Compound Inequalities. . 25712-6 Absolute Value Inequalities . . . . 26012-7 Graphing Inequalities in

Two Variables. . . . . . . . . . . . . . . . 262Study Guide . . . . . . . . . . . . . . . . . . . . . 265

Foldables . . . . . . . . . . . . . . . . . . . . . . . . 269Vocabulary Builder . . . . . . . . . . . . . . . . 27013-1 Graphing Systems of Equations . . 27213-2 Solutions of Systems of

Equations . . . . . . . . . . . . . . . . . . . 27413-3 Substitution . . . . . . . . . . . . . . . . . 27713-4 Elimination Using (�) and (�) . . . 28013-5 Elimination Using (�) . . . . . . . . . 28413-6 Solving Quadratic-Linear

Systems of Equations . . . . . . . . . . 28813-7 Graphing Systems of Inequalities . 291Study Guide . . . . . . . . . . . . . . . . . . . . . 293

Foldables . . . . . . . . . . . . . . . . . . . . . . . . 297Vocabulary Builder . . . . . . . . . . . . . . . . 29814-1 The Real Numbers . . . . . . . . . . . . 29914-2 The Distance Formula . . . . . . . . . 30214-3 Simplifying Radical Expressions . . 30514-4 Radical Expressions: (�) and (�) . . 30814-5 Solving Radical Equations . . . . . . 310Study Guide . . . . . . . . . . . . . . . . . . . . . 313

Foldables . . . . . . . . . . . . . . . . . . . . . . . . 317Vocabulary Builder . . . . . . . . . . . . . . . . 31815-1 Simplify Rational Expressions . . . 31915-2 Rational Expressions: (�)

and (�). . . . . . . . . . . . . . . . . . . . . 32215-3 Dividing Polynomials . . . . . . . . . . 32515-4 Combining Rational Expressions

with Like Denominators . . . . . . . 32815-5 Combining Rational Expressions

with Unlike Denominators . . . . . 33115-6 Solving Rational Equations . . . . . 334Study Guide . . . . . . . . . . . . . . . . . . . . . 338

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Algebra: Concepts and Applications v

Organizing Your Foldables

Make this Foldable to help you organize andstore your chapter Foldables. Begin with onesheet of 11" � 17" paper.

FoldFold the paper in half lengthwise. Then unfold.

Fold and GlueFold the paper in half widthwise and glue all of the edges.

Glue and LabelGlue the left, right, and bottom edges of the Foldableto the inside back cover of your Noteables notebook.

Reading and Taking Notes As you read and study each chapter, recordnotes in your chapter Foldable. Then store your chapter Foldables insidethis Foldable organizer.

Foldables Organizer

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vi Algebra: Concepts and Applications

This note-taking guide is designed to help you succeed in Algebra: Conceptsand Applications. Each chapter includes:

Inequalities

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Use the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.

NOTE-TAKING TIP: When you take notes, definenew terms and write about the new concepts inyour own words. Write your own examples thatuse the new terms and concepts.

C H A P T E R

12

Algebra: Concepts and Applications 245

Ch

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FoldFold each sheet in halffrom top to bottom.

CutCut along fold. Staplethe eight half-sheetstogether to form a booklet.

LabelLabel each page with alesson number and title.

Begin with four sheets of grid paper.

12–1Inequalities

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246 Algebra: Concepts and Applications

C H A P T E R

12BUILD YOUR VOCABULARY

Vocabulary TermFound

Definition Description or

on Page Example

boundary

compound inequality

half-plane

intersection

quadratic inequalities

set-builder notation

union

This is an alphabetical list of new vocabulary terms you will learn in Chapter 12.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.

The Chapter Openercontains instructions andillustrations on how to makea Foldable that will help youto organize your notes.

A Note-Taking Tipprovides a helpfulhint you can usewhen taking notes.

The Build Your Vocabularytable allows you to writedefinitions and examplesof important vocabularyterms together in oneconvenient place.

Within each chapter,Build Your Vocabularyboxes will remind youto fill in this table.

Algebra: Concepts and Applications vii

Write x � 0 and x � �3 as a compound inequalitywithout using and.

x � 0 and x � �3 can be written as � x � or

� x � .

Write x � �2 and x � 5 as a compoundinequality without using and.

A veterinarian has a scale for weighing dogs and catsthat weigh more than 10 pounds but no more than 65 pounds. The weights w that can be measured on thisscale can be written as 10 � w � 65. Graph the solutionof this inequality.

Rewrite the compound inequality using . 10 � w � 65

is the same as w � 10 and .

Your Turn

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12–5


Solving Compound Inequalities

Two or more inequalities that are connected by the

words or form a compound inequality.

An intersection is the set of elements common

to inequalities.

A union is the set of elements in each of

inequalities.

BUILD YOUR VOCABULARY (page 246)

Summarize thedifference between“intersection” and“union” under the tabfor Lesson 12–5. Then,give examples of whento use each one.

12–1Inequalities

ORGANIZE IT

• Solve compoundinequalities.

WHAT YOU’LL LEARN

GLE 14. Graph and interpret linear inequalities in one or two variables andsystems of linear inequalities (A-2-H, A-4-H)

12-2

Solving Addition and Subtraction Inequalities

BRINGING IT ALL TOGETHER

C H A P T E R

12STUDY GUIDE


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BUILD YOURVOCABULARY

Use your Chapter 12 Foldable tohelp you study for your chaptertest.

To make a crossword puzzle,word search, or jumble puzzle of the vocabulary wordsin Chapter 12, go to:

www.glencoe.com/sec/math/t_resources/free/index.php

You can use your completedVocabulary Builder (page 246) to help you solve the puzzle.

VOCABULARYPUZZLEMAKER

12-1

Inequalities and Their Graphs

Write the letter of the graph that matches each inequality.

1. x � �1 a.

2. x � �1 b.

3. x � �1 c.

4. x � �1 d.

Write an inequality for each statement. Then solve.

5. A number subtracted from 16. A number added to 12 is a 21 is no less than �2. minimum of �1.

7. 5 more than a number is 18. 18 less than a number isat least 15. at most 45.

1 2 3�1 0�2�3

1 2 3�1 0�2�3

1 2 3�1 0�2�3

1 2 3�1 0�2�3

Karl’s point totals in the first four of five basketballgames were 15, 12, 19, and 18. How many points t musthe score in the fifth game to have a mean point total ofmore than 16?

The sum of Karl’s points, divided by , will give the

mean score. The mean must be more than .

�

( ) � (16)

15 � 12 � 19 � 18 � t �

� t � 80

64 � t � � 80 �

t �

Karl must score more than points in the fifth game to

have a mean point total of more than .

Lien’s score on the first four of five 100-pointtests were 82, 85, 95, and 91. What score s on the fifth testwill give her a mean score of at least 90 for all five tests?

Your Turn

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12–4


Page(s):Exercises:

HOMEWORKASSIGNMENT

15 � 12 � 19 � 18 � t

15 � 12 � 19 � 18 � t Multiply each

side by .

Subtract.

Lessons cover the content ofthe lessons in your textbook.As your teacher discusses eachexample, follow along andcomplete the fill-in boxes.Take notes as appropriate.

Examples parallel theexamples in your textbook.

Foldables featurereminds you to takenotes in your Foldable.

Your Turn Exercises allowyou to solve similarexercises on your own.

Bringing It All TogetherStudy Guide reviews themain ideas and keyconcepts from each lesson.

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Each lesson iscorrelated to theLouisiana GLEs.

viii Algebra: Concepts and Applications

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Your notes are a reminder of what you learned in class. Taking good notescan help you succeed in mathematics. The following tips will help you takebetter classroom notes.

• Before class, ask what your teacher will be discussing in class. Reviewmentally what you already know about the concept.

• Be an active listener. Focus on what your teacher is saying. Listen forimportant concepts. Pay attention to words, examples, and/or diagramsyour teacher emphasizes.

• Write your notes as clear and concise as possible. The following symbolsand abbreviations may be helpful in your note-taking.

• Use a symbol such as a star (★) or an asterisk (*) to emphasis importantconcepts. Place a question mark (?) next to anything that you do notunderstand.

• Ask questions and participate in class discussion.

• Draw and label pictures or diagrams to help clarify a concept.

• When working out an example, write what you are doing to solve theproblem next to each step. Be sure to use your own words.

• Review your notes as soon as possible after class. During this time, organizeand summarize new concepts and clarify misunderstandings.

Note-Taking Don’ts• Don’t write every word. Concentrate on the main ideas and concepts.

• Don’t use someone else’s notes as they may not make sense.

• Don’t doodle. It distracts you from listening actively.

• Don’t lose focus or you will become lost in your note-taking.

NOTE-TAKING TIPS

Word or Phrase Symbol orAbbreviation Word or Phrase Symbol or

Abbreviation

for example e.g. not equal

such as i.e. approximately �

with w/ therefore �

without w/o versus vs

and � angle �

The Language of Algebra

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NOTE-TAKING TIP: When you take notes, it isimportant to record important concepts. Be sureto refer to your notes when reviewing for tests.

Ch

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C H A P T E R

1


Stack sheets of paper with edges �

34

� inch apart.

Fold up bottom edges.All tabs should be thesame size.

Staple along the fold.

Label the tabs with topics from the chapter.

Begin with four sheets of 8�12

� " � 11" paper.

Use algebraic expressions and equationsUse the order of operations to evaluate expressionsUse properties of real numbers to simplify expressionsUse the four-step plan to solve problemsUse sampling and frequency tables.

Algebra

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C H A P T E R




on Page Example

algebraic expression[al-juh-BRAY-ik]

coefficient[CO-i-FISH-unt]

conclusion

conditional

counterexample

data

deductive reasoning[dee-DUK-tiv]

equation[EE-KWAY-zhun]

equivalent expressions[ee-KWIV-a-lunt]

evaluate[ee-val-yoo-WAYT]

factors

formula[FOR-myu-la]

frequency table

histogram


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Chapter BUILD YOUR VOCABULARY1



DefinitionDescription or

on Page Example

hypothesis[hi-PA-the-sis]

if-then statement

inductive reasoning[in-DUK-tiv]

like terms

line graph

numerical expression[noo-MARE-ik-ul]

order of operations

population

product

quotient

sample

sampling

simplest form

simplify

stem-and-leaf plot

term

variable[VARE-ee-a-bul]

whole numbers

Writing Expressions and Equations

Write an algebraic expression for each verbalexpression.

the sum of p and 12

the product of k and q

Write an algebraic expression for eachverbal expression.

a. b decreased by 6 b. the product of 9 and z

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1–1

• Translate words intoalgebraic expressionsand equations.

WHAT YOU’LL LEARNAn algebraic expression contains and

along with �, �, �, and/or �.

A variable is a used to represent an unknownnumber.

A numerical expression contains along with�, �, �, and/or �.

A factor is a quantity that is being .

A product is the result of that are multiplied.

An equation is a mathematical that containsan equals sign (�).

In your notes, writeseveral words or phrases for each of the operations,addition, subtraction,multiplication, anddivision.

ORGANIZE IT


Algebra

GLE 9. Model real-life situations using linear expressions, equations, and inequalities(A-1-H, D-2-H, P-5-H)

(pages 2–3)BUILD YOUR VOCABULARY

A python eats 4 pounds of meat each month.

Write a numerical expression to represent the amountit eats in 5 months.

The numerical expression is � or 4(5).

Write an algebraic expression to represent the amountit eats in d months.

The algebraic expression is 4 � d or .

A spider has eight legs. Write anexpression for each situation.a. the number of b. the number of

legs on 3 spiders legs on m spiders

Write a verbal expression for each algebraic expression.

37 � s

the of and

37 by s

5(b � 3)

5 the difference of and

5 by the of b and 3

Write a verbal expression for eachalgebraic expression.a. x � 7

b. 4 � (2n)

Your Turn

Your Turn

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1–1


Write one other verbalexpression forExamples 5 and 6.

WRITE IT

Write an equation for each sentence.

The quotient of t and 8 equals 20.

t 8 20 or � 20

Seven less than three times g is 31.

� � 31

Write an equation for each sentence.a. A number x increased by 12 is the same as 27

b. The product of 8 and d decreased by 9 equals 23.

Write a sentence for each equation.

j � 4 � 21

more than j is 21.

3z � 12 � 11

The product of decreased by 12 11.

Write a sentence for each equation.a. r � 10 � 5

b. �n3

� � 6 � 30

Your Turn

Your Turn

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1–1


Page(s):Exercises:

HOMEWORKASSIGNMENT

Find the value of each expression.

14 � 10 � 2

14 � 10 � 2 � 14 � 10 and 2.

� 14 and 5.

4 � (6 � 7)4 � (6 � 7) � 4 � 13 6 and 7.

� 52 4 and 13.

Find the value of each expression.a. (2 � 4) � 2 b. 3 � 5 � 2 � 1

Name the property of equality shown by eachstatement.

If k � 7, then k � 3 � 7 � 3.

Property of Equality

If a � 4 � 9, then 9 � a � 4.

Property of Equality

Name the property of equality shown byeach statement.a. If a � 2 � 5 and 5 � 3 � 2, then a � 2 � 3 � 2.

b. If x � 7, then 10 � x � 10 � 7.

Your Turn

Your Turn

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1–2


Order of Operations

Order of Operations1. Find the values of

expressions insidegrouping symbols,such as parentheses( ), brackets [ ], andas indicated byfraction bars.

2. Do all multiplicationsand/or divisions fromleft to right.

3. Do all additionsand/or subtractionsfrom left to right.

In your notes,summarize the Order ofOperations.

KEY CONCEPT

• Use the order ofoperations to evaluate expressions.

WHAT YOU’LL LEARN

GLE 8. Use order of operations to simplify or rewrite variable expressions (A-1-H, A-2-H)

Find the value of [25 � 8(12 � 11)] � 11. Identify theproperty used in each step.

[25 � 8(12 � 11)] � 11

� (25 � 8( )] � 11 Property

� (25 � ) � 11 Identity

� � 11 Property

� 3 Property

Find the value of 30 � (6 � 4) � 1 � 5. Identifythe property used in each step.

Evaluate xy � 8 if x � 4 and y � 3.

xy � 8 � � 8 Replace x with 4 and y with 3.

� � 8 Substitution Property

� 20 Substitution Property

Evaluate each expression if a � 7 and b � 1.a. 10 � a � b b. (a � 2) � 4b

Your Turn

Your Turn

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1–2


Properties of EqualitySubstitution If a � b,then a may be replacedby b.

Reflexive a � a

Symmetric If a � b, thenb � a.

Transitive If a � b, andb � c, then a � c.

Properties of NumbersAdditive Identity When0 is added to any numbera, the sum is a.

Multiplicative IdentityWhen a number a is multiplied by 1, the product is a.

Multiplicative Propertyof Zero If 0 is a factor,the product is 0.

KEY CONCEPTS

To evaluate an expression is to find the of an

expression by replacing the variables with numbers.


Page(s):Exercises:

HOMEWORKASSIGNMENT

Name the property shown by each statement.

8 � (3 � 4) � (3 � 4) � 8 7 � (8 � k) � (7 � 8) � k

Property Property

of of

Name the property shown by each statement.

a. 5 � 4 � 3 � 5 � 3 � 4 b. a � (2 � 8) � (a � 2) � 8

Simplify the expression (4 � m) � 9. Identify theproperties used in each step.

(4 � m) � 9

� (m � ) � 9 Property (�)

� m � (4 � ) Property (�)

� m �

� Property (�)

Your Turn

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1–3


Commutative and Associative Properties

To simplify an expression, all

parentheses first and then add, subtract, ,

or divide.


• Use the commutativeand associativeproperties to simplifyexpressions.

WHAT YOU’LL LEARN

Commutative Propertyof Addition andMultiplication The orderin which two numbersare added or multiplieddoes not change theirsum or product.

Associative Property of Addition andMultiplication The wayin which three numbersare grouped when theyare added or multiplieddoes not change theirsum or product.

KEY CONCEPTS

Standard 1.1 Studentsuse properties of numbersto demonstrate whetherassertions are true orfalse.

Standard 24.3 Studentsuse counterexamples toshow that an assertion isfalse and recognize that a single counterexampleis sufficient to refute an assertion.

Simplify the expression (12 � z) � 7. Identifythe properties used in each step.

State whether the statement Subtraction of wholenumbers is commutative is true or false. If false, providea counterexample.

Write two subtraction expressions using the CommutativeProperty and check to see whether they are equal.

7 � 5 � 5 � 7 Evaluate each expression separately.

7 � 5 � 5 � 7 7 � 5 � and 5 � 7 �

We found a , so the statement

is .

State whether the statement Subtraction ofwhole numbers is associative is true or false. If false, provide acounterexample.

Your Turn

Your Turn©

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1–3


Page(s):Exercises:

HOMEWORKASSIGNMENT

Closure Property ofWhole Numbers Becausethe sum or product oftwo whole numbers isalso a whole number, theset of whole numbers isclosed under additionand multiplication.

KEY CONCEPT

Whole numbers are the numbers 0, 1, 2, 3, 4, and so on.

A counterexample is an that shows that the

statement is .

BUILD YOUR VOCABULARY (pages 2–3)

Simplify each expression.

5(2 � m)

5(2 � m) � (5 � ) � (5 � ) Distributive Property

� � 5m Substitution Property

3(4x � 2)

3(4x � 2) � (3 � ) � (3 � ) Distributive Property

� � 6 Substitution Property


a. 7(b � 3) b. 3(4t � 8)

Your Turn

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Distributive Property

A term is a number, , product, or quotient

of and variables.

The numerical part of a that contains a variable is

a coefficient.

Like terms are terms that contain the same

such as 2a and 5a.

Expressions whose are the same are equivalent

expressions.

An algebraic expression is in simplest form when it has nolike terms and no parentheses.


• Use the DistributiveProperty to evaluateexpessions.

WHAT YOU’LL LEARN

Distributive PropertyFor any numbers a, b,and c, a(b � c) �ab � ac and a(b � c) �ab � ac.

KEY CONCEPTS

In your notes, write three examples of termsand three examples ofitems that are not termsand label each group.Circle the coefficient ineach term.

ORGANIZE IT


Algebra


8p � 5p

8p � 5p � p Distributive Property

� Substitution Property

10k � 6m � 5k � 2m

10k � 6m � 5k � 2m

� 10k � � 6m � Comm. Property (�)

� (10k � ) � (6m � ) Assoc. Property (�)

� ( )k � ( )m Distributive Property

� � Substitution Property

Simplify each expression.a. 3d � 5d b. 6n � 7m � 4n

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

Suppose you deposit $350 into an account that pays2% interest. How much money would you have in theaccount after five years?

EXPLORE You know the amount of money deposited, theinterest rate, and the time.You need to find the amount of money, includinginterest, at the end of five years

PLAN Use the formula I � prt and substitute the knownvalues. Add this amount to the original deposit.

Estimate: 1% of $350 is . So, 2% of $350 is

about 2 � or per year. This will

be in five years. You should have about

� � in five years.

SOLVE I � prt Interest formula

I � � � p � ,

r � ,

t �

I �

You will earn in interest, so the total

amount after five years is �

or .

EXAMINE Since is the same as the ,

the answer is reasonable.

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A Plan for Problem Solving

• Use a four-step plan tosolve problems.

WHAT YOU’LL LEARN

In your notes, list theseven Problem-SolvingStrategies.


Algebra

ORGANIZE IT

REMEMBER ITAlways check tomake sure the answeris reasonable.

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Page(s):Exercises:

HOMEWORKASSIGNMENT

Suppose you deposit $270 into an account thatpays 3% interest. How much money would you have in theaccount after ten years?

How many ways can you make 50¢ using quarters,dimes, and nickels?

EXPLORE You need to know how many ways you can make

without using .

PLAN Make a chart listing every possible combination.

SOLVE

There are ways to make .

EXAMINE Check that each combination totals 50¢ and thatthere are no other possible combinations.

How many ways can you make 30¢ usingquarters, dimes, and pennies?

Your Turn

Your Turn

Coin Number

Quarters 2 1 1 1 0 0 0 0 0 0

Dimes 0 2 1 0 5 4 3 2 1 0

Nickels

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1–6 Collecting Data

One hundred cable-television subscribers are surveyedto find how much time the average American spendsreading. Is this a good sample? Explain.

No; many of those surveyed would prefer

to reading.

Twenty-five hundred people across the state ofGeorgia were randomly surveyed to find the average level ofcompleted education. Is this a good sample? Explain.

Your Turn

• Collect and organizedata using samplingand frequency tables.

WHAT YOU’LL LEARNSampling is a convenient way to data, or

information, so that can be made about

a population.

A sample is a small used to represent a much

larger .


Sampling Criteria A goodsample is representativeof the larger population,selected at random, andlarge enough to provideaccurate data.

KEY CONCEPT

A frequency table is a table that uses tally marks to

and display the of events.



GLE 28. Identify trends in data and support conclusions by using distribution characteristicssuch as patterns, clusters, and outliers (D-1-H, D-6-H, D-7-H)

Make a frequencytable to organizethe data in thechart. Use intervalsof 5.

STEP 1 Make a table with three . Add a title.

STEP 2 Use so there are fewer categories.

In this case, the intervals are of size .

STEP 3 Use marks to record the times in each

.

STEP 4 Count the tally marks in each row and record this

number in the column.

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Record High Temperaturesfor Selected U.S. States (°C)

44 38 53 49 5748 41 43 41 4345 38 48 47 4748 49 46 46 41

Record High Temperaturesfor Selected U.S. States (°C)

� � 2

40–44

45–49

� 1

55–59 �

A museum featureda musical laser show. The table atthe right shows the number ofpeople who attended each of theshows. Make a frequency table toorganize the data.

The owners of a bookstore specializing in travel booksare looking for a new location. They counted thenumber of people who passed by the proposed locationduring one afternoon. The frequency table below showsthe results of their sampling.

Your Turn

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Attendance

96 22 57 61 92 8025 34 76 65 82 3788 67 42 45 94 6492 29 33 57 44 8838 27 75 26 49 9172 36 89 54 45 90

People Tally Frequency

under 13 � � � � 5

teens � � � � � � � 8

20s � � � � � � � � � � � � 25� � � � � � � �

30s � � � � � � � � � � � � 32� � � � � � � � � � � � � �

40s � � � � � � � � � � � � � 36� � � � � � � � � � � � � � � �

50s � � � � � � � � � � � � � � � 18

60s � � � � � � � � � 11

a. Which group of people passed by the location mostfrequently?

b. Is this a good location for the shoe store? Explain.

a. Which group of people passed by the location mostfrequently?

in their 40s

b. Is this a good location for the bookstore? Explain.

Yes, people in their are likely to be interested

in .

The owners of a children’s shoe store arelooking for a new location. They counted the number ofpeople who passed by the proposed location during oneafternoon. The frequency table below shows the resultsof their sampling.

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

People Tally Frequency

under 13 � � 2

teens � � � � � � � � � � � � � 16

20s � � � � � � � � � 11

30s � � � � � � � � 10

40s� � � � � � � � � � � �

34� � � � � � � � � � � � � � � �

50s � � � � � � � � � � � � � � � � 19

60s� � � � � � � � � �

22� � � � � � � �

Construct a line graph of the data given in the table.Use the graph to predict the percent of the labor forcein farming in the year 2010.

Draw a axis and a vertical axis and label them

as shown below. Include the . Plot the .

Draw a by connecting the .

You can see from the graph that the general trend is that

the percent of the labor force in farming is .

A good prediction for the year 2010 might be between

.

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Displaying and Interpreting Data

• Construct and interpretline graphs, histograms,and stem-and-leaf plots.

WHAT YOU’LL LEARNA line graph is one where is displayed to

show .


In your notes, list fourtypes of graphs that areused to organize data.

ORGANIZE IT


Algebra

1940 20001960 1980

20

15

10

5

0

Year

Percent of the LaborForce in Farming

Perc

ent

Percent of the LaborForce in Farming

Year Percent

1940 17

1950 12

1960 6

1970 3

1980 2

1990 2

GLE 28. Identify trends in data and support conclusions by using distribution characteristicssuch as patterns, clusters, and outliers (D-1-H, D-6-H, D-7-H)

Construct a line graph of the data given in thetable. Use the graph to predict the years of life expected atbirth for men born in the year 2010.

Your Turn©

Glencoe/M

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1–7


Life Expectancy for Men, 1910–1990

Year Years of Life

1910 48.4

1930 58.1

1950 65.6

1970 67.1

1990 71.8

A histogram uses data from a frequency table and displays

it over intervals.


The table shows the number of people in different agegroups who entered a new store during the first hour ofits grand opening. Construct a histogram of the data.

Age Tally Frequency

1–10 � � � � 5

11–20 � � � � � � � 8

21–30 � � � � � � � � � � � � � � � � 25� � � �

31–40 � � � � � � � � � � � � � � 32� � � � � � � � � � � �

41–50 � � � � � � � � � � � � � � � � 36� � � � � � � � � � � � �

51–60 � � � � � � � � � � � � � � � 18

61–70 � � � � � � � � � 11

Explain why it isimportant to use equalintervals when labelinga histogram.

REVIEW IT

STEP 1 Draw a horizontal axis and a vertical axis and label themas shown below. Include the title.

STEP 2 Label equal intervals given in the frequency table on thehorizontal axis. Label equal intervals of 5 on

the axis.

STEP 3 For each time interval, draw a whose height

is given by the .

The table shows the number of people in different agegroups who attended a play on opening night. Construct ahistogram of the data.

Your Turn

1–1011–20

21–3031–40

41–5051–60

61–70

4035302520151050

Age

Ages of People Entering Store

Frequency

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Age Tally Frequency

1–10 � � 2

11–20 � � � � � � � � 10

21–30 � � � � � � � � � � � � 15

31–40� � � � � � � � � � � �

23� � � � � � �

41–50� � � � � � � � � � � �

35� � � � � � � � � � � � � � � �

51–60� � � � � � � � � � � � � � � �

29� � � � � � � �

61–70 � � � � � � � � � � 12

The table shows the record high temperatures forseveral states. Make a stem-and-leaf plot of thetemperatures.

The digits are the , so the stems are

. The digits are the .

Arrange the leaves in numerical to make the

results easier to observe and analyze.

Make a stem-and-leaf plot of the heights ininches of the following 3rd graders.48, 46, 39, 38, 45, 46, 49, 42, 50, 52, 42, 48, 47, 44, 51, 43

Your Turn

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When data is organized into two , stems and

leaves, the result is a stem-and-leaf plot.


Record High Temperatures for Several U.S. States (°C)

44 38 53 4957 48 41 4341 43 45 3848 47 47 48

Stem Leaf

3

4 1 1 3 3 4 5 7 7 8 8 8 9

5 3�8 � 38

REMEMBER ITThe leaves in astem-and-leaf plot are always single-digitvalues.

Page(s):Exercises:

HOMEWORKASSIGNMENT


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Write the letter of the algebraic expression that best matcheseach phrase.

1. three more than a number n

2. five times the difference of x and 4

3. one half the number r

4. the product of x and y divided by 2

5. Translate two times the sum of x and 3 equals 4.

Find the value of each expression.

6. 400 � 5[12 � 9] 7. 69 � 57 � 3 � 16 � 4

8. 17 � 3 � 6 9. �24

6��

32� 4

�





You can use your completedVocabulary Builder (pages 2–3) to help you solve the puzzle.


C H A P T E R

1STUDY GUIDE


1-1

Writing Expressions and Equations

1-2

Order of Operations

a. 5(x � 4)

b. �12

�r

c. n � 3

d. �x2y�

Chapter BRINGING IT ALL TOGETHER©

Glencoe/M

cGraw

-Hill

Write the letter of the term that best matches eachequation.

10. 3 � 6 � 6 � 3

11. 2 � (3 � 4) � (2 � 3) � 4

12. 2 � (3 � 4) � (2 � 3) � 4

13. 2 � (3 � 4) � 2 � (4 � 3)

14. Tell how you can use the Distributive Property to write12m � 8m in simplest form. Use the word coefficient inyour explanation.

15. Explain how the Distributive Property could be used torewrite 5(6 � 4).

16. A 1-ounce serving of chips has 140 calories. There are14 servings of chips in a bag. How many calories arethere in a bag of chips?

1


a. Associative Property of Addition

b. Associative Property of Multiplication

c. Commutative Property of Addition

d. Commutative Property of Multiplication

e. Closure Property

1-3

Commutative and Associative Properties

1-4

Distributive Property

1-5

A Plan for Problem Solving

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The principal needs to decide how to select the students tobe polled. Determine whether each is a good sample.Explain why or why not.

17. All of the students are asked to enter through the main doors.Every twenty-fifth student is selected to be polled.

18. Only those students who are in the four classrooms closest tothe principal’s office are selected for the poll.

The manager of Fox Music tallied thenumber of people who applied for jobs at the store.

19. In which months was the number of applicants the least?

20. In which months was the number of applicants the most?

Mrs. Anderson’s science class recorded the daily high temperature every day for 30 days.21. Make a stem-and-leaf plot of the data.

Chapter BRINGING IT ALL TOGETHER1


1-6

Collecting Data

1-7

Displaying and Interpreting Data

1–2 3–4 5–6 7–8 9–10 11–12

5

10

15

20

25

30

0

Months

Freq

uen

cy

Job Applicants, Fox Music

High Temperature (°F)

68 65 68 70 70 70

65 66 68 72 73 76

75 78 81 78 74 71

72 78 77 74 70 68

67 69 68 66 65 63

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Check the one that applies. Suggestions to help you study are given with each item.

ARE YOU READY FOR THE CHAPTER TEST?

I completed the review of all or most lessons without usingmy notes or asking for help.

• You are probably ready for the Chapter Test.

• You may want take the Chapter 1 Practice Test on page 47 ofyour textbook as a final check.

I used my Foldable or Study Notebook to complete the reviewof all or most lessons.

• You should complete the Chapter 1 Study Guide and Reviewon pages 44–46 of your textbook.

• If you are unsure of any concepts or skills, refer back to the specific lesson(s).

• You may also want to take the Chapter 1 Practice Test on page 47.

I asked for help from someone else to complete the review ofall or most lessons.

• You should review the examples and concepts in your StudyNotebook and Chapter 1 Foldable.

• Then complete the Chapter 1 Study Guide and Review onpages 44–46 of your textbook.



Visit algconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 1.

Student Signature Parent/Guardian Signature

Teacher Signature

C H A P T E R

1Checklist


Integers

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NOTE-TAKING TIP: When taking notes, be sure to focus on what the teacher is saying. Listen forimportant concepts being emphasized.

Ch

apte

r 2

C H A P T E R

2


Stack Stack sheets of paper with edges �

34

� inch apart.

FoldFold up bottom edges. All tabs should be the same size.

StapleStaple along the fold.

LabelLabel the tabs as shown.

Begin with four sheets of plain 8�12

�" � 11" paper.

Systems of Equationsand Inequalities

13-1 Graphing systems of equations13-2 Solutions systems of equations13-3 Substitution13-4 Elimination using addition and subtraction13-5 Elimination using multiplication13-6 Solving quadratic-linear systems of equations 13-7 Graphing systems of inequations

Integers2-1 Graph2-2 Compare and Order2-3 Add2-4 Subtract2-5 Multiply2-6 DivideVocabulary

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C H A P T E R




on Page Example


absolute value

additive inverse [A-duh-tiv]

coordinate [co-OR-duh-net]

coordinate plane

coordinate system

dimensions

element

graph

integers [IN-tah-jerz]

matrix [MAY-triks]

natural numbers

negative numbers

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Chapter BUILD YOUR VOCABULARY




on Page Example

number line

opposites

ordered array

ordered pair

origin [OR-a-jin]

quadrants [KWA-druntz]

scalar multiplication [SKAY-ler]

Venn diagram

x-axis

x-coordinate

y-axis

y-coordinate

zero pair

2

Graphing Integers on a Number Line©

Glencoe/M

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2–1

• Graph integers on anumber line and compareand order integers.

WHAT YOU’LL LEARNA number line is a line with distances markedoff to represent numbers.

A negative number is a number less than .

Diagrams that use or ovals inside a rectangle

to show relationships are called Venn diagrams.

Natural numbers are the set of whole numbers without

the element .

To plot points named by numbers on a number lineis to graph.

The that corresponds to a point on a number

line is a coordinate.


Integers Integers arethe negative numbers�1, �2, �3, �4, . . .and whole numbers 0, 1,2, 3, 4, . . .

KEY CONCEPT Name the coordinates of G, H, and J.

The coordinate of G is , H is , and J is .

Graph points K, L, and M on a number line if K hascoordinate �4, L has coordinate 2, and M has coordinate �1.

Find each number on a . Place a

on the mark above the number. Then write the

above the dot.

4–3 –2 –1 0 1 2 3 5–5 –4

K M L

4–3 –2 –1 0 1 2 3 5–5 –4

G J H

GLE 1. Identify and describe differences among natural numbers, whole numbers,integers, rational numbers, and irrational numbers (N-1-H, N-2-H, N-3-H)

a. Name the coordinates of P, S, and Z.

b. Graph points A, B, and C on a number line if A hascoordinate �3, B has coordinate �2, and C has coordinate 5.

Replace in 3 �4 with � or � to make a truesentence.

3 is to the of �4 on the number line. So, 3 �4.

The table shows the hightemperature each day forone week in January in amidwestern city. Order the temperatures from greatest to least.

Graph each on

a number line.

The order from greatest to least is , 5°, , ,

�2°, , and �7°.

8–6 –5 –4 –3 –2 –1 0 21 4 53 6 7–8 –7

4–3 –2 –1 0 1 2 3 5–5 –4

4–3 –2 –1 0 1 2 3 5–5 –4

Z P S

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Write an example of anegative number in areal world application.

WRITE ITDay Temp. (°C)

Monday 5

Tuesday �1

Wednesday 8

Thursday �3

Friday 0

Saturday �7

Sunday �2

Copy the Venn diagramfrom page 52 of yourtextbook under the tabfor Lesson 2-1. Be sureto explain the diagram.




ORGANIZE IT

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2–1


Page(s):Exercises:

HOMEWORKASSIGNMENT

Replace with � or � to make a truesentence.a. �4 2 b. 5 �6

c. The table shows the average low temperature each day for one week in January in a northern city.Order the temperatures from least to greatest.

Evaluate each expression.

��1�

��1� � The graph of �1 is unit away

from .

�4� � ��5�

�4� � ��5� � � The absolute value of 4 is .

� 9 The absolute value of �5 is .

Evaluate each expression.a. ��4� b. ��3� � �7�

Your Turn

Your Turn

Absolute Value Theabsolute value of anumber is the distance it is from 0 on thenumber line.

KEY CONCEPT

City Temp. (°C)

Monday 7

Tuesday 5

Wednesday �3

Thursday �5

Friday �1

Saturday 0

Sunday 4

Write the ordered pair that names point H.

The x-coordinate is ,

and the y-coordinate is .

The ordered pair for point H

is .

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The Coordinate Plane2–2


• Graph points on acoordinate plane.

WHAT YOU’LL LEARNThe coordinate system is the formed by the

intersection of two perpendicular number lines that meet attheir zero points.

The point of intersection of the two in the

coordinate plane is called the origin.

The number line on a coordinate plane is

the y-axis.The horizontal number line is the x-axis.

The planecontaining the x- and y-axes is the coordinate plane.

An ordered pair is a pair of numbers used to locate any

on a .

The number in a coordinate pair is the x-coordinate.

The number in a coordinate pair is the

y-coordinate.


O x

y

1 2 3 4 5

654321

–1–2

–1–2–3J

H

K

GLE 23. Use coordinate methods to solve and interpret problems (e.g., slope as rate ofchange, intercept as initial value, intersection as common solution, midpoint as equidistant) (G-2-H, G-3-H)

Write the ordered pair that names each point.

a. A

b. M

c. D

Graph V(2, 4) on the coordinate plane.

• Start at the , O.

• The x-coordinate is . So, move

2 units to the .

• The y-coordinate is . So, move

4 units and draw a .

Graph each point on a coordinateplane.

a. B(5, 0)

b. G(�5, 3)

Name the quadrant in which F(0, 1) is located.

Point F lies on the . It is not located in a .

Name the quadrant in which each point is located.a. S(�2, �4) b. H(�8, 1)

Your Turn

Your Turn

Your Turn©

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2–2


Vy

xO




ORGANIZE ITUnder the tab for Lesson2-2, draw a coordinateplane and label each ofthe four quadrants aswell as the x- and y-axes.

D

A

M

y

xO

Page(s):Exercises:

HOMEWORKASSIGNMENT

Find each sum.

6 � 7

6 � 7 � Both numbers are , so

the sum is .

�5 � (�8)

�5 � (�8) � Both numbers are , so

the sum is .

Find each sum.a. 7 � 15 b. �14 � (�17)

Your Turn

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Adding Integers2–3


• Add integers.

WHAT YOU’LL LEARN

Adding Integers withthe Same Sign To addintegers with the samesign, add their absolutevalues. Give the resultthe same sign as theintegers.

Additive InverseProperty The sum ofany number and itsadditive inverse is 0.

Adding Integers withDifferent Signs To addintegers with differentsigns, find the differenceof the absolute values.Give the result the samesign as the integer withthe greater absolutevalue.

KEY CONCEPTS

When one algebra tile is paired with one

algebra tile, the result is a zero pair.

If the of two numbers is , the numbers are

called opposites or additive inverses.


Find each sum.

�9 � 5��9� � �5� � � or

��9� � �5�, so the sum is .

Therefore, �9 � 5 � .

GLE 5. Demonstrate computational fluency with all rational numbers (e.g., estimation, mental math, technology, paper/pencil) (N-5-H)

Page(s):Exercises:

HOMEWORKASSIGNMENT

(�9) � 8

��9� � �8� � � or

��9� � �8�, so the sum is .

Therefore, (�9) � 8 � .

Find each sum.a. �5 � 2 b. (�1) � 9

Simplify �7y � 6y.

�7y � 6y � [ � 6]y Use the

Property.

� �7 � 6 �

Simplify 3a � (�5a).Your Turn

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2–3





ORGANIZE ITWrite an expression thatis the sum of onenegative integer and onepositive integer underthe tab for Lesson 2-3.Tell how you knowwhether the sum ispositive or negative.

Find each difference.

10 � 3

10 � 3 � 10 � To subtract 3, add .

�

�7 � (�6)

�7 � (�6) � �7 � To subtract �6, add .

�

�1 � 8

�1 � 8 � �1 � To subtract 8, add .

�

3 � (�5)

3 � (�5) � 3 � To subtract �5, add .

�

4 � 6

4 � 6 � 4 � To subtract 6, add .

�

�7 � (�10)

�7 � (�10) � �7 � To subtract �10, add .

�

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Subtracting Integers2–4


• Subtract integers.

WHAT YOU’LL LEARN

Subtracting Integers Tosubtract an integer, addits additive inverse.

KEY CONCEPT




ORGANIZE ITWrite an expression that is the difference of two negative integersunder the tab for Lesson 2-4. Rewrite the expression as anaddition expression.


Find each difference.a. 9 � 6 b. �8 � (�2) c. �4 � 6

d. 7 � (�2) e. 1 � 8 f. �2 � (�5)

Evaluate a � b if a � �8 and b � �2.

a � b � �8 � (�2) a � �8, b � �2.

� Write �8 � (�2) as .

� �8 � 2 �

Evaluate p � q � r if p � �7, q � �3, and r � 2.

p � q � r � �7 � (�3) � 2 p � �7, q � �3, and r � 2.

� � 2 �7 � (�3) �

� Write �10 � 2 as

.

� �10 � (�2) �

a. Evaluate m � n if m � �5 and n � �4.

b. Evaluate a � b � c if a � �3, b � �8, and c � 5.

Your Turn

Your Turn©

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2–4


Page(s):Exercises:

HOMEWORKASSIGNMENT

Find each product.

4(�3)

4(�3) � The factors have signs.

The product is .

�2(7)

�2(7) �

Find each product.a. 2(�2) b. �3(10)

Find each product.

10(4)

10(4) � The factors have the sign.

The product is .

�8(�6)

�8(�6) �

Find each product.a. 5(3) b. �1(�8)

Your Turn

Your Turn

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Multiplying Integers2–5


• Multiply integers.

WHAT YOU’LL LEARN

Multiplying Two Integerswith Different SignsThe product of twointegers with differentsigns is negative.

Multiplying Two Integerswith the Same SignsThe product of twointegers with the samesigns is positive.

KEY CONCEPTS

GLE 5. Demonstrate computational fluency with all rational numbers (e.g., estimation, mentalmath, technology, paper/pencil) (N-5-H)

GLE 26. Perform translations and line reflections on the coordinate plane (G-3-H)

Find 7(�3)(�6).

7(�3)(�6) � (�6) 7(�3) �

� �21(�6) �

Find each product.a. 3(�4)(�2) b. 2(�1)(�9)(5)

Evaluate 4ab if a � �3 and b � �5.

4ab � 4(�3)(�5) Replace a with �3 and b with �5.

� (�5) 4(�3) �

� �12(�5) �

Simplify (4m)(�7n).

(4m)(�7n) � (4) (�7) 4m � (4) ;

�7n � (�7)

� (4)(�7) Commutative Property

� (4)(�7) � ;

(m)(n) �

a. Evaluate 3xy if x � �7 and y � �3.

b. Simplify (�3n)(2x).

Your Turn

Your Turn

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2–5


Page(s):Exercises:

HOMEWORKASSIGNMENT




ORGANIZE ITWrite a multiplicationexpression using twonegative integers underthe tab for Lesson 2-5.Will the solution bepositive or negative?How do you know?

Find each quotient.

�12 � 3

�12 � 3 � The signs are .

The quotient is .

�50 � (�10)

�50 � (�10) � The signs are the .

The quotient is .

Find each quotient.a. �10 � 5 b. �35 � (�7)

Evaluate �3ba� if a � �6 and b � 9.

�3ba� � �

3(�9

6)� Replace a with and b with .

� 3(�6) �

� Divide by .

Evaluate �5yx� if x � �4 and y � 2.Your Turn

Your Turn

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Dividing Integers2–6


• Divide integers.

WHAT YOU’LL LEARN

9

Dividing Integers Thequotient of two integerswith the same sign ispositive. The quotient oftwo integers withdifferent signs isnegative.

KEY CONCEPT


In the last 5 years at a high school, the number ofstudents with no tardies during the entire school yeardropped from 315 to 95. What was the average changein the number of students without a tardy for each ofthose 5 years?

First, find the change in the number of students with notardies.

� � �220 There were fewer

students with no tardies at the

end of the years.

To find the average change, divide �220 by .

�220 � �

The change in the number of students with no

tardies for the entire school year was per .

In 1990, the population of Washington, D.C.,was 606,900. By 1998, the population of the city had fallen to523,124. Find the average change in population for each ofthose eight years.

Your Turn

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2–6


Page(s):Exercises:

HOMEWORKASSIGNMENT




ORGANIZE ITIn your own words,summarize the rules fordividing integers underthe tab for Lesson 2-6.


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Refer to the Venn diagram shown. Write true or false foreach of the following statements.

1. All whole numbers are integers.

2. All natural numbers are integers.

3. All whole numbers are natural numbers.

4. All natural numbers are whole numbers.

Replace each with � or � to make a true sentence.

5. 6 �3 6. 4 �5 7. ��3� �2� 8. ��1� ��8�

Whole Numbers

Integers

Natural Numbers


Use your Chapter 2 Foldable to help you study for yourchapter test.



You can use your completedVocabulary Builder (pages 28–29) to help you solve the puzzle.


C H A P T E R

2STUDY GUIDE


2-1

Graphing Integers on a Number Line


Glencoe/M

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Graph the given points on the coordinate plane shown.9. A(0, 3)

10. B(1, 4)

11. C(�4, 2)

12. D(�6, �1)

13. Explain how to add integers with the same sign.

14. Explain how to add integers with opposite signs.

15. If two numbers are additive inverses, what must be true abouttheir absolute values?

Find each sum.

16. �3 � (�6) 17. 4 � (�1)

18. 7 � (�10) 19. �1 � 10

2


2-2

The Coordinate Plane

2-3

Adding Integers

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Write each subtraction problem as an addition problem.

20. 12 – 4 21. �15 � 7

22. 0 � 11 23. �15 � (�4)

Describe how to find each difference. Then find each difference.

24. 8 � 11

25. 5 � (�8)

26. 17 � 14

27. �8 � 19

Find each product.

28. (�4)(9) 29. (�2)(�13)

30. 5(�8) 31. 6(3)

32. 4(�3) 33. �7(�10)

Find each quotient.

34. 15 � 12 35. 9 � �1

36. ��35

7� 37. �

��

7183

�



2-5

Multiplying Integers

2-4

Subtracting Integers

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• You may want take the Chapter 2 Practice Test on page 89 ofyour textbook as a final check.












Teacher Signature

C H A P T E R

2Checklist


Addition and Subtraction Equations

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NOTE-TAKING TIP: It is often helpful to reviewyour notes as soon as you can after class.

Ch

apte

r 3

C H A P T E R

3


FoldFold the short sides tomeet in the middle.

FoldFold the top to the bottom.

OpenCut along second fold to make four tabs.

LabelLabel each tab as shown.

Begin with a sheet of 11" � 17" paper.

RationalNumbers

Mean Median Mode Range

Equations Absolute Value

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C H A P T E R




on Page Example


cross products

empty set

equivalent equations

inequality[IN-ee-KWAL-a-tee]

mean

measure of centraltendency

measure of variation

median

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on Page Example

mode

open sentence

range

rational numbers[RA-shun-ul]

replacement set

sequence[SEE-kwens]

solution

solving

statement

unit cost

Rational Numbers

Replace each with �, �, or � to make a true sentence.

�1 ��35

�

Since �1 is to the of ��35

� on a number line,

�1 ��35

�.

�3(2)(0) 7 � (�8)

�3(2)(0) 7 � (�8)

Find the value of each side.

0 is to the of �1 on a number line and 0 �1.

So, �3(2)(0) 7 � (�8).

Replace each with �, �, or � to make a true sentence.a. �2 �

13

� b. 7(0) 9 � (�9)

Your Turn

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3–1

• Compare and orderrational numbers.

WHAT YOU’LL LEARNAn inequality is a mathematical sentence that uses

and to compare two expressions.

Cross products are the products of the

terms of two .


Rational Number Arational number is anynumber that can beexpressed as a fractionwhere the numeratorand denominator are integers and thedenominator is not zero.

Comparison PropertyFor any two numbers a and b, exactly one ofthe following sentencesis true.a � b a�b a � b

KEY CONCEPTS


Replace each with�, �, or � to make each sentence true.

�172� �

58

� ��145� � �

130�

�172� �

58

� ��15

4� �

�10

3�

(7) 12 (�4) 15

� 60 �40 � �45

So, �172� �

58

�. So, ��145� ��

130�.

Replace each with �, �, or � to makeeach sentence true.a. �

23

� �57

� b. ��191� ��

78

�

Write �56

�, �79

�, and �45

� in order from least to greatest.

�56

� � 0.8333 . . . or This is a decimal.

�79

� � 0.7777 . . . or This is a decimal.

�45

� � This is a decimal.

In order from least to greatest, the decimals are

. So, the fractions in order from least

to greatest are .

Write �23

�, �58

�, and �35

� in order from least togreatest.

Your Turn

Your Turn

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3–1


Write three examples ofrational numbers underthe tab for RationalNumbers. Include adecimal and onenegative number.

ORGANIZE IT

RationalNumbers



Find the cross products.

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3–1


Page(s):Exercises:

HOMEWORKASSIGNMENT

Latisha needs to buy snacks for her art club. A packageof 12 granola bars costs $2.69 and a package of 18 granolabars costs $3.55. Which is the better buy? Explain.

Find the unit cost of each package. In each case, the unit cost isexpressed in cents per bar.

unit cost of package of 12:

$2.69 � � 0.2241 or about per bar.

unit cost of package of 18:

$3.55 � � 0.1972 or about per bar.

Since $0.20 $0.22, the package of granola bars

is the better buy.

John needs to buy cookies. The cost of a packageof 24 cookies is $3.79. A package of 36 cookies costs $4.29.Which is the better buy? Explain.

Your Turn

Unit cost is the cost per unit.

unit cost � total � number of


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3–2


Adding and Subtracting Rational Numbers

Find each sum.

0.41 � (�1.3)

0.41 � (�1.3)

� �(� �� ) The signs differ. Subtract

� � ( � )the lesser absolute valuefrom the greater.

�

�5�130� � 2�

35

�

�5�130� � 2�

35

�

� ��5�130�� 2 The LCD is 10. Replace

�35

� with .

� �� The signs differ. Subtract the lesser absolute value from the greater.

� ��

�3.7 � 12.5 � (�1.3)

�3.7 � 12.5 � (�1.3)

� � � � (�1.3)� Commutative &Associative Properties (�)

� � � � Add.

�

• Add and subtractrational numbers.

WHAT YOU’LL LEARN

Explain the procedurefor finding the LCD of two fractions. (previouscourse)

REVIEW IT


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3–2


Find each sum.

a. (�3.2) � (�1.1)

b. �2�34

� � 5�58

�

c. �20.5 � 10.8 � (�5.4)

Find �4.7 � (�5.9)

�4.7 � (�5.9)

� �4.7 � To subtract �5.9 add .

� The signs differ. Subtract the lesserabsolute value from the greater.

Find �8.1 � (�2.4).

Evaluate c � d if c � 5�16

� and d � �3�58

�.

c � d � 5�16

� � ��3�58

�� c � 5�16

�, d � �3�58

�

� 5�16

� � 3�58

� To subtract �3�58

�, add 3�58

�.

� 5 � 3 The LCD is .

� �

� or

Evaluate x � y if x � 8�14

� and y � �4�25

�.Your Turn

Your Turn

Your Turn

Page(s):Exercises:

HOMEWORKASSIGNMENT


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• Find the mean, median,mode, and range of aset of data.

WHAT YOU’LL LEARN A measure of central tendency is a number used to

describe a set of data because it represents a

centralized, or , value.


The stem-and-leaf plot shows the number of childrenenrolled in each of 9 gymnastics classes offered at alocal recreation center.

Find the mean of the gymnastics data.

First, find the of the number of children enrolled.

Then by the number of items of data.

mean � 4 � 5 � 8 � 10 � 12 � 15 � 15 � 18 � 21

� or

Find the median of the gymnastics data.

Arrange the numbers in order from to .

4 8 10 15 15 18

Since there is an number of data items, the

number is the median.

The median is students.

Stem Leaf

0 4 5 81 0 2 5 5 82 1 2|1 � 21

Mean The mean, oraverage, of a set of datais the sum of the datadivided by the numberof pieces of data.

Median The median of aset of data is the middlenumber when the datain the set are arrangedin numerical order.

Mode The mode of a setof data is the numberthat occurs most often inthe set.

KEY CONCEPTS

9

3–3 Mean, Median, Mode, and RangeGLE 27. Determine the most appropriate measure of central tendency for a setof data based on its distribution (D-1-H)

Find the mode of the gymnastics data.

Look for the number that occurs most often.

4 5 8 10 12 15 15 18 21

In this set appears . So, the set of data

has one mode, students.

The stem-and-leaf plot below shows the numberof people enrolled in each of 9 aerobics classes offered at alocal recreation center. Find the mean, median, and mode ofthe aerobics data.

Your Turn

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3–3


Page(s):Exercises:

HOMEWORKASSIGNMENT

Using the stem-and-leaf plot from Examples 1 through 3,find the range of the gymnastics data.

To find the range of the gymnastics data, subtract the

value of the data from the value.

The greatest value is .

The least value is .

So, the range is � or students.

Refer to Example 1. Find the range of theaerobics data.

Your Turn

Measures of variation are used to describe the

of the data.


Range The range of aset of data is thedifference between thegreatest and the leastvalues of the set.

Include mean,median, mode, andrange in your notes. Be sure to include anexample of each.

KEY CONCEPT

Stem Leaf

0 5 7 81 0 3 3 5 62 1 2|1 � 21

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Find the solution of 13 � 33 � 4d if the replacement setis {�6, �5, �4, �3}.

Since makes the sentence 13 � 33 � 4d true,

the solution is .

• Determine whether agiven number is asolution of an equation.

WHAT YOU’LL LEARNA statement is any sentence that is or ,but not both.

A mathematical sentence like m � 5 � 12 is called an opensentence.

A set of numbers from which replacements for a

may be chosen is called a replacement set.

Finding the replacements for the variable that results in a

sentence is called solving.

The solutions of an open sentence are

values that make the sentence true.


Value of d 13 � 33 � 4d True or False?

�6 13 � 33 �

�5 13 � 33 �

�4 13 � 33 �

�3 13 � 33 �

REMEMBER ITA replacement setcontains numbers thatmay result in a falsesentence.

3–4 Equations



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3–4


Find the solution of each equation if thereplacement set for x is {3, 4, 5, 6} and for n is {0, 1, 2, 3}.

a. 4x � 7 � 13 b. n � 2 � �n �

21

�

Solve each equation.

h � [24 � (�3)(2)] � 1

h � [24 � (�3)(2)] � 1

h � (24 � ) � 1 �3 and 2.

h � �1 Divide 24 by .

h � 1 from .

The solution is .

� c

� c Evaluate the numerator and thedenominator separately.

� c 4 � 5 � 6 � � 6 or

� c (3 � 4) � 5 � � 5 or

The solution is .

Solve each equation.a. x � [20 � (�4)(1)] �3 b. 3(7) � 4 � z

Your Turn

4 � 5 � 6��(3 � 4) �5

4 � 5 � 6��(3 � 4) �5

Your Turn

Page(s):Exercises:

HOMEWORKASSIGNMENT

Explain the differencebetween an expressionand an equation.(Lesson 1-1)

REVIEW IT

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3–5


Solving Equations By Using Models

Use algebra tiles to solve each equation.

t � 3 � 1

STEP 1 Model t � 3 � 1.

Place variable

tile and positive

square tiles on one

side of the mat to

represent . Place positive square tile

on the other side of the mat to represent .

STEP 2 To get the variable tile

by itself, add

negative square

tiles to each side.

On the left side

you have zero pairs, and on the right you have

zero pair.

STEP 3 Remove the

.

STEP 4 The variable tile is

matched with

negative square tiles.

So, .

• Solve addition andsubtraction equationsby using models.

WHAT YOU’LL LEARN

t

�

� ��

�

t � 3 � 1

t

�

�

�

�

�

�

�

�

�

�

�

t � 3 � 1

t

�

�

�

�

�

�

�

�

�

�

�

t � 3 � 3 � 1 � 3

t�

��

t � �2

GLE 11. Use equivalent forms of equations and inequalities to solve real-lifeproblems (A-1-H)

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3–5


b � 4 � �3

STEP 1 Write the equation as

b � � �3.

Place variable

tile and negative

square tiles on one side of the mat to represent

. Place negative square tiles on

the other side of the mat to represent .

STEP 2 To get the variable tile by itself,

add

square positive tiles to each side.

STEP 3 Remove the

.

STEP 4 The variable tile is

matched with

positive square tile.

So,

Use algebra tiles to solve each equation.a. x � 4 � 6 b. p � 4 � 2

Your Turn

Page(s):Exercises:

HOMEWORKASSIGNMENT

b

�

��

�

� �

�

�

b � (�4) � �3

b

�

��

�

�

�

�

��

�

�

� �

�

�

�

b � (�4) � �3

b�

��

�

�

�

�

��

�

�

�

�

�

� �

b � (�4) � 4 � (�3) � 4

b ��

b � 1



r � (�13) � 15

r � (�13) � 15

r � 13 � 15 Rewrite the equation.

r � 13 � � 15 � Add to each side.

r � 0 �

r �

Check:r � (�13) � 15

� (�13) � 15 Replace with .

� 15 28 � (�13) �

�4.8 � y � �13.7

�4.8 � y � �13.7

�4.8 � y � � �13.7 � Add to each side.

y � 0 �

y � Check your solution.

Solve each equation. Check your solution.a. z � 12 � 5 b. 8 � m � �6

Your Turn

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Solving Addition and Subtraction Equations3–6

• Solve addition andsubtraction equations by using the propertiesof equality.

WHAT YOU’LL LEARN When the same number is added to each side of anequation, the result is an equivalent equation.


Addition Property ofEquality If you add thesame number to eachside of an equation, thetwo sides remain equal.

Subtraction Property ofEquality If you subtractthe same number fromeach side of an equation,the two sides remainequal.

Write theproperties under the tabfor Equations.

KEY CONCEPTS

GLE 9. Model real-life situations using linear expressions, equations, andinequalities (A-1-H, D-2-H, P-5-H)

Solve k � 12 � �6.

k � 12 � �6

k � 12 � � �6 � Subtract from each side.

k � Check your solution.

Solve x � ��25

�� 170�.

x � � �170� Rewrite the equation.

x � �25

� � � �170� � Subtract from each side.

x � �170� � Rewrite �

25

� as .

x � Check your solution.

Solve each equation. Check your solution.

a. y � 10 � �2 b. x � � � �34

�� 1121�

Your Turn

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3–6


Page(s):Exercises:

HOMEWORKASSIGNMENT

Solve �d � 4 � � 3. Check your solution.

Method 1: Use a number line

�d � 4� � 3 means the distance between d and 4 is units.So, to find x on the number line, start at 4 and move

units in either direction.

d � d �

Method 2: Write and solve a compound sentence.

�d � 4� � 3 also means d � 4 � or d � 4 � .

d � 4 � or d � 4 �

d � 4 � � 3 � d � 4 � � �3 �

d � d �

Check:

Replace d with . Replace d with .

�d � 4� � 3 �d � 4� � 3

� � 4� � 3 � � 4� � 3

� � � 3 � �� 3

� 3 � 3

The solution set is .

43210�1�2 5 6 7 8 9 10 11 12

3 units3 units

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Solving Equations Involving Absolute Value3–7


Write an equation thatcontains an absolutevalue under the tab forAbsolute Value. Solvethe equation usingmethods 1 and 2.

ORGANIZE IT

• Solve equationsinvolving absolute value.

WHAT YOU’LL LEARN

RationalNumbers



GLE 11. Use equivalent forms of equations and inequalities to solve real-lifeproblems (A-1-H)

Solve �g � 3� � 2 � 6

To solve the equation, first rewrite the equation.

�g � 3� � 2 � 6

�g � 3� �2 � � 6 � Add to each side.

�g � 3� �

Next, write a compound sentence and solve it.

g � 3 � or g � 3 �

g � 3 � � 8 � g � 3 � � �8 �

g � g �


Solve and check your solution.a. �a � 2� � 5 b. �x � 4� � 3 � 8

Your Turn

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3–7


Page(s):Exercises:

HOMEWORKASSIGNMENT

Write an example of anabsolute value equationthat has no solution.

WRITE IT

A set with members is called an empty set.


Solve �y � 5� � 2 � �7.

First simplify the expression.

�y � 5� � 2 � �7

�y � 5� � 2 � �7 Add to each side.

�y � 5� � �5 �y � 5� � �5 is nevertrue.

The solution is the or .

Solve �x � 3� � 1 � �8.Your Turn


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1. Explain why ��73�, 0.6� and 15 are rational numbers.

Match the correct inequality symbols at the right with their correct description on the left.

2. less than or equal to

3. greater than

4. less than

Find each sum or difference.

5. �2.6 � 5.8 6. 7.3 � 12.4 7. �10.1 � (�14.9)

8. �14

� � ��18

�� 9. �56

� � ��78

�� 10. �5�18

� � ��2�1156��





You can use your completedVocabulary Builder (pages 48–49) to help you solvethe puzzle.


C H A P T E R

3STUDY GUIDE


3-1

Rational Numbers

3-2

Adding and Subtracting Rational Numbers

a. �

b. �

c. �

d.


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Match the measure of central of tendency at the right with its description on the left

11. the score used most often

12. the average score

13. the middle score

Use the stem-and-leaf plot shown.

14. What is the median of the

data set?

15. What is the mean? Roundto the nearest tenth if

necessary.

16. What is the mode?

17. How can you tell whether an equation is an open sentence?

18. Consider the equation 3n � 6 � 15 with the replacement set{0, 1, 2, 3, 4, 5}. Complete the chart to find the solution(s) ofthe equation.

3


3-3

Mean, Median, Mode, and Range

3-4

Equations

Value for n 3n � 6 � 15 True or False?

0

1

2

3

4

5

a. mean

b. median

c. mode

d range

Stem Leaf

2 0 1 1 2 53 2 2 2 7 9 94 1 3 35 6 6 8 96 0 1 8 8 8 4|2 � 42

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19. For each algebra tile, write the part of the equation that itrepresents.

a. b.


20. �3 � m � 7 21. z � 2 � �5

22. n � 4 � �1 23. 6 � k � 1

24. To solve x � 17 � 46 using the Subtraction Property of

Equality, you would subtract from each side.

25. To solve y � 9 � �30 using the Addition Property of Equality,

you would add to each side.

26. Write an equation that you could solveby subtracting 32 from each side.

Determine whether each sentence is sometimes, always, or never true. Explain.

27. �x � 3� � 12

28. �3 � �m � (�6)�

29. �n� � 8 � 4

30. �6 � �y� � 7

�x



3-5

Solving Equations by Using Models

3-6

Solving Addition and Subtraction Equations

3-7

Solving Equations Involving Absolute Value

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• You may want take the Chapter 3 Practice Test on page 135of your textbook as a final check.










Visit algconcepts.com toaccess your textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 3.


Teacher Signature

C H A P T E R

3Checklist


Multiplication and Division Equations

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NOTE-TAKING TIP: When you take notes, recordimportant ideas and examples from each lesson. Be sure to include examples that will help youunderstand the concepts.

C H A P T E R

4


Ch

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FoldFold each sheet in halfalong the width.

UnfoldUnfold and cut four rows from the left side of each sheet, from the top to the crease.

StackStack the sheets and stapleto form a booklet.


Begin with seven sheets of grid paper.

Mult

iplica

tion a

nd

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ns 4-1 Multiplying Rational Numbers

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C H A P T E R




on Page Example


combination

consecutive integers[con-SEC-yoo-tiv]

event

Fundamental Counting Principle

factorial[fak-TOR-ee-ul]

grouping symbols

identity

multiplicative inverses[mul-tah-PLIK-uh-tiv]

outcomes

permutation[PUR-myu-TAY-shun]

reciprocal[ree-SIP-ruh-kul]

sample space

tree diagram

Find each product.

�3.2(5)

–3.2(5) � The factors have different signs.

The product is .

�4.7(�0.4)

�4.7(�0.4) � The factors have the same sign.

The product is .

Find each product.a. �8.5(3) b. �9.1(�0.8)

A skydiver jumps from 12,000 feet. Solve the equation h � 12,000 � (0.5)(�32.1)(144) to find the skydiver’sheight after he free-falls for 12 seconds.

h � 12,000 � (0.5)(�32.1)(144)

� 12,000 � (144) Multiply 0.5 and �32.1.

� 12,000 � Multiply �16.05 and 144.

� Add 12,000 and �2311.2.

After 12 seconds, the skydiver’s height is feet.

A skydiver jumps 12,000 feet. Solve the equationh � 12,000 � (0.5)(�32.1)(196) to find the skydiver’s heightafter he free-falls for 14 seconds.

Your Turn

Your Turn

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Multiplying Rational Numbers4–1


• Multiply rationalnumbers.

WHAT YOU’LL LEARN

Multiplying TwoRational Numbers with Different SignsThe product of tworational numbers withdifferent signs isnegative.

Multiplying TwoRational Numbers with the Same Sign Theproduct of two rationalnumbers with the samesign is positive.

KEY CONCEPTS


Find each product.

�27

��35

��

�27

��35

�� Multiply the numerators and multiplythe denominators.

� The factors have different signs.

The product is .

�8 � ��37

��8 � ��

37

�� 37

�� Rewrite �8 as an improperfraction.

� ��81

��

37

�� Multiply the numerators andmultiply the denominators.

� or The factors have the samesign. The product is

.

�3�13

� � �17

�

�3�13

� � ��17

�� 17

� Rewrite �3�13

� as an improperfraction.

� ��

310

� 7� 1

� Multiply the numerators andmultiply the denominators.

� The factors have different signs.

The product is .

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4–1


Multiplying Fractions To multiply fractions,multiply the numeratorsand multiply thedenominators.

Write therules for multiplyingrational numbers in yournotes. Be sure to includeexamples.

KEY CONCEPT

Find each product.

a. ��23

� � �49

� b. ��34

� � (�7)

c. �2�12

� � ��56

��

Simplify 5b(�2.2y).

5b(�2.2y) � (5)(b)(�2.2)(y) 5b � (5)(b);(�2.2y)� (�2.2)(y)

� (5)(�2.2) Commutative Property

� (b � y) Associative Property

� Simplify.

Simplify ��14

�p��35

�r�.Your Turn

Your Turn

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4–1


Page(s):Exercises:

HOMEWORKASSIGNMENT

Multiplicative Propertyof –1 The product of –1and any number is thenumber’s additiveinverse.

KEY CONCEPT

Counting Outcomes

Brooke is shopping for a new computer system. She has a list of 2 different CPUs, 3 different monitors, and 3 different printers. How many different ways can shechoose one CPU, one monitor, and one printer?Make a tree diagram to find the number of combinations.

There are different ways.

CPUs Monitors Printers Outcomes

CPU1

M1

M2

M3

P1P2P3

P1P2P3

P1P2P3

CPU1, M1, P1CPU1, M1, P2CPU1, M1, P3



CPU2

M1

M2

M3

P1P2P3

P1P2P3

P1P2P3




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4–2

• Use tree diagrams andthe FundamentalCounting Principle tocount outcomes.

WHAT YOU’LL LEARN

Write an example of acounting problem anddraw a tree diagram forit under the tab forCounting Outcomes.

Mult

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ORGANIZE IT

Outcomes are all possible of a

counting problem or the results of an experiment.

A tree diagram is used to show the total number of

possible .

The sample space is the list of all possible outcomes.


Suppose you can order a pizza with 2 differenttypes of crust, 2 different types of sauce, and 4 different typesof toppings. How many ways can you order a pizza?

How many different kinds of photo processing arepossible?

There are processing times, paper types, and

photo sizes, so the number of different kinds of photo

processing is � � or .

A concession stand offers the choices shown inthe table below. Suppose one item is selected from eachcolumn. How many different choices are possible?

Your Turn

Your Turn

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4–2


Page(s):Exercises:

HOMEWORKASSIGNMENT

Fundamental CountingPrinciple If event M canoccur in m ways and isfollowed by event N thatcan occur in n ways, then the event M followed by event N can occur inm � n ways.

KEY CONCEPT

An event is the subset of the possible outcomes, or

.


Process Time Paper Type Photo Size

1 hour regular 3 by 51 day glossy 4 by 6

deluxe

Meat Topping Bun

hot dog ketchup wheathamburger mustard white

veggie burger onions

Dividing Rational Numbers

Find each quotient.

8 � (�2.5)

8 � (�2.5) � Numbers have different signs.

The quotient is .

�9.3 � (�0.3)

�9.3 � (�0.3) � Numbers have the

sign. The quotient is positive.

Find each quotient.a. �14.4 � (�0.6) b. 8.8 � (�1.1)

Your Turn

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4–3

Two numbers whose product is are called

multiplicative inverses or reciprocals.


• Divide rationalnumbers.

WHAT YOU’LL LEARN

Dividing RationalNumbers with DifferentSigns or the Same SignThe quotient of twonumbers with differentsigns is negative.

The quotient of twonumbers with the samesign is positive.

Multiplicative InverseProperty The product of a number and itsmultiplicative inverse is 1.

Dividing Fractions Todivide a fraction by anynonzero number,multiply by thereciprocal of a number.

KEY CONCEPTS

Find each quotient.

�12 � �25

�

�12 � �25

� � �12 � To divide by �25

�, multiply by itsreciprocal.

� ��620� The numbers have different signs.

The product is negative.

�


2�37

� � �12

�

2�37

� � �12

� � � �12

� Rewrite 2�37

� as an improper

fraction.

� � To divide by �12

�, multiply by its

reciprocal.

� or The numbers have the same sign.

The product is .

Find each quotient.

a. ��38

� � (�6) b. 3�12

� � ��34

��

Two paintings are to be hung on a wall so that thedistances between the centers of the paintings is thesame as the distance from either center to the end ofthe wall. The paintings are both 6 feet wide and the

wall measures 22�14

� feet across. How far from the end

of the wall closest to it should the center of eachpainting be located?

Draw a picture of the 2 paintings and the walls on either side.Draw a dotted line down the center of each painting. Notice howthis divides the wall into 3 pieces. Divide the wall length by 3.

22�14

� � 3 � � 3

� �

� or

The center of each painting should be feet from the wall.

Your Turn

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4–3


22 ft14

REMEMBER ITIf one of thenumbers in a divisionproblem is an integer,write it as a fractionwith a denominator of 1.

Under the tab forDividing RationalNumbers, write anexample showing howto divide two fractions.

Mult

iplica

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ORGANIZE IT

Larry needs to saw a board into 3 equal pieces.

The board measures 18�34

� inches long. How many inches from

each end should he make the cuts?

Evaluate �3x

� if x � ��34

�.

�3x

� � Replace x with .

� 3 � Rewrite the fraction as adivision sentence.

� �31

� � Replace 3 by �31

�. To divide by

��34

�, multiply by its reciprocal.

� ��132� Multiply the numerators and

multiply the denominators.

� Simplify.

Evaluate �5x

� if x � ��27

�.Your Turn

Your Turn©

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4–3


Page(s):Exercises:

HOMEWORKASSIGNMENT

3


5b � 30

5b � 30

� Divide each side by .

b �

Check: 5b � 30

5 � 30 Substitute b � .

� 30

�24 � 3g

�24 � 3g


� g Check the solution.

�5.5z � �22

�5.5z � �22


z � Check the solution.

Solve each equation. Check your solution.a. �3x � 39

Your Turn

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Solving Multiplication and Division Equations4–4


• Solve multiplication anddivision equations byusing the properties ofequality.

WHAT YOU’LL LEARN

Division Property ofEquality If you divideeach side of an equationby the same nonzeronumber, the two sidesremain equal.

KEY CONCEPT

5b 30

�24 3g

�5.5z �22


b. �18t � �90

c. 4.2a � �21

Brian received a $25 gift certificate from hisgrandparents for his birthday. How many $2.35packages of trading cards can he buy with the giftcertificate?

Let c represent the number of packages of trading cards.Write an equation to represent the problem.

Price per number of totalpackage times packages equals cost.

� c �

Solve the equation for c.

c �

�

c �

Since Brian cannot buy part of a package, he has enough

money to buy packages.

Allison is planning her birthday party. Shewants to take her guests to a movie that costs $7.25 perperson. She has $40 to spend on her guests. How many guests can she invite?

Your Turn

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4–4


2.35c 25

Explain how you canrecognize when todivide both sides of anequation by a number.

WRITE IT

Any word problem canbe solved using thefour-step plan. List thesteps below. (Lesson 1-5)

REVIEW IT


�w7

� � 6

�w7

� � 6 Multiply each side by .

��w7

�� (6)

w � Check the solution.

�9 � �12

�m

�9 � �12

�m

(�9) � ��12

�m� Multiply each side by .

� m Check the solution.

��25

�x � �8

��25

�x � �8

��25

�x� � (�8) Multiply each side by .

x � Check the solution.


a. ��m

5� � 12 b. ��

13

�b � 7

c. 24 � ��34

�y

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4–4


Page(s):Exercises:

HOMEWORKASSIGNMENT

Multiplication Propertyof Equality If youmultiply each side of anequation by the samenumber, the two sidesremain equal.

KEY CONCEPT

Write examples showing how to solvemultiplication or divisionequations under the tabfor Solving Multiplicationand Division Equations.

Mult

iplica

tion a

nd

Divis

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ORGANIZE IT

Solving Multi-Step Equations©

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4–5

• Solve equationsinvolving more thanone operation.

WHAT YOU’LL LEARN

Write and solve anequation that involvesmultiple steps. Writeyour equation under the tab for SolvingMulti-Step Equations.

Mult

iplica

tion a

nd

Divis

ion Eq

uatio


ORGANIZE IT


�x6

� � 4 � 2

�6x

� � 4 � 2

�6x

� � 4 � � 2 � Add to each side.

�6x

� � 6

��6x

�� (6) Multiply each side by .

x � Check the solution.

3m � 12 � 27

3m � 12 � 27

3m � 12 � � 27 � Subtract from

each side.

3m � 15


m � Check the solution.

Solve each equation. Check your solution.a. �

h5

� � 6 � 13 b. 9 � 4a � 45

Your Turn

3m 15


Solve �6.2 � �3

1�5

n�.

�6.2 � �3

1�5

n�

15(�6.2) � 15��3 1�5

n�� Multiply each side by 15.

�93 �

�93 � � 3 � � n Subtract.

� n

Solve ��x7� 3� � �3. Check your solution.

In a city, the tallest building is 1268 feet tall. This is 35 feet greater than 3 times the height of the fifth tallest building. How tall is the fifth tallest building?Let x represent the height of the fifth tallest building.Translate the information into an equation and solve.

Height of 3 times the height oftallest building equals 35 feet plus the fifth tallest building.

1268 � 35 � 3x

1268 � 35 � 3x

1268 � � 35 � 3x � Subtract from

each side.1233 � 3x

�Divide each side by .

� x

The fifth tallest building is feet tall.

Your Turn

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4–5


REMEMBER ITWhen solving wordproblems, always checkto make sure youranswer is reasonable.Ask yourself: Does thisanswer make sense?

3x1233

The Parker family recently purchased a new car.Their old car had 105,000 miles on its odometer. This is 50,000more than four times the number of miles on the new car. Howmany miles does the new car have on its odometer?

Your Turn©

Glencoe/M

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4–5


Page(s):Exercises:

HOMEWORKASSIGNMENT

Consecutive integers are integers in counting order such as3, 4, and 5.


Find four consecutive odd integers whose sum is �8.

Let n represent the first odd integer. Then n � 2 representsthe second odd integer, n � 4 represents the third, and n � 6represents the fourth.

� � � � �8

4n � 12 � �8

4n � 12 � � �8 �

4n � �20

�

n �

The numbers are , n � 2 or , n � 4 or ,

and n � 6 or .

Find four consecutive even integers whose sumis �28.

Your Turn

4n �20

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Variables on Both Sides4–6


8y8

• Solve equations withvariables on both sides.

WHAT YOU’LL LEARN

Write and solve anequation that involvesvariables on both sidesof the equation. Writeyour equation under thetab for Variables onBoth Sides.

Mult

iplica

tion a

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Divis

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ORGANIZE IT

An that is for every value of the

variable is called an identity.



y � 8 � 9y

y � 8 � 9y

y � 8 � � 9y � Subtract from each side.

8 � 8y


� y

Solve each equation.a. 4x � 10x � 3 b. �

13

�t � 4 � ��23

�t


3 � 4y � 6 � 4y

3 � 4y � 6 � 4y

3 � � 4y � 6 � 4y � Add to each side.

3 � 6

The equation has . 3 � 6 is never true.

Your Turn


8m � 2 � 2 � 3m � 11m

8m � 2 � 2 � 3m � 11m

8m � 2 � 2 � �3m � 11m �

8m � 2 � 8m � 2 Property

The equation is an . 8m � 2 � 8m � 2 is true for all values of m.

Solve each equation.a. 9 � 5h � 4 � �5h � 13

b. 3y � 11 � 8y � 4 � 5y � 6

Your Turn

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4–6


Page(s):Exercises:

HOMEWORKASSIGNMENT


5(2x � 1) � �255(2x � 1) � �25

10x � 5 � �25 Distributive Property

10x � 5 � � �25 � Add to each side.

10x � �20

� Divide.

x � �2

5(h � 6) � 6 � 3(5h �2)

5(h � 6) � 6 � 3(5h � 2)

� 6 � Distributive Property

5h � � 15h � 6 Add like terms.

5h � 24 � � 15h � 6 � Subtract.

24 � 10h � 6

24 � � 10h � 6 � 6 Add to each side.

30 � 10h

� Divide.

3 � h Check the solution.

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Grouping Symbols4–7


Symbols that group terms together in an expression orequation are called grouping symbols. Parentheses andbrackets are examples of grouping symbols.


10x �20

• Solve equations withgrouping symbols.

WHAT YOU’LL LEARN

Write and solve anequation that involvesgrouping symbols. Writeyour equation under thetab for GroupingSymbols.

Mult

iplica

tion a

nd

Divis

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ORGANIZE IT

30 10h


Solve each equation. Check your solution.a. 4(3x � 7) � �8 b. 7(3 � p) � 7 � �4(p � 5)

The area of the trapezoid is 64 square millimeters.Find the value of x.

A � �12

�(b1� b2)h Area of a trapezoid

� �12

� [ � ]

64 � �12

�(3x � 1)8 Add 2x � 1 and x � 2.

64 � �12

� � 8 � (3x � 1) Commutative Property

64 � (3x � 1)

64 � 12x � 4 Distributive Property

64 � � 12x � 4 � Subtract.

60 � 12x

�Divide.

� x

The value of x is .

The area of a trapezoid is 52 square inches.Find the lengths of the bases if one base is 2 inches more than the other base, and the altitude is 4 inches.

Your Turn

Your Turn©

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4–7


Page(s):Exercises:

HOMEWORKASSIGNMENT

(2x – 1) mm

(x + 2) mm

8 mm

60 12x

REMEMBER ITWhen you addintegers with the samesign, add their absolutevalues. The sign is thesame as the sign of the integers.

When you add integerswith different signs, first find the differenceof their absolute values.The sign will be thesame as the sign of the integer with thegreater absolute value.(Lesson 2-3)


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Multiply.

1. �6.2 � (�0.3) 2. 4(�8.2)

3. ��34�(6) 4. �

�83� � �

�75�

Kynda is playing in a chess tournament. The tree diagram for herpossible outcomes is shown below.

5. Name two different outcomes.

6. Three different outcomes result in a win-loss record of 2-1.What are they?

7. How many outcomes are possible inchoosing a sundae with 7 different types of ice cream, 4 different types of toppings,and 3 different types of sauce?







C H A P T E R

4STUDY GUIDE


4-1

Multiplying Rational Numbers

4-2

Counting Outcomes

Game 1 Game 2 Game 3 Outcomes

win

win

lose

win

losewin

lose

win-win-win

win-win-losewin-lose-win

win-lose-lose

lose

win

lose

win

losewin

lose

lose-win-win

lose-win-loselose-lose-win

lose-lose-lose


Glencoe/M

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Write negative or positive to describe each quotient. Then find thequotient.

8. �4186..62

�

9. �13.4 � 2

10. �37

� � ��45

��11. �

�65� � ��

89

��

Complete the sentence after each equation to tell how you would solve the equation.

12. �7x

� � 16

each side by .

13. 5x � 125

each side by , or multiply each side by .

14. �8k � 96

Divide each side by , or multiply each side by .

15. Explain how rewriting 4�13

�x � 2�18

� as �133�x � �

187� helps you solve

the equation.

4


4-3

Dividing Rational Numbers

4-4

Solving Multiplication and Division Equations

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16. Suppose you want to solve �x �5

3� � 6.

a. What is the first step in solving the equation?

b. What is the next step in solving the equation?

17. The sum of two consecutive odd integers is �36.

a. Write an equation for this situation.

b. What are the two consecutive odd integers?

State the first step in solving each equation.

18. �3x � 6 � �10x � 10

19. �3.6 � 4.2z � 3 � 2.1z

20. �19

� y � 3 � �59

� y

21. Suppose you want to help a friend solve 6k � 7 � 3k � 8.What would you advise her to do first? Why?


22. �8(x � 5) � 4(2 � x) 23. 2(x � 1) � 3(x � 2) � 7



4-5

Solving Multi-Step Equations

4-6

Variables on Both Sides

4-7

Grouping Symbols

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Teacher Signature

C H A P T E R

4Checklist


Proportional Reasoning and Probability

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NOTE-TAKING TIP: When you take notes, it helpsto write about when you would use the conceptin your daily life. For example, percents are usedwhen calculating the tax on an item purchased.

Ch

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C H A P T E R

5

Algebra: Applications and Concepts 93

FoldFold lengthwise to the holes.

Cut Cut four tabs.

Label Label the tabs using lesson concepts as shown.

Solve proportions

Solve problems

involving scale

drawings and models

Solve problems

by using the percent

proportion and the

percent equation

Find the probability of

simple events, mutually

exclusive events, and

inclusive events

Begin with a sheet of notebook paper.

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C H A P T E R




on Page Example

base

box-and-whisker plot

circle graph

complement[kahm-PLU-ment]

compound event

dimensional analysis[duh-MEN-shun-ul]

empirical probability[im-PEER-i-kul]

experimental probability[ek-speer-uh-MEN-tul]

extremes

inclusive[in-KLOO-siv]

independent events

lower quartile[KWAR-tile]

mutually exclusive[MYOO-chew-a-lee]

odds


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on Page Example

percent equation

percent of decrease

percent of increase

percent proportion

percentage

percentile

probability[PRA-buh-BIL-i-tee]

proportion[pro-POR-shun]

random

rate

ratio

scale

scale drawing

scale model

theoretical probability[thee-uh-RET-i-kul]

unit rate

upper quartile


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Solve each proportion.

�m9� � �

6335�

�m9� � �

6335�

9 � 63 Find the cross products.

315 � 63m


� m Simplify.

�z �

49

� � �5z

�

�z �

49

� � �5z

�

(z � 9) � 4 Find the cross products.

5z � � 4z Distributive Property

5z � 45 � � 4z � Subtract from each side.

�45 � �z


� z Simplify.

5–1

• Solve proportions.

WHAT YOU’LL LEARNA ratio is the of two numbers.

An equation stating that two ratios are

is a proportion.


Property of ProportionsThe cross products of aproportion are equal.

KEY CONCEPT

Write a proportion andfind its cross productsunder the tab for SolveProportions.

Solve proportions

Solve problems

involving scale

drawings and models

Solve problems


proportion and the

percent equation




inclusive events

ORGANIZE IT

315 63m

�45 �z

GLE 21. Determine appropriate units and scales to use when solving measurementproblems (M-2-H, M-3-H, M-1-H)

GLE 22. Solve problems using indirect measurement (M-4-H)

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Convert 15 pints to quarts.

Recall that 2 pints � 1 quart. Let x represent the number of quarts. Write a proportion.

�x15

qupainrttss

� � �12

qpuinatrst

�

�1x5� � �

21

�

(1) � 2 Find the cross products.

� 2x

� Divide each side by 2.

� x So, 15 pints � quarts.

Solve each proportion.a. �

246� � �

n6

� b. �bb

��

25

� � �92

�

c. Convert 3 pounds to ounces.

Your Turn

15 2xREMEMBER ITThe denominator ina proportion cannot beequal to zero. Divisionby zero is undefined.

The of two measurements having different units

of measure is called a rate.

A simplified with a denominator of is a

unit rate.

The process of carrying units throughout a

is dimensional analysis.


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5–1


The density of copper is 8.96 grams per cubiccentimeter. Suppose you have a piece of copper whosevolume is 25 cubic centimeters. How many grams ofcopper do you have?

8.96 grams per cubic centimeter �

Multiply the unit rate by the number of cubic

centimeters of copper.

� � or grams

So, the piece of copper contains grams of copper.

The density of wood is 0.71 gram per cubiccentimeter. Suppose you have a piece of wood whose volume is60 cubic centimeters. How many grams of wood does it contain?

A trucker drove 210 miles in 5 hours. At this rate, howfar will she travel in 8 hours?

Write the rate 210 miles in 5 hours as a unit rate. Thenmultiply by 8.

�2510

homuirless

� � �412

hmoiulers

� The unit rate is 42 miles per hour.

8 hours � �412

hmoiulers

� � �81

� � �42 m

1iles� Note that the units cancel.

� miles

Damien drove 220 miles in 4 hours. At this rate,how far will he travel in 6 hours?

Your Turn

Your Turn

8.96 grams��1 cubic centimeter

1

grams

11

Page(s):Exercises:

HOMEWORKASSIGNMENT

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5–2


Scale Drawing and Models

The scale on a map of the upper Midwest is 1 inch �15 miles. Find the distance between Chicago andMilwaukee on the map if the distance between the two cities is 90 miles.

Use the and the distance between the cities to write

a .

�


� 15x


� x

The distance between Chicago and Milwaukee on the map is

about .

A scale drawing or scale model is used to represent an

object that is too or too to be drawn

or built at actual size.

A scale is the of the length of a model to the

corresponding length of the object.


1 inch x inches map distance

actual distance

90 15x

• Solve problemsinvolving scale drawingsand models.

WHAT YOU’LL LEARN

Give examples of scaledrawings or modelsunder the tab for SolveProblems involving ScaleDrawings and Models.

Solve proportions

Solve problems

involving scale

drawings and models

Solve problems


proportion and the

percent equation




inclusive events

ORGANIZE IT

GLE 22. Solve problems using indirect measurement (M-4-H)

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5–2


Page(s):Exercises:

HOMEWORKASSIGNMENT

The scale on a map of Alaska is �78

� inch � 150 miles.

Find the actual distance between Fairbanks and Barrow if the

distance between them on the map is 3�18

� inches.

A railroad car is 36 feet long and a scale model of the railroad car is 1.5 feet long. What is the scale for the model?

Write the ratio of the length of the model to the length of therailroad car. Then solve a proportion in which the length of themodel is 1 foot and the length of the railroad car is x feet.

� �1x f

feoeott

�

1.5x � (1) Find the products.

1.5x �


x �

The scale is 1 foot � feet or .

The height of the Statue of Liberty, from the heels to the top of the head, is about 111 feet. If a model of thestatue is 5.5 feet from the heel to the top of the head, find thescale of the model.

Your Turn

Your Turn

1.5 feet model lengthactual length

1.5x 36

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5–3


The Percent Proportion

A percent is a that compares a number to .

The number that is divided into the

is the base.

The number that is divided by the is the

percentage.


Express the fraction as a percent.

�58

� of the square is shaded.

�58

� � �10

r0

� r is the percent.


� 8r

�5080

� � �88r� Divide each side by 8.

� r So, �58

� � .

Express each fraction or ratio as a percent.a. �

45

� of the circle is shaded.

b. On Friday, 12 out of 60 students brought a sack lunch to school.

Your Turn

• Solve problems by usingthe percent proportion.

WHAT YOU’LL LEARN

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What percent of 175 is 35?

�BP

� � �10

r0

� Use the percent proportion.

� �10

r0

� Replace P with and

B with .

(100) � r Find the cross products.

3500 � 175r

�3157050

� � �117755r


� r So, of 175 is 35.

20 is 40% of what number?

�BP

� � �10

r0

� Use the percent proportion.

�2B0� � �

14000

� Replace P with 20 and r with 40.

(100) � B Find the products.

� 40B


� r So, 20 is of 50.

a. What percent of 80 is 60? b. 25% of what number is67.5?

Your Turn

Percent ProportionIf P is the percentage, B is the base, and r isthe percent, the percent

proportion is �BP

� � �10

r0

�.

KEY CONCEPT

2000 40r

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5–3


Page(s):Exercises:

HOMEWORKASSIGNMENT

A family recently moved to a newhome. The table shows how much time the family spent packing boxes,cleaning the two homes, andunpacking boxes. What percent of the time did the family spend on each activity?

The family worked for 7 � 5 � 8 or 20 days. This is the base. To find each percent, write and solve the percentproportion for each activity.

Packing: � �10

r0

� 7 100 20

Cleaning: � �10

r0

� 5 100 20

Unpacking: � �10

r0

� 8 100 20

The family spent of the time packing, of

the time cleaning, and of the time unpacking.

The table shows thenumber of hours Timothy spent on threeactivities. What percent of the time didTimothy spend on each activity?

Your Turn

=��

=��

=��

A circle graph shows the relationship between partsand the whole.


ActivityTime(days)

packing 7

cleaning 5

unpacking 8

Activity Time (h)

reading 1

sports 4

homework 3

The Percent Equation

Find 17% of $250.

P � RB Use the percent equation.

� ( ) Replace R with and

B with .

� 0.17 250

So, 17% of $250 is .

35% of what number is 105?


� B Replace P with and

R with .


� B 105 0.35

So, 35% of is 105.

a. Find 12% of 360. b. 19 is 25% of what number?

Your Turn

ENTER�

ENTER�

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5–4

• Solve problems by usingthe percent equation.

WHAT YOU’LL LEARN

105 0.35B

Percent EquationThe percentage is equal to the rate timesthe base.

KEY CONCEPT


Write theformulas for percentproportion and thepercent equation underthe tab for SolveProblems by Using thePercent Proportion andthe Percent Equation.

Riona serves food at a restaurant where she is paid 18% of the diners’ bills. She earned $126 last weekend.What was the total of the diners’ bills?


� B Replace P with and

R with .


� B 126 0.18

The total of the diners’ bills was .

A restaurant collects 7% sales tax on all itemssold. If $35 is collected in one day, what are the total sales forthat day?

Your Turn

ENTER�

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5–4


126 0.18B

Mixture problems involve combining or more

parts into a . The parts that are combined

usually have a different price or a different percent ofsomething.


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5–4


Page(s):Exercises:

HOMEWORKASSIGNMENT

Explain why 40 issubtracted from eachside before each side is divided by 56 inExample 4. (Lesson 4-5)

REVIEW IT

All 208 freshmen at a school went on a field trip. Fortransportation, buses that each hold 64 students andvans that each hold 8 students were used. Every bus and van was completely filled, and there were 5 vehiclesused. How many buses and vans were used?

students on buses � students on vans � total students

� �

64b � � � 208 Distributive Property

� 40 � 208 64b � 8b �

56b �

� Divide.

b �

There were buses and � or vans.

A T-shirt shop sells adult shirts for $15 eachand children’s shirts for $8 each. If $324 was collected for 30 shirts, how many of each type were sold?

Your Turn

Number ofCapacity

TotalVehicles Capacity

Buses b 64 64b

Vans 5 � b 8 8(5 � b)

16856b

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Percent of Change5–5


• Solve problemsinvolving percent ofincrease and decrease.

WHAT YOU’LL LEARNWhen an increase or decrease is expressed as a ,

the percent is called the percent of increase or percent of

decrease.


Find the percent of increase or decrease. Round to thenearest percent.

original: 110

new: 140

This is an increase. The amount of increase is

– or . Use the percent proportion.

�BP

� � �10

r0

�

� �10

r0

�

(100) � r Cross products

� 110r

�3100000

� � �111100r


� r

The percent of increase is about .

Find the percent of increase or decrease. Roundto the nearest percent. original: 180, new: 153

Your Turn

REMEMBER ITReplace the equalsign with � to indicatethat your answer isapproximate.


A family bought a home computer for $890. A sales tax of 5.5% on the purchase was then added. What was thetotal price?

First, use the percent equation to find the sales tax.P � RB

�

� 48.95

Then add to $890.

890 � �

The total price was .

Suppose the family also bought a printer. The originalprice of the printer was $140, but they received a 15% discount. What was the sale price of the printer(before the sales tax was added)?

A discount of 15% means that the family will pay

� 15% or of the price of the printer.

Use the percent equation to find the sale price.

P � RB

� (140)

�

The sale price of the printer is .

a. What is the total cost of a dress that sells for $60 if the salestax rate is 5%?

b. All hamsters are on sale for 20% off. What is the sale price ofa hamster that normally sells for $9.95?

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

REMEMBER ITThere may be morethan one way to solve a problem. Refer topage 213 in yourtextbook for anotherway to solve Examples 2and 3.

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5–6


Probability and Odds

Use the graph shown. If a person is chosen at random,what is the probability that the person is age 5–17?

There are million people who are between the ages of 5

and 17. The total population is million.

P (age 5–17) �

� or

The probability is or .

number of people age 5–17��

total population

When all possible have an equally likely

chance of happening, the outcomes are said to be random.

Theoretical probability is what should occur.

What actually occurs when conducting an

is called the experimental probability.


Population of California

Age

Under 55–17

18–2425–3435–4445–5455–64

65 and over

Number (millions)

3

4

24

55

36

Source: U.S. Census Bureau

3

• Find the probability andodds of a simple event.

WHAT YOU’LL LEARN

Probability Theprobability of an event is a ratio that comparesthe number of favorableoutcomes to the numberof possible outcomes.

KEY CONCEPT

GLE 31. Define probability in terms of sample spaces, outcomes, and events (D-4-H)

GLE 33. Explain the relationship between the probability of an event occurring, and the odds of an event occurring and compute one given the other (D-4-H)

A bag contains 1 yellow crayon, 3 red crayons,4 blue crayons, and 7 green crayons. Suppose a crayon ischosen at random. What is the probability that it is yellow?

Your Turn©

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5–6


Page(s):Exercises:

HOMEWORKASSIGNMENT

A that compares the number of ways an event

can occur to the number of it cannot occur

is odds.


A coin is randomly removed from a change purse thatcontains 7 pennies, 8 nickels, and 5 quarters. What arethe odds that the coin is a nickel?

There are 8 nickels. So, there are 8 favorable outcomes.

There are � or coins that are not nickels.

So there are unfavorable outcomes.

odds of choosing a nickel � or

A bag contains 1 yellow crayon, 3 red crayons,4 blue crayons, and 7 green crayons. Find the odds of choosinga green crayon.

Your Turn

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5–7


Compound Events

Two or more simple events connected by the words

or are called compound events.

The outcome of one event does not affect the outcome of

the other event when they are independent events.

Two events that cannot occur at the same time are

mutually exclusive.

Two events that can occur at the time are

inclusive.


The two spinners are spun. Findthe probability that the leftspinner lands on green and theright spinner lands on a numbergreater than 2.

P (green) � (P � 2) � or

P (green and � 2) � � or

Two dice are rolled. Find the probability that aneven number is rolled on the first die and a number less than3 is rolled on the second.

Your Turn

Probability ofIndependent EventsThe probability of twoindependent events isfound by multiplying theprobability of the firstevent by the probabilityof the second event.

Probability of MutuallyExclusive EventsThe probability of twomutally exclusive eventsis found by adding theprobability of the firstevent and the probabilityof the second event.

KEY CONCEPTS

green

bluered2

36

5 4

1

3 6 3

3 3 9

• Find the probability ofmutually exclusive andinclusive events.

WHAT YOU’LL LEARN

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5–7


Page(s):Exercises:

HOMEWORKASSIGNMENT

If there is a 90% chance of snow in January and a 95% chance of snow in February, find the probabilitythat it will snow sometime in January or February.

Since it is possible to snow in both months, these events are inclusive.

P (January) � P (February) �

These events are independent since the weather in January does not affect the weather in February.P (January or February)� P (January) � P (February) � P(January and February)

� 0.9 � 0.95 �

� 1.85 �

� or

a. Refer to the spinners in Example 1. Find the probabilitythat the left spinner lands on red or the right spinner landson 3.

b. If there is a 50% chance of rain on Monday and a 20%chance of rain on Tuesday, find the probability that it willrain sometime on Monday or Tuesday.

Your Turn

A marble is selected at random from a bag that contains5 red marbles, 3 blue marbles, and 2 yellow marbles.What is the probability that the marble is either red oryellow?

A marble cannot be both red and yellow, so the events are mutually exclusive. Find the sum of the individualprobabilities.P (red or yellow) � P (red) � P (yellow)

� � or10 1010

Probability of InclusiveEvents The probabilityof two inclusive events is found by adding theprobabilities of theevents, then subtractingthe probability of bothevents.

Explaincompound events,independent events andinclusive events, underthe tab for probability.

KEY CONCEPT



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For each proportion, write the cross products.

1. �1345� � �

165�

2. �68

� � �1126�

3. A jet flying at a steady speed traveled 825 miles in 2 hours.How far did the jet travel in 1.5 hours?

4. A model car is 12 centimeters long. The real car is 12 feet long.What is the scale of the model?

Write a proportion and solve for each of the following.

5. A 24-inch tall model was made in a scale of 1:3.What is the height of the actual object?

6. A flower that is 18 inches long is drawn to a scale of 1 centimeter to 1 inch.What is the height of theflower in the drawing?





You can use your completedVocabulary Builder (pages 94–95) to help you solvethe puzzle.


C H A P T E R

5STUDY GUIDE

5-1

Solving Proportions

5-2

Scale Drawings and Models



Glencoe/M

cGraw

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Use the percent proportion to find each number.7. 12 is what percent of 36?

8. 16 is 20% of what number?

9. 75% of 28 is what number?

10. The table shows how Lavonne spendsher day. What percent of the time doesshe spend on each activity? Round each answer to the nearest percent.

11. Explain how to check that thepercentages are correct.

Use the percent equation to find each number.

12. 25 is 30% of what number?

13. What is 15% of 200?

14. Find 25% of 15.

5

5-3

The Percent Proportion

5-4

The Percent Equation

Activity Time (hr)

sleep 8

school 6

work 3

homework 2

other 5

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Find the percent of increase or decrease. Round to the nearest percent.

15. original: 50 16. original: 50new: 42 new: 58

17. When you find a discount price, do you add to or subtract from the original price?

Write whether each statement is true or false. If false,replace the underlined word or number to make a true statement.

18. can be written as a fraction, a decimal, or a percent.

19. The outcomes happen at when all outcomes are equally likely to happen.

20. The probability of an impossible event is .

21. The odds against an event occurring are the odds that the event occur.

22. Two dice are rolled. Find the probability that an even number isrolled on the first die and the number 5 is rolled on the second die.

23. A sock contains 2 red marbles, 2 yellow marbles, and 6 bluemarbles. One marble is chosen at random. What is theprobability that the marble is either red or yellow?

will

1

random

Probability



5-5

Percent of Change

5-6

Probability and Odds

5-7

Compound Events

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Teacher Signature

C H A P T E R

5Checklist


Functions and Graphs

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NOTE-TAKING TIP: When you take notes, it ishelpful to write definitions and examples for eachof the vocabulary terms.

Ch

apte

r 6

C H A P T E R

6


Fold Fold lengthwise to the holes.

Cut Cut along the top lineand then cut 10 tabs.

Label Label the tabs using the vocabulary words as shown.


Ordered Pair

Relation

Domain and Range

Range

Solution Set

Linear Equation

Function

Functional Notation and Value

Direct Variation

Inverse Variation

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C H A P T E R




on Page Example


constant of variation

[VARE-ee-AY-shun]

dependent variable

direct variation

domain

equation in

two variables

function

functional notation

functional value

functional variable

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on Page Example

independent variable

inverse variation

linear equation

[LIN-ee-ur]

range

rate problem

relation

solution set

vertical line test

x-coordinate

y-coordinate

Relations

Express the relation {(�4, 5), (�3, 2), (0, 1), (1,�1),(3, �2)} as a table and as a graph. Then determine thedomain and range.

The domain is

and the range is

Express the relation {(�5, 2), (�3, 1),(�2, �1), (0, �2), (2, �2)} as a table and as a graph.Then determine the domain and range.

Your Turn

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6–1

• Show relations as setsof ordered pairs, astables, and as graphs.

WHAT YOU’LL LEARNThe first number in an ordered pair is the x-coordinate.

The number in an ordered pair is the

y-coordinate.

A set of pairs is a relation.


x y

�4 5

0 1

3 �2

x

y

O

(�4, 5)

(�3, 2)

(1, �1)

(3, �2)

(0, 1)

Domain and Range of aRelation The domain ofa relation is the set of allfirst coordinates fromthe ordered pairs of therelation.

The range of the relationis the set of all secondcoordinates from theordered pairs of therelation.

Write thedefinition for a relationand give an exampleunder the tab forRelation.

KEY CONCEPTS

GLE 23. Use coordinate methods to solve and interpret problems (e.g., slope as rateof change, intercept as initial value, intersection as common solution, midpoint asequidistant) (G-2-H, G-3-H)

Express the relation shown on the graph as a set ofordered pairs and in a table. Then find the domain and range.

The set of ordered pairs for the relation

is {(�5, 1), ( ), ( ),

( , 3)}.

The domain is , and

the range is

Express the relation shown on the graph as a set ofordered pairs and in a table.Find the domain and the range.

(0, 4)

(3, 1)

(�4, 3)

(�1, 2)x

y

O

Your Turn

(�5, 1)

(�2, �2) (0, �3)

(2, 3)

x

y

O

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Write a set of orderedpairs under the tab forOrdered Pair. Under thetab labeled Domain andRange, name thedomain and range ofthe set of ordered pairsyou chose.

Ordered Pair

Relation

Domain and Range

Range

Solution Set

Linear Equation

Function


Direct Variation

Inverse Variation

ORGANIZE IT

x y

�5 1

�2 �2

0 �3

2 3

The table shows the population of New York City since 1920.

a. Determine the domain and range of the relation.

The domain is

The range is

b. Graph the relation.

The x-coordinate goes from

to . The

y-coordinates include values from

from to . You can

include 0 and use units of 2.

The table showsColonial Population estimatesfrom 1730 to 1780.

a. Determine the domain andrange of the relation.

b. Graph the relation.

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

YearPopulation

( in millions)

1730 0.6

1740 0.9

1750 1.2

1760 1.6

1770 2.1

1780 2.8Source: www.infoplease.com

YearPopulation(millions)

1920 5.6

1930 6.9

1940 7.5

1950 7.9

1960 7.8

1970 7.9

1980 7.1

1990 7.3

19401920 20001960 1980

1086420

Year

Population

Population(millions)

Explain how you can tellwhich decade had thegreatest increase inpopulation.

WRITE IT

Which of the ordered pairs (0, 0), (1, 4), (2, 1), or (�1, 2)are solutions of y � �x � 3?

Make a table. Substitute the x- and y-values of each orderedpair into the equation.

A statement results when the ordered pair

is substituted into the equation. So, the ordered pair

is a solution of the equation y � �x � 3.

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Equations as Relations6–2


• Solve linear equationsfor a given domain.

WHAT YOU’LL LEARNA set of to a problem is a solution set.

An equation that contains two values is

an equation in two variables.


x y y � �x � 3 True or False?

0 0 0 � (0) � 3

�

1 4 4 � �(1) � 3

�

2 1 1 � �(2) � 3

�

�1 2 2 � �(�1) � 3

�

Solution of an Equationin Two Variables If atrue statement resultswhen the numbers in an ordered pair aresubstituted into anequation in twovariables, then theordered pair is asolution of the equation.

In your ownwords, tell what asolution set is under the tab for Solution Set.

KEY CONCEPT

GLE 15. Translate among tabular, graphical, and algebraic representations of functions andreal-life situations (A-3-H, P-1-H, P-2-H)

GLE 36. Identify the domain and range of functions (P-1-H)

Which of the ordered pairs (0, 2), (1, �3), (2, �5),or (�1, �2) are solutions of y � �2x – 1?

Solve y � 2x � 1 if the domain is {�2, �1, 0, 1, 2}. Graphthe solution set.

Make a table. Substitute each value of into the equation

to determine the corresponding values of .


x

y

O

(2, 5)

(1, 3)

(0, 1)

(�1, �1)

(�2, �3)

Your Turn©

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6–2


x 2x � 1 y (x, y)

�2 2(�2) � 1

�1 2(�1)� 1

0 2(0) � 1

1 2(1) � 1

2 2(2) � 1

Solve y � 3x � 2 if the domain is {�2, �1, 0, 1, 2}. Graph the solution set.

Find the domain of y � 10 � 4x if the range is {�6, �2, 2, 6, 10}.

Make a table. Substitute each value of into the

equation. Then solve each equation to determine the

corresponding values of .

The domain is .

Find the domain of y � 8 � 3x if the range is{�4, �1, 2, 5, 8}.

Your Turn

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

REMEMBER ITSometimes you cansolve an equation for ybefore substituting eachdomain value into theequation. This makescreating a table ofvalues easier.

y y � 10 � 4x x (x, y)

�6 �6 � 10 � 4x

�2 �2 � 10 � 4x

2 2 � 10 � 4x

6 6 � 10 � 4x

10 10 � 10 � 4x

Graphing Linear Relations

Determine whether each equation is a linear equation.Explain. If an equation is linear, identify A, B, and Cwhen written in standard form.

4xy � 4

Since the term has two variables, the

equation cannot be written in the form .

So, this is not a .

y � x

First, rewrite the equation so that both variables are on thesame side of the equation.

y � x

0 � x � y Subtract from each side.

This equation can be written as Therefore,

it is a linear equation in the form Ax � By � C, where

A � , B � , and C � .

Determine whether each equation is alinear equation. Explain. If an equation is linear,identify A, B, and C when written in standard form.

a. 2x � 3 � y b. 2x � 3xy

Your Turn

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6–3

• Graph linear relations.

WHAT YOU’LL LEARN

Linear Equation inStandard Form A linearequation is an equationthat can be written inthe form Ax � By � C,where A, B, and C areany numbers, and A andB are not both zero. Ax � By � C is called thestandard form if A, B,and C are integers.

Write thestandard form of a linearequation under the tabfor Linear Equation. Besure to include examples.

KEY CONCEPT

An equation with a graph that is a

is a linear equation.


GLE 15. Translate among tabular, graphical, and algebraic representations offunctions and real-life situations (A-3-H, P-1-H, P-2-H)

Graph 3x � y � 2.

In order to find values for y more easily, solve the equation

for .

3x � y � 2

�y � 2 � Subtract from each side.

y � �2 � Multiply each side by .

Now make a table and draw the graph.

Graph each equation.a. y � 2x � 2 b. x � y � �5

Your Turn

x

y

O

3x � y � 2

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x �2 � 3x y (x, y)

�2 –2 � 3(�2)

�1 –2 � 3(�1)

0 –2 � 3(0)

1 –2 � 3(1)

2 –2 � 3(2)

Graph y � �2.

In standard form, this equation is

written as � y � �2. So, for

any value of x, y � . For

example, if x � 0, y � ; if x � 1,

y � ; if x � 3, y � .

The graph of y � �2 is a horizontal line.

Graph y � 3x.

Make a table and draw the graph.

In standard form, y � 3x is written as 3x � y � 0. In anyequation where C � 0, the graph passes through the origin.

Graph each equation.a. y � �4 b. y � �4x

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Page(s):Exercises:

HOMEWORKASSIGNMENT

x

y

O

y � �2

x 3x y (x, y)

�2 3(�2)

�1 3(�1)

0 3(0)

1 3(1)

2 3(2)

x

y

O

y � 3x

Determine whether each relation is a function. Explainyour answer.

{(1, 1), (3, 4), (4, 5), (4, 6)}This is not a function because one member of the domain, ,

is paired with two members of the range, and .

The table represents a function since, for each member of the

, there is only corresponding member of

the .

The graph represents a relation that is not a

because there are many members of the

(the x-values) that are paired with members of the

(the y-values).

Determine whether each relation is afunction. Explain your answer.a. {(1, 2), (2, 3), (3, 5), (4, 5)}

Your Turn

x

y

O

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Functions6–4


• Determine whether a given relation is a function.

WHAT YOU’LL LEARN

x �2 �1 0 1 2

y 0 1 2 3 4Function A function is arelation in which eachmember of the domain ispaired with exactly onemember of the range.

Write threeexamples of relationsthat are functions underthe tab for Function. Write each function in a different form.

KEY CONCEPT

GLE 35. Determine if a relation is a function and use appropriatefunction notation (P-1-H)

Standard 16.0 Studentsunderstand the conceptsof a relation and afunction, determinewhether a given relationdefines a function, andgive pertinent informationabout given relations andfunctions.

Standard 18.0 Studentsdetermine whether arelation defined by agraph, a set of orderedpairs, or a symbolicexpression is a functionand justify the conclusion.

b.

c.

Use the vertical line test to determine whether eachrelation is a function.

This relation is since each

vertical line passes through no more than

point of the of the

relation.

This relation is since

a line passes through

point of the graph.

Use the vertical line test to determinewhether each relation is a function.a. b.

x

y

Ox

y

O

Your Turn

O x

y

O x

y

x

y

O

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6–4


x �3 �1 1 1 2

y �1 �2 �3 �4 �5

Vertical Line Test for aFunction If each verticalline passes through nomore than one point ofthe graph of a relation,then the relation is afunction.

KEY CONCEPT

If ƒ(x) � x � 4, find each value.

ƒ(2)

f(2) � x � 4

f(2) � � 4 Replace x with .

� Add.

ƒ��12

��f(x) � x � 4

f��12

�� 4 Replace x with .

� Add.

ƒ(c)

f(x) � x � 4

f(c) � � 4 Replace x with .

If ƒ(x) � �x � 4, find each value.a. f(�1) b. f(1.5) c. f(a)

Your Turn

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6–4


Page(s):Exercises:

HOMEWORKASSIGNMENT

Give an example of anequation in functionalnotation under the tabfor Functional Notationand Value. Then choosea number and find thefunctional value of yourfunction for the chosennumber.

Ordered Pair

Relation

Domain and Range

Range

Solution Set

Linear Equation

Function


Direct Variation

Inverse Variation

ORGANIZE IT Writing equations of the form “y � . . .” as “f(x) � . . .” iscalled functional notation.

A functional value is a that corresponds

to a specific .


Determine whether the equation is a direct variation.

y � �12

�x

Graph the equation. The graph

passes through the .

So, the equation is

.

The constant of variation is .

y � x � 2

Graph the equation. The graph

does not pass through the .

So, the equation is

.

Determine whether each equation is adirect variation.a. y � �

23

�x � 1 b. y � �2x

Your Turn

x

y

O

y � x � 2

x

y

O

y � x12

Direct Variation©

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6–5

• Solve problemsinvolving directvariations.

WHAT YOU’LL LEARN The variable whose value depends on the value of the

independent variable is the dependent variable.

The independent variable is the whose value

is chosen.

The constant of variation is the number in an

equation of the form y � kx.


Direct Variation A directvariation is a linearfunction that can bewritten in the form y � kx, where k � 0.

Write thestandard form of anequation that is a directvariation under the tabfor Direct Variation. Alsodraw the graph of alinear equation thatvaries directly.

KEY CONCEPT

GLE 7. Use proportional reasoning to model and solve real-life problemsinvolving direct and inverse variation (N-6-H)

The length of a trip varies directly as the amount ofgasoline used. How many gallons of gasoline would be needed for a 550-mile trip if a 66-mile trip used 3 gallons of gasoline?

Let � represent the length of the trip and let g represent theamount of gasoline used. The statement the length variesdirectly as the amount of gasoline translates into an equation � � kg in the same way as y varies directly as x translates intoy � kx. Find the value of k.

� � kg Direct variation

� k � Replace � with and g

with .


� k

Next, find the amount of gasoline needed for a 550-mile trip.

� � kg

550 � � g Replace � with 550 and

k with .


� g

A 550-mile trip would use gallons of gasoline.

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6–5


In Example 3, explainthe meaning of k withinthe context of theproblem.

WRITE IT

A problem involving the formula �

� or d � rt is a rate problem.


550 22 � g

How many gallons of gasoline would be neededfor a 630-mile trip if a 126-mile trip used 7 gallons of gasoline?

Suppose y varies directly as x and y � 27 when x � 18.Find x when y � 15.

Use �y

x1

1� � �

y

x2

2� to solve the problem.

� Let y1 � , y2 � , and

x1 � .

x2 � (15) Find the cross products.

27x2 �


x2 �

So, x � when y � 15.

Suppose y varies directly as x and y � 35 whenx � 14. Find x when y � 15.

Your Turn

Your Turn©

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6–5


Page(s):Exercises:

HOMEWORKASSIGNMENT

REMEMBER ITThere are severalcombinations forproportions that can beused in direct variation.In addition to theproportion used in

Example 4, �y

y1

2� � �

x

x1

2� and

�x

y1

1� � �

x

y2

2� can also be used.

x2

27x2 270

The number of bricklayers needed to build a brick wall varies inversely as the number of hours needed.If 4 bricklayers can build a brick wall in 30 hours,how long would it take 5 bricklayers to do it?

Let x � the number of bricklayers. Let y � the number of hours.First find the value of k.

xy � k Definition of inverse variation

� � � k Replace x with and

y with .

� k The constant of variation is .

Next, find the number of hours for 5 bricklayers to build abrick wall.

y � �kx

� Divide each side of xy � k by .

� Replace k with and

x with .

�

A crew of 5 bricklayers can build the wall in hours.

The number of painters needed to paint abedroom varies inversely as the number of hours needed.If 3 painters can paint a bedroom in 8 hours, how long would ittake 4 painters to do it?

Your Turn

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Inverse Variation6–6


• Solve problems involvinginverse variations.

WHAT YOU’LL LEARN

Inverse Variation An inverse variation isdescribed by anequation of the form xy � k, where k � 0.

KEY CONCEPT

GLE 7. Use proportional reasoning to model and solve real-life problems involvingdirect and inverse variation (N-6-H)

Suppose y varies as x and y � �2 when x � �12.Find y when x � 8.

�xx

1

2� � �

yy

2

1� Inverse variation proportion

� Let x1 � , y1 � ,

and x2 � .

(�2) � y2 Find the cross products.

� 8y2

� y2

Therefore, when x � 8, y � .

Suppose y varies inversely as x and y � �5when x � �9. Find y when x � 6.

Your Turn

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6–6


Page(s):Exercises:

HOMEWORKASSIGNMENT

Why does the productof �12 and �2 result ina positive answer?(Lesson 2-5)

REVIEW IT

y2

Match the relation on the left to its other form on the right.

1.

2. {(1, 5), (1, 3), (1, 0), (1, �2)}

3.

4. Name the domain and range of the relation in Exercise 1.

Domain � Range �

Solve each equation if the domain is {�1, 0, 1, 2, 3}.

5. y � 3x 6. y � 2 � x 7. x � y � 1

(2, 1)

(3, 4)(4, 3)

(�1, �1)

x

y

O

a.

b. {(�1, 3), (1, 2), (0, �1), (2, 4)}c.

d. {(3, 4), (2, 1), (4, 3), (�1, �1)}

(1, 0)

(1, 3)

(1, 5)

(1, �2)

x

y

O

x 4 1 3 �1

y 3 2 4 �1


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You can use your completedVocabulary Builder (pages 118–119) to help yousolve the puzzle.


C H A P T E R

6STUDY GUIDE


6-1

Relations

x �1 1 0 2

y 3 2 �1 4

6-2

Equations as Relations


Glencoe/M

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-Hill

Complete the table.

8.

9.

10.

11.

Determine whether each relation is a function. Explain youranswer.

12. 13.

If ƒ(x) � 2x � 3, find the following:

14. ƒ(0) 15. ƒ(�2) 16. ƒ(3) 17. ƒ(b)

x

y

O

6


6-4

Functions

6-3

Graphing Linear Relations

EquationLinear or

ReasonNonlinear?

2x � 3y � 1

4xy � 2y � 7

2x2� 4y � 3

�5x

� � �43y� � 2

x �2 �1 0 4

y 1 2 3 7

For each situation, write an equation with the properconstant of variation.

18. The distance d varies directly as time t.

19. The wages W earned by an employee vary directly with thenumber of hours h that are worked.

20. The length of a trip varies directly as the amount of gasolineused. Pedro’s car used 4 gallons of gasoline in the first 112 miles of his trip. How much gasoline should he expect to use in the remaining 84 miles of the trip?

Write direct variation, inverse variation, or neither todescribe the relationship between x and y described byeach equation.

21. y � 3x 22. xy � 5 23. y � 28x

For each problem, y varies inversely as x. Write an equationyou can use to solve for k. Then write a proportion youcould use to solve the problem.

24.

25.

Problem Equation Proportion

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6-5

Direct Variation

6-6

Inverse Variation

If y � 8 when x � 12,find y whenx � 4.

If x � 50 when y � 6,find x when y � 30.

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Teacher Signature

C H A P T E R

6Checklist


Linear Equations

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NOTE-TAKING TIP: When you take notes, don’tlose focus as to what your teacher is saying. If youdo, you will become lost in your note-taking.Write important equations and/or sketch graphsusing the methods presented in each lesson.

C H A P T E R

7


Fold Fold each sheet in halffrom top to bottom.

Cut Cut along fold. Staple the eight half-sheetstogether to form abooklet.

Cut Cut tabs into margin. The top tab is 4 lines wide, the next tab is 8 lines wide, and so on.

Label Label each page with a lesson number and title. The last tab is for the vocabulary.


7–2

7–1Linear Equations

Ch

apte

r 7

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C H A P T E R




on Page Example


best-fit line

correlation coefficient[CORE-uh-LAY-shun]

extrapolation[ek-STRA-puh-LAY-shun]

family of graphs

interpolation[in-TER-puh-LAY-shun]

linear regression

median-median line

parallel lines[PARE-uh-lel]

parent graph

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on Page Example

perpendicular lines[PER-pun-DI-kyoo-lur]

point-slope form

rate of change

residual

rise

run

scatter plot

slope

slope-intercept form[IN-ter-SEPT]

x-intercept

y-intercept

Slope

Determine the slope of each line.

slope � �cchh

aann

ggee

iinn

yx

�

�

� The slope is .

slope � �cchh

aann

ggee

iinn

yx

�

�

� The slope is .

Determine the slope of each line.a. b.

(�1, �2) (2, �1)x

y

Ox

y

(1, 1)

(2, �1)O

Your Turn

(1, 4)

(3, 4)

O x

y

(1, 3)

(2, 1)

O x

y

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7–1

• Find the slope of a linegiven the coordinates oftwo points on the line.

WHAT YOU’LL LEARNSlope is the ratio of the , or the vertical change,

to the , or the horizontal change.


� 1

� 2

�1

� 4

� 3

�2

GLE 25. Explain slope as a representation of “rate of change” (G-3-H, A-1-H)

Slope The slope of a lineis the ratio of the changein y to the correspondingchange in x.

Name twopoints and find the slopebetween them under the tab for Lesson 7-1.Then give examples of positive, negative,zero and undefinedslopes.

KEY CONCEPT

Page(s):Exercises:

HOMEWORKASSIGNMENT

A line contains the points whose coordinates are listedin the table. Determine the slope of the line.

Each time x increases unit, y decreases units.

slope � �

The slope of the line containing these points is .

A line contains the points whose coordinatesare listed in the table. Determine the slope of the line.

Find the slope of a line that passes through (3, 8) and (3, 4).

m �

�

� The slope is .

Determine the slope of the line that passesthrough (5, 2) and (3, �1).

Your Turn

Your Turn

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7–1


Determining Slope GivenTwo Points The slope mof a line containing anytwo points (x1, y1) and(x2, y2) is given by thefollowing formula.

Slope � �y

x2

2

�

�

y

x1

1�

KEY CONCEPT

y2 � y1

x2 � x1

4 �

3 �

x �2 �1 0 1

y 7 4 1 �2

�3 �3 �3

�1 �1 �1

x �3 �2 �1 0

y 2 4 6 8

change in ychange in x

0

Writing Equations in Point-Slope Form

Write the point-slope form of an equation for each line passing through the given point and having thegiven slope.

(�2, 7), m � ��13

�

y � y1 � m(x � x1) Point-Slope Form

y � � ��13

�(x � ) Replace x1, y1, and m.

y � � ��13

�(x � ).

An equation of the

line is .

(4, 0), m � 4


y � � 4(x � ) Replace x1, y1, and m.

� 4(x � 4)

An equation of the

line is .

Write the point-slope form of an equationfor each line passing through the given point andhaving the given slope.a. (0, �4), m � ��

13

� b. (1, �3), m � 3

Your Turn

(4, 0)x

y

O

(�2, 7)

x

y

O

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• Write a linear equationin point-slope formgiven the coordinates ofa point on the line andthe slope of the line.

WHAT YOU’LL LEARN

Point-Slope Form For anonvertical line throughthe point at (x1, y1) withslope m, the point-slopeform of a linearequation is y � y1 �m(x � x1)

Write anequation in Point-Slopeform and graph theequation under the tabfor Lesson 7-2.

KEY CONCEPT

GLE 24. Graph a line when the slope and a point or when two points are known(G-3-H)

7–2

Write the point-slope form of an equation of the line shown.

First, determine the slope of the line.

m � � or

The slope is . Use the slope and either point to write

an equation.

Method 1 Use (�2, �2).

y � � m(x � ) Point-Slope Form

y � � �14

�[x � ] (x1, y1) �

y � � �14

�(x � )

Method 2 Use (2, �1).

y � � m(x � ) Point-Slope Form

y � � �14

�(x � ) (x1, y1) �

y � � �14

�(x � )

Both and

are point-slope forms of an equation for the line passingthrough (�2, �2) and (2, �1).

Write the point-slope form of an equation of the line shown.

(�2, 5)

(1, �1)x

y

O

Your Turn

O x

y

(–2, –2)(2, –1)

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7–2


How can you tell thatthe two methods usedin the example providethe same equation?

WRITE IT

y2 �

� x1

� (�2)

2 �

1

Page(s):Exercises:

HOMEWORKASSIGNMENT

Writing Equations in Slope-Intercept Form

Write an equation in slope-intercept form of each linewith the given slope and y-intercept.

m � 3, b � �1

y � mx � b Slope-Intercept Form

� x � Replace m with and

b with .

An equation of the line is .

m � ��32�, b � 0


� x � Replace m with and

b with .


Write an equation in slope-intercept formof each line with the given slope and y-intercept.a. m � 0, b � 5 b. m � �

12

�, b � �3

Your Turn

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7–3

• Write a linear equationin slope-intercept formgiven the slope and y-intercept.

WHAT YOU’LL LEARN The y-coordinate of the point at which a graph crosses the

is the y-intercept.

The x-coordinate of the point at which a graph crosses the

is the x-intercept.


Slope-Intercept FormGiven the slope m and y-intercept b of a line,the slope-intercept formof an equation of theline is y � mx � b.

Write anequation in slope-intercept form and graphthe equation under thetab for Lesson 7-3.

KEY CONCEPT

GLE 13. Translate between the characteristics defining a line (i.e., slope, intercepts,points) and both its equation and graph (A-2-H, G-3-H)

GLE 24. Graph a line when the slope and a point or when two points are known (G-3-H)

Write an equation of a line in slope-intercept form foreach situation.

slope �3 and passes through (1, �4)

Using the point-slope form, replace (x1, y1). Then simplify.

y � � m(x � )

y � � �3(x � )

y � 4 � �3x � 3

y � 4 � � �3x � 3 �

y � �3x � 1


passing through (6, 2) and (3, �2)

First, determine the slope of the line.

m � � or �43

�.

Now substitute the known values into the point-slope form.


y � � (x � 6) Replace (x1, y1) with

(6, 2) and m with �43

�.

y � 2 � �43

�x � Distributive Property

Then write in slope-intercept form.

y � 2 � 2 � �43

�x � 8 � 2 Add 2 to each side.

y � �43

�x � Slope-Intercept Form

An equation of the line is . You can see from

the graph that the y-intercept is .

(6, 2)

(3, �2)

x

y

O

(1, �4)

x

y

O

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7–3


3 �

�2 �

�3

REMEMBER ITThe formula for

slope is m � �y

x2

2

�

�

y

x1

1�.

A plumber charges $25 for a service call plus $50 per hour of service. The graph represents the plumber’s charges. Write an equation of the line in slope-intercept form.

The y-intercept of the line is 25.

Determine the slope.

m � � or

Now substitute these values into the -intercept form.


� x � Replace m with and b with .


A taxi drivercharges $5 for each ride plus $2 per mile. The graph representsthe taxi driver’s charges. Writean equation of the line in slopeintercept form.

Your Turn

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7–3


Page(s):Exercises:

HOMEWORKASSIGNMENT

1 �

75 � y2 �

� x1

1 2 3

175

150

125

100

75

50

25

Hours of Service

Plumber’s Charges

Charge(dollars)

y-intercept(0, 25)

y

xO

Ch

arg

e (d

olla

rs)

5

1 2 3 4 5 6 7

10

15

20

25

30

35

Miles

y-intercept (0, 5)

x

y

Write an equation in slope-intercept formof each line.a. a line whose slope is �

12

� b. passing through and passes through (2, �6) (�1, 4) and (3, 0)

Your Turn

The scatter plot showsthe number of yearsof experience and thesalary for eachemployee in a smallcompany. Determinewhether the scatterplot shows a positiverelationship, negativerelationship, or norelationship. If thereis a relationship,describe it.

As the number of years of experience increase, the salary

does not seem to increase or . Thus, there is

between experience and .

Use thescatter plot shown.Determine whether thescatter plot shows apositive relationship,negative relationship, or norelationship. If there is arelationship, describe it.

Your Turn

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Scatter Plots7–4


• Graph and interpretpoints on scatter plots.

WHAT YOU’LL LEARN A graph in which data is displayed as ordered pairs is a scatter plot.


Scatter Plots

Positive relationship: Asx increases, y increases.

Negative relationship: Asx increases, y decreases.

No relationship: Noobvious pattern.

KEY CONCEPTS

GLE 10. Identify independent and dependent variables in real-life relationships (A-1-H)

GLE 29. Create a scatter plot from a set of data and determine if the relationship is linear ornonlinear (D-1-H, D-6-H, D-7-H)

4 8 12 16

80,000

60,000

40,000

20,000

Years of Experience

Experience vs. Salary

Salary(dollars)

y

xOTe

st G

rad

e

10

2 4 6 8 10

20

0

30

40

50

60

70

80

90

Number of Absences

Absences vs. Grades

The table shows the averagenumber of minutes a pediatricdentist spends during eachappointment instructing thepatient in proper dental care,and the number of cavities foreach patient.

A. Make a scatter plot of the data.

Let the horizontal axis represent the instruction time and let the vertical axis represent the number of cavities. Plot the data as shown.

B. Does the scatter plot show a relationship betweeninstruction time and cavities? Explain.

; it appears that a number of cavities

is directly related to a amount of instruction

time. There is a relationship.

C. Describe the independent and dependent variables.Then state the domain and the range.

The number of depends on the

, so the number of

cavities is the variable. The

is the set of all instruction times and the is the

set of all numbers of cavities.

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7–2


ORGANIZE ITGive an example of ascatter plot that has apositive relationshipunder the tab for Lesson7-4. Also give an exampleof one that has anegative relationshipand one that has norelationship.

4 8 12O

654321

Instruction Time (min)

Dental Care

Numberof

Cavities

y

x

Instruction Number of Time (min) Cavities

6 1

4 3

7 2

10 1

1 5

1 6

5 3

2 4

2 3

The table shows the gestation period andaverage longevity for various animals.

A. Make a scatter plot of the data. Let the horizontal axisrepresent gestation time and let the vertical axis representthe longevity of the animal.

B. Does the scatter plot show a relationship between gestationperiod and longevity? Explain.

C. Describe the independent and dependent variables. Thenstate the domain and the range.

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

REMEMBER ITThe domain is theset of all of the firstcoordinates in a set of ordered pairs and the range is the set ofall of the secondcoordinates in the set of ordered pairs.

AnimalGestation Longevity

(days) (years)

kangaroo 36 7

leopard 98 12

tiger 105 16

baboon 187 20

hippopotamus 238 41

gorilla 258 20

Graphing Linear Equations

Determine the x-intercept and y-intercept of the graphof 2y � x � 8. Then graph the equation.

To find the x-intercept, let y � 0.

2y � x � 8

2(0) � x � 8 Replace y with 0.

� 8

��

1x� � �

�81� Divide each side by �1.

x �

To find the y-intercept, let x � 0.

2y � x � 8

2y � 0 � 8 Replace x with 0.

� 8

�22y� � �

82


y �

The x-intercept is , and the

y-intercept is . This means

that the graph intersects the x-axis

at and the y-axis at

. Graph the equation.

Determine the x-intercept and y-intercept of the graph. Graph the equation. 3y � 2x � �6

Your Turn

O x

y

(–8, 0)

(0, 4)

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7–5

• Graph linear equationsby using the x- and y- intercepts or theslope and y-intercept.

WHAT YOU’LL LEARN

Draw the graph of a lineand label the x and yintercepts under the tabfor Lesson 7-5.

7–2


ORGANIZE IT

GLE 13. Translate between the characteristics defining a line (i.e., slope, intercepts,points) and both its equation and graph (A-2-H, G-3-H)

Graph each equation by using the slope and y-intercept.

y � ��15

�x � 2

y � mx � b

y � ��15

�x � 2

The slope is , and the

y-intercept is . Graph the point at .

Then go 1 unit and 5 units. This will be

the point at . Then draw the line through points

and .

3x � y � 4

First, write the equation in slope-intercept form.

3x � y � 43x � y � 3x � 4 � 3x Subtract 3x from each side.

�y � � 4


y � � 4

The slope is . The y-intercept

is . Graph the point at

. Then go 3 units

and 1 unit. This will be

the point at . Then

draw the line through points and .

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REMEMBER ITA negative slopemeans that the graph ofthe line decreases whentracing the line from leftto right. A positive slopemeans that the graph ofthe line increases whentracing the line fromleft to right.

O x

y

(0, 2) (5, 1)

(1, –1)

(0, –4)

O x

y

�y �3x � 4

Graph each equation by using the slopeand y-intercept.

a. y � �13

�x � 1 b. 2x � 4y � 8

Graph y � �3.


y � 0x � 3 slope � , y-intercept �

No matter what the value of x,

y � . So, all ordered pairs

are of the form (x, ). Some

examples are (0, ) and

(2, ).

Graph each equation.a. x � �4 b. y � 2

Your Turn

Your Turn

(–3, –3) (2, –3)

O x

y

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Page(s):Exercises:

HOMEWORKASSIGNMENT

Graph each pair of equations. Describe any similaritiesor differences. Explain why they are a family of graphs.

y � ��12

�x � 2

y � ��12

�x � 1

The graphs have y-intercepts of 2,and �1. They are a family of graphs

because each slope is .

y � 5x � 1y � �x � 1

Each graph has a slope.

Each graph has a y-intercept of �1.Thus, they are a family of graphs.

Graph each pair of equations. Describe anysimilarities or differences. Explain why they are a familyof graphs.a. y � x � 3 b. y � 3x � 2

y � x � 2 y � �23

�x � 2

Your Turn

y = 5x – 1

y = –x – 1O x

y

y = –1–2x + 2

y = –1–2x – 1

Ox

y

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Families of Linear Graphs7–6


• Explore the effects ofchanging the slopesand y-intercepts oflinear functions.

WHAT YOU’LL LEARN

Summarize the twocategories of families ofgraphs under the tabfor Lesson 7-6. What isthe effect of a change inthe slope and y-intercepton the graph of a line?

7–2


ORGANIZE IT

Graphs and equations of graphs that have at least one characteristic in common are a family of graphs.


GLE 38. Identify and describe the characteristics of families of linear functions, with and withouttechnology (P-3-H) GLE 40. Explain how the graph of a linear function changes as the coefficients or constants are changed in the function’s symbolic representation (P-4-H)

Gretchen and Max each have a savings account and plan to save $20 per month. Thecurrent balance in Gretchen’saccount is $150 and thebalance in Max’s account is$100. Then y � 20x � 150 and y � 20x � 100 represent howmuch money each has in theiraccount, respectively, afterx months. Compare and contrast the graphs of the equations.

The equations have the same , but the

of Gretchen’s graph is than the

y-intercept of Max’s graph. Gretchen’s account will alwayshave more money than Max’s.

Tyler and Yingboth babysit for Mrs. Hernandez.Tyler charges $4 per hour andYing charges $5 per hour.Suppose x represents thenumber of hours. Then y � 4x and y � 5x represent how much money each personwill make, respectively, after x hours. Compare and contrastthe graphs of the equations.

Your Turn

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Bal

ance

50

01 2 3 4 5 6 7

100

150

200

250

300

Months

y � 20x � 100

y � 20x � 150

Co

st (

$)

1

0.5 1 1.5 2

2

3

4

5

6

7

8

Time (hours)

y � 4x

y � 5x

Change y � �3x � 1 so that the graph of the newequation fits each description.

same y-intercept, less steep positive slope

The y-intercept is , and the slope

slope is . The new equation will

also have a of �1. In

order for the slope to be less steep and positive, its value

must be than 3, such as 1. The new equation

is .

same slope, y-intercept is shifted down 2 units

The slope of the new equation will be . Since the current

y-intercept is , the new y-intercept will be �1 � 2, or

. The new equation is . Check by

graphing.

Change y � �x � 3 so that the graph of thenew equation fits each description.a. same slope, y-intercept is shifted up 4 units

b. same y-intercept, steeper negative slope

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

Parent Graphs

y � x: As the value of mincreases, the line getssteeper.

y � �x: As the value ofm decreases, the linegets steeper.

y � 2x: As the value of b increases, the graphshifts up on the y-axis.As the value of bdecreases, the graphshifts down on the y-axis.

KEY CONCEPT

The simplest of graphs in a of graphs is a parent graph.


y � x � 1

y � �3x � 1

x

y

O

Parallel and Perpendicular Lines

Determine whether the graphs of the equations areparallel.

y � 3x � 49x � 3y � 12

First, determine the slope of each line. Write each equation in slope-intercept form.

y � 3x � 4 Slope-Intercept Form

The slope is .

9x � 3y � 12

9x � 3y � � 12 �


y � x �

The slope is .

The slopes are not the same so the lines are not .

Determine whether the graphs of the equationsare parallel.

3x � y � 2 2y � �6x � 8

Your Turn

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7–7

• Write an equation of a line that is parallel or perpendicular to the graph of a givenequation and thatpasses through agiven point.

WHAT YOU’LL LEARN

Parallel Lines If two lines have the sameslope, then they are parallel.

Give twoexamples of equationswhose graphs are paralleland two examples ofequations whose graphsare perpendicular underthe tab for Lesson 7-7.

KEY CONCEPT

3y �9x � 12

GLE 23. Use coordinate methods to solve and interpret problems (e.g., slope as rateof change, intercept as initial value, intersection as common solution, midpoint asequidistant) (G-2-H, G-3-H)

Write an equation in slope-intercept form of the line that is parallel to the graph of y � �

23

�x � 3 and passes through the point at (�3, 1).

The slope of the given line is . So, the slope of the new

line will also be . Find the new equation by using the

point-slope form.


y � � �23

�(x � ) x1 � , y1 � ,

and m � .

y � 1 � �23


y � 1 � 1 � �23

�x � 2 � 1 Add 1 to each side.

y � �23

�x �

An equation whose graph is parallel to the graph of y � �23

�x � 3

and passes through (�3, 1) is .

Write an equation in slope-intercept form of the line that is parallel to the graph of y � �

12

�x � 4 and passes

through the point at (�6, 2).

Determine whether the graphs of the equations areperpendicular.

y � 2x � 4

y � �12

�x � 3

Your Turn

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7–7


Perpendicular Lines If the product of theslopes of two lines is �1,then the lines areperpendicular.

KEY CONCEPT

First, determine the slopes of the lines. Write each equation inslope-intercept form.

y � 2x � 4 y � �12

�x � 3

y � 2x � � 4 �

y � � 4

The graphs are perpendicular becausethe product of their slopes is �2 � �

12

�

or .

Determine whether thegraphs of x � y � 1 and x � y � �4 are perpendicular.

Write an equation in slope-intercept form of the linethat is perpendicular to the graph of y � �2x � 5 andpasses through the point at (2, �3).

The slope is �2. A line perpendicular to the graph of

y � �2x � 5 has slope �12

�. Find the new equation by using

the point-slope form.


y � � �12

�(x � ) x1 � , y1 � ,

and m � .

y � � �12


y � 3 � 3 � �12

�x � 1 � 3 Subtract 3 from each side.

y � �12

�x �

Write an equation in slope-intercept form of the line that is perpendicular to the graph of y � �

14

�x � 2 and passes

through the point at (�1, �2).

Your Turn

Your Turn

y � �2x � 4

y � x � 312

x

y

O

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7–7


Page(s):Exercises:

HOMEWORKASSIGNMENT


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Find the slope between each set of points.

1. (2, �3), (5, 1) 2. (0, 5), (�1, �4) 3. (2, 0), (2, �1)

4. In the formula y � y1 � m(x � x1), what do x1 and y1 represent?

Match each equation with the correct graph.

5. y � 3 � �12

�(x � 2) 6. y � 2x 7. y � 1 � ��23

�x

a. b. c.

x

y

Ox

y

Ox

y

O







C H A P T E R

7STUDY GUIDE


7-1

Slope

7-2

Writing Equations in Point-Slope Form


Glencoe/M

cGraw

-Hill

Complete the chart below by writing the formula for each form ofequation. Then write an example of each equation.

8.

9.

Refer to the scatter plot shown at the right.

10. Which quantity is the independent quantity? the dependent quantity?

11. What conclusion can you draw from the scatter plot?

12. Explain how to find the x- or y-intercept of an equation.

7


7-4

Scatter Plots

7-3

Writing Equations in Slope-Intercept Form

Len

gth

(m

inu

tes)

05 10 15 20 25 30 35

10

20

30

Distance from Workplace (mi)

x

y

Form of Equation Formula Example

standard form

slope-intercept form

7-5

Graphing Linear Equations

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Refer to the graph shown at the right.

13. What is the y-intercept of the graph?

14. What is the x-intercept of the graph?

Tell whether each set of graphs is a family of graphs.

15. 16.

17. Parallel lines (always/never) intersect.

Tell whether the graphs are parallel, perpendicular, orneither. Explain.

18. 19.

x

y

Ox

y

O

x

y

Ox

y

O

x

y

O



7-6

Families of Linear Graphs

7-7

Parallel and Perpendicular Lines

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Teacher Signature

C H A P T E R

7Checklist


Powers and Roots

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NOTE-TAKING TIP: When taking notes in mathclass, be sure to write down important rules andproperties. It is also a good idea to recordexamples of any rules and properties.

C H A P T E R

8


Fold Fold each sheet of gridpaper in half along thewidth. Then cut along the crease.

Staple Staple the eight half-sheetstogether to form a booklet.

Cut Cut even lines from thebottom of the top sheet,six lines from the secondsheet, and so on.

LabelLabel the tabs with lesson topics as shown.


Powers and rootsUse powers in expressions

Multiply and divide PowersSimplify expressions containing negative exponents

Scientific NotationSimplify Radicals

Estimate square rootsPythagorean Theorem

Ch

apte

r 8



on Page Example


This is an alphabetical list of new vocabulary terms you will learn in Chapter 8.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description on thesepages. Remember to add the textbook page number in the second column forreference when you study.

C H A P T E R


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base

composite number[kahm-PA-zit]

converse

exponent[ek-SPO-nent]

hypotenuse[hi-PA-tin-oos]

irrational numbers[i-RA-shun-ul]

leg

negative exponent



on Page Example

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perfect square

power

prime factorization[FAK-tor-i-ZAY-shun]

prime number

Pythagorean Theorem[puh-THA-guh-REE-un]

radical[RAD-ik-ul]

radical expression

radical sign

scientific notation

square root

Write each expression using exponents.

5 � 5 � 5

The base is . It is a factor times. 5 � 5 � 5 �

d � d � d � d � d � d

The base is . It is a factor times.

d � d � d � d � d � d �

Write (6)(6)(�7)(�7)(�7)(�7)(�7) using exponents.

Use the Associative Property to group factors with like bases.

(6)(6)(�7)(�7)(�7)(�7)(�7) � [(6)(6)][(�7)(�7)(�7)(�7)(�7)]

�

Write each expression using exponents.a. c b. (�3)(�3)(�3)(�3)

Your Turn

Powers and Exponents©

Glencoe/M

cGraw

-Hill


8–1

• Use powers inexpressions.

WHAT YOU’LL LEARNA perfect square is the of an integer and itself.

An exponent tells how many times a number is used as a

.

The number that is raised to a is called a base.

A power is a number that is expressed when using an

.


GLE 2. Evaluate and write numerical expressions involving integer exponents (N-2-H)

GLE 12. Evaluate polynomial expressions for given values of the variable (A-2-H)

Write each power as a multiplication expression.

64

64� The exponent 4 means that

6 is a factor 4 times.

h5

h5� The exponent 5 means that

h is a factor 5 times.

7a3b2

7a3b2� 7 is used as a factor once,

a is used 3 times, and b is used twice.

Write each power as a multiplicationexpression.

a. 83 b. b5

c. 3yz4

Evaluate 5a2 if a � 4.

5a2� 5� �

2

Replace a with 4.

� 5� � Evaluate the power 4 � 4 � 16.

� Multiply.


a. 2x3 if x � 4 b. 4b2� c3 if b � �2 and c � �2

Your Turn

Your Turn

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8–1


Page(s):Exercises:

HOMEWORKASSIGNMENT

Order of Operations

1. Do all operationswithin groupingsymbols first; startwith the innermostgrouping symbols.

2. Evaluate all powers inorder from left toright.

3. Do all multiplicationsand divisions from leftto right.

4. Do all additions andsubtractions from leftto right.

KEY CONCEPT

Write three examples ofexpressions withexponents under theappropriate tab. Then,write them again inexpanded form.





ORGANIZE IT

Multiplying and Dividing Powers©

Glencoe/M

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-Hill


8–2

• Multiply and dividepowers.

WHAT YOU’LL LEARN Simplify each expression.

52 � 57

52� 57

� 5 To multiply powers that have the samebase, write the common base, then add

�the exponents.

s10 � s5

s10� s5

� s To multiply powers that have the same base, write the common base,

�then add the exponents.

(10a)(5a4)

(10a)(5a4) � � � 5�� a4� Use the Commutativeand AssociativeProperties. a � a1

� 50a

�

(m5n4)(m3n7)

(m5n4)(m3n7) � (m5� m3)(n4

� n7) Use the Commutativeand Associative

� m � n Properties.

�


a. 53� 56 b. a � a4

c. (6x)(3x2) d. (a2b3)(a8b5)

Your Turn

Product of Powers Youcan multiply powers withthe same base by addingthe exponents.

Quotient of Powers Youcan divide powers withthe same base bysubtracting the exponents.

KEY CONCEPTS

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8–2



�77

6

2�

�77

6

2� � 7 � 2 To divide powers that have the same base,write the common base. Then subtract the

� exponents

�pp10�

�pp

10� � p � 1 Write the common base. Then subtract the

�

exponents.

�1

3

5

aa4

6

bb3

4�

�135aa4

6

bb3

4

� � ��aa

6

4�� Group the powers that have the same base.

� 5a b

� 5a b

�


a. �1100

8

5� b. �cc

7

�

c. d. 21y8z9�3y2z8

20x5y6�

5x2y

Your Turn

Write three examples ofmultiplying terms withpowers and threeexamples of dividingterms with powersunder the tab forLesson 8-2.





ORGANIZE IT

3b4� � � �

Page(s):Exercises:

HOMEWORKASSIGNMENT

Write 6�2 using positive exponents. Then evaluate theexpression.

6�2� Definition of negative exponent.

� 6 � 6 �


q3r�4

q3r�4� q3

� r�4

� q3� Definition of negative exponent.

�

�mm

2

5nn

1

2

0�

�mm

2

5nn

1

2

0

� � ��mm

2

5�� nn

1

2

0

�� m � n Quotient of powers.

� m � n

� � n8 Definition of negative exponent.

�


a. x�3 b. a�3b c. �cc

2

3dd

5�

Your TurnPowers and roots

Use powers in expressionsMultiply and divide Powers

Simplify expressions containing negative exponentsScientific Notation

Simplify RadicalsEstimate square rootsPythagorean Theorem

Negative Exponents©

Glencoe/M

cGraw

-Hill


8–3

• Simplify expressionscontaining negativeexponents.

WHAT YOU’LL LEARN

Write three examples ofvariables with negativeexponents under the tabfor Lesson 8-3. Then,write those examplesusing positiveexponents.

ORGANIZE IT

6

1

1

1

Negative Exponents

a�n � �a1n�; �

a�

1n� � an

The value of a cannot be 0.

KEY CONCEPT

Simplify ��

2150hh

3

�

k

6

7k4

� .

��

2150hh

3

�

k

6

7k4

� � ��

2150

�� hh

�

3

6

�� kk

4

7��

hh

�

3

6

�� kk

4

7�� Write ��

2150

� in simplest form.

� Quotient of powers.

� Simplify the powers.

�

A large archery target has a diameter of 1 meter.An arrow tip has a diameter of 10�2 meter. How manyarrows could fit across the diameter of the target?

To find the number of arrows, divide 1 by 10�2.

�10

1�2� � �

1100�

0

2� 1 � 100

� 100 �

� or

a. Simplify ��188aa

�

4

2

kk

8

5�.

b. If a pencil’s eraser had a diameter of 1 centimeter, how manypencil’s sharpened lead tips could fit across the diameter ofthe eraser if each tip was 10�3 centimeters?

Your Turn

HOMEWORKASSIGNMENT

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8–3

Page(s):Exercises:

HOMEWORKASSIGNMENT


�2h k5

�2h k5

�2� �

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Scientific Notation

• Express numbers inscientific notation.

WHAT YOU’LL LEARN Express each measurement in standard form.

8 kilobytes

8 kilobytes

� 8 � The prefix kilo- means .

� bytes Move the decimal point

places right.

2.5 microseconds

2.5 microseconds

� 2.5 � The prefix micro- means .

� second Move the decimal places

to the left.

Express each measurement in standardform.a. 5 gigabytes b. 4.5 milliseconds

Express each number in scientific notation.

325,000

325,000 � � 10? The decimal point moves 5 places.

� Since 325,000 is greater thanone, the exponent is positive.

Your Turn

Multiplying by Powersof 10

• If the exponent ispositive, move thedecimal point to theright.

• If the exponent isnegative, move thedecimal point to theleft.

Scientific Notation Anumber is expressed inscientific notation when

it is the form a � 10n, 1 � a � 10 and n is an integer.

KEY CONCEPTS

GLE 3. Apply scientific notation to perform computations, solve problems, andwrite representations of numbers (N-2-H)

8–4

HOMEWORKASSIGNMENT

0.00028

0.00028 � � 10? The decimal point moves

places.

� Since 0.00028 is between zero and one, the exponent is negative.

Express each number in scientific notation.a. 78,000,000 b. 0.0032

Evaluate 30 � 30,000,000.

First express each number in scientific notation. Then use theAssociative and Commutative Properties to regroup factors.

30 � 30,000,000 � (3 � 10 )(3 � 10 )

� (3 � 3)� � � Associative andCommutative Properties.

� 9 � 10

�


a. 20 � 40,000 b. �94.6

�

�

11005

9

�

Your Turn

Your Turn

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8–4


Write three largenumbers and three verysmall numbers underthe tab for Lesson 8-4.Then, write eachnumber using scientificnotation.





ORGANIZE IT

Page(s):Exercises:

HOMEWORKASSIGNMENT

8–5 Square Roots©

Glencoe/M

cGraw

-Hill



�100�Since 102

� , �100� � .

��81�Since 92

� , ��81� � .

�256�Find the prime factorization of 256.

�256� �

� �16� � �16� Use the Product Property of Square Roots.

� Simplify the radical.

• Simplify radicals byusing the Product andQuotient Properties ofSquare Roots.

WHAT YOU’LL LEARN

Square Root A squareroot of a number is oneof its two equal factors.

Product Property ofSquare Roots The squareroot of a product ofpositive numbers is equalto the product of eachsquare root.

Quotient Property ofSquare Roots The squareroot of a quotient ofpositive numbers is equalto the quotient of eachsquare root.

KEY CONCEPTS

The symbol, , called a radical sign, is used to

indicate a .

An that contains a square root is a

radical expression.

A whole number that has only two , one

and itself, is a prime number.

A that has more than two factors is a

composite number.

The prime factorization of a number is an expression of the

of the prime factors.


2 � 2 � 2 � 2 � 2 � � �

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HOMEWORKASSIGNMENT

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8–5


�400��400� �

� �16� � �25�

� � or

Simplify each expression.a. �4� b. ��121�

c. �225� d. �289�

Simplify ��1090

��.

��1090

�� Use the Quotient Property of Square Roots.

�

Simplify ��130204

��.Your Turn

Your Turn

2 � 2 � 2 � 2 � �

Write three radicalexpressions under thetab for Lesson 8-5. Then,simplify the expressions.If one cannot besimplified, explain why.





ORGANIZE IT

�

Estimating Square Roots


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Pages(s):Exercises:

HOMEWORKASSIGNMENT

• Estimate square roots.

WHAT YOU’LL LEARN

Write the square of eachinteger from 1–20 andwrite the squares underthe tab for Lesson 8-6.For example, 12 = 1, 22 = 4, etc.





ORGANIZE IT

Estimate each square root.

�48�

List some perfect squares to find the two perfect squaresclosest to 48.

1, 4, 9, 16, 25, 36, 49, ... 48 is between and .

� 48 �

�36� � �48� � �49�

� �48� �

Since 48 is closer to than to , the best

whole number estimate for �48� is .

�200�

� 200 � 144, 196, 225, 256, 289...

�196� � �200� � �225�

� �200� �

Since 200 is closer to than to , the best

whole number estimate for �200� is .

Estimate each square root.

a. �15� b. �250�

Your Turn

Irrational numbers are not rational or integers because

their decimal values do not terminate or repeat.


8–6

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The Pythagorean Theorem8–7


Find the length of the hypotenuse of the right triangle.

c2� a2

� b2 Pythagorean Theorem

c2�

2�

2Replace a and b.

c2� � 81

c2� Find the square root of each side.

c � �225�

c �

The length is inches.

Find the length of thehypotenuse of the right triangle.

10 ftc ft

24 ft

Your Turn

9 in.

12 in.

x in.

• Use the PythagoreanTheorem to solveproblems.

WHAT YOU’LL LEARNThe side opposite the right of a right triangle is

called the hypotenuse.

The two sides that form the right angle of a

triangle are called the legs.


Pythagorean Theorem In a right triangle, thesquare of the length ofthe hypotenuse, c, isequal to the sum of thesquares of the lengths ofthe legs, a and b.

Draw a righttriangle and label theright angle, hypotenuse,and the legs under thetab for Lesson 8-7. Underthe triangle, write thePythagorean Theorem.

KEY CONCEPT

HOMEWORKASSIGNMENT

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Find the length of one leg of a right triangle if the length of thehypotenuse is 22 centimeters and the length of the other leg is 15 centimeters. Round to the nearest tenth.

c2� a2


222�

2� b2 Replace c with 22 and

a with 15.

� � b2

484 � 225 � 225 � 225 � b2 Subtract.

� b2

�259� � b Find the square root of each side.

� b Use a calculator.

The length of the leg is about centimeters.

The measures of the three sides of a triangle are 6, 11,and 13. Determine whether this triangle is a righttriangle.

c2� a2


132� 62

� 112 Replace the variables.

� �

169 ≠ 157

Since c2 ≠ a2� b2, the triangle is a right triangle.

a. Find the length of one leg of a right triangle if the length ofthe hypotenuse is 28 inches and the length of the other legis 20 inches. Round to the nearest tenth.

b. The measures of the three sides of a triangle are 15, 18,and 21. Determine whether this triangle is a right triangle.

Your Turn

8–7

Page(s):Exercises:

HOMEWORKASSIGNMENT

Converse of thePythagorean Theorem Ifc is the measure of thelongest side of a triangle

and c2 � a2 � b2, thenthe triangle is a righttriangle.

KEY CONCEPT

22 cm 15 cm

y cm

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To make a crossword puzzle,word search, or jumble puzzle of the vocabulary words inChapter 8, go to


You can use your completedVocabulary Builder(pages 168–169) to help yousolve the puzzle.


C H A P T E R

8STUDY GUIDE

8-1

Powers and Exponents

a. b6

b. b

c. b9

d. b15

e. b8

1. Using 5x, fill in the boxes with the correct terms. Five is the

and x is the .

2. Complete the table.

Write the letter of the correct answer at the right that bestmatches each expression.

3. b3� b5

4. b3� b3

5. b4� b5

� b6

6. �bb

4

3�

8-2

Multiplying and Dividing Powers

22 32 42 52 62 72 82 92 102

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103 102 101 100 10�1 10

�2 10�3

7. Complete the table.

True or False?

8. 5�4 is equal to �51�4�.

Complete each sentence to change from scientific notationto standard notation.

9. To express 3.64 � 106 in standard notation, move the decimal

point places to the .

10. To express 7.825 � 10�3 in standard notation, move the

decimal point places to the .

Complete each sentence to change from standard notationto scientific notation.

11. To express 0.0007865 in scientific notation, move the decimal

point places to the right and write .

12. To express 54,000,000,000 in scientific notation, move the

decimal point places to the left and write .


13. ��49� 14. �121�

15. ��2851�� 16. ��

13060

��

8-3

Negative Exponents

8-4

Scientific Notation

8-5

Square Roots

Complete the chart of the following square roots.

17.

18.

19.

20.

Complete the sentence.

21. The side opposite the right angle of a right triangle is called the

.

22. In a right triangle, each of the two sides that form the right

angle is a of the right triangle.

Write an equation that you could solve to find the missingside length of each right triangle.

23. 24. 25.

9 ft

10 ft6 ft10 ft

9 ft

10 ft


8-7

The Pythagorean Theorem

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8-6

Estimating Square Roots

Square Root �?� � x � �?� Estimate

�45� �36� � �45� � �49� 7

�18�

�88�

�112�

�125�

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C H A P T E R

8Checklist





• You may want to take the Chapter 8 Practice Test on page 377 of your textbook as a final check.












Teacher Signature

Polynomials

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NOTE-TAKING TIP: When you take notes, be sureto write clear and concise notes so that you canstudy them more easily.

Ch

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C H A P T E R

9


FoldFold lengthwise to the holes.

Cut Cut along the top line and then cut three tabs.

Label Label the tabs using the lesson concepts as shown.

Identify and

classify

polynomials

Add and

subtract

polynomials

Multiply

polynomials




C H A P T E R


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on Page Example

binomial [by-NO-mee-ul]

degree

FOIL method

like terms

monomial[ma-NO-mee-ul]

polynomial[PA-lee-NO-mee-ul]

trinomial[try-NO-mee-ul]


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Polynomials9–1

• Identify and classifypolynomials and findtheir degree.

WHAT YOU’LL LEARNA monomial is a number, a , or a

of numbers and variables that have only positive

.

A monomial or the of one or more monomials

is called a polynomial.

A polynomial with terms is a binomial.

A polynomial with terms is a trinomial.


Determine whether each expression is a monomial.Explain why or why not.

a2b3c

a2b3c is a monomial because it is a of variables.

�1x

�

�1x

� is not a monomial because it includes .

Determine whether each expression is amonomial. Explain why or why not.a. �10x2y3z

b. 5x

Your Turn

Write three examples ofexpressions that are notmonomials.

WRITE IT


State whether each expression is a polynomial. If it is apolynomial, identify it as a monomial, binomial, ortrinomial.

�4x � 2

Yes. Since the expression is the sum of monomials,

it is a .

5 � 3x2 � x � 2

Yes. The expression 5 � 3x2� x � 2 can be written as

. Since the expression can be written as

the sum of monomials, it is a .

3x�2 � 4x3

No. Since the expression contains a exponent,it is not a polynomial.

State whether each expression is apolynomial. If it is a polynomial, identify it as amonomial, binomial, or trinomial.a. 9x2

� 2x � 4 b. �3x2� 2x�3

c. a � 3b

Your Turn

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9–1

The degree of a monomial is the of the

of the variables.


Write an example of amonomial, binomial,and trinomial under thetab for Identify andClassify Polynomials.Then write a polynomialwith more than 3 terms.

Identify and

classify

polynomials

Add and

subtract

polynomials

Multiply

polynomials

ORGANIZE IT


Find the degree of each polynomial.

8b4 � 92

The degree of 8b4� 92 is .

2ab � 3a2b � 5a4b2

The degree of 2ab � 3a2b � 5a4b2

is .

Find the degree of each polynomial.a. 3a3b5

� 2ab b. �2x2� x � xy4

Your Turn

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9–1

Term Degree

8b4

92

Page(s):Exercises:

HOMEWORKASSIGNMENT

Term Degree

2ab 1 � 1 or

3a2b 2 � 1 or

5a4b2 4 � 2 or

Describe how amonomial can havedegree zero. Write anexample of a monomialwith degree zero.

WRITE IT


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• Add and subtractpolynomials.

WHAT YOU’LL LEARN Find each sum.

(3s � 4t) � (6s � 2t)

Method 1 Group the like terms together.

(3s � 4t) � (6s � 2t)

� � (4t � 2t) Group the like terms.

� s � t Distributive Property

�

Method 2 Add in column form.

3s � Align the like terms.

(�) � 2t

(b2 � 4b � 6) � (3b2 � 3b � 1)

To add, use either method. Let’s use Method 1. That is, groupthe like terms together.

(b2� 4b � 6) � (3b2

� 3b � 1)

� � (4b � 3b) �

� b2� (4 � 3) �

� b2� 1 �

�

REMEMBER ITTerms that have thesame variable(s) andpower(s) are like terms.


(2d2 � 7de � 8e2) � (�d2 � 8e2)

To add, let’s use Method 2. That is, add in column form.

2d2� 7de �

(�) �d2� 8e2

Find each sum.a. (3x2

� y) � (5x2� 2y)

b. (x2� 5x � 2) � (5x2

� 4x � 5)

c. (3y2� 4yz � 2z2) � (�2y2

� 4yz � 8z2)

Find each difference.

(2g � 7) � (g � 2)

Method 1 Find the additive inverse of g � 2. Then group thelike terms together and add. The additive inverseof g � 2 is �(g � 2) or �g � 2.

(2g � 7) � ( g � 2) � (2g � 7) � Add the additiveinverse.

� (2g � g) � (7 � 2) Group the liketerms.

� g � (7 � 2) DistributiveProperty

�

Method 2 Arrange like terms in column form.

2g � 7

(�) g � 2

2g � 7

Your Turn

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REMEMBER ITSubtracting aninteger is the same asadding its inverse.

(�) � Add the additiveinverse.

List an example of amonomial, binomial,and trinomial under thetab for Add andSubtract Polynomials.Then, name theiradditive inverses.

Identify and

classify

polynomials

Add and

subtract

polynomials

Multiply

polynomials

ORGANIZE IT

2g � 7

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9–2

Pages(s):Exercises:

HOMEWORKASSIGNMENT


(4a2 � 3a � 4) � (a2 � 6a � 1)

The additive inverse of (a2� 6a � 1) is �(a2

� 6a � 1).

or �a2� � .

(4a2 � 3a � 4) � (a2 � 6a � 1)

� (4a2 � 3a � 4) �

� (4a2 � 1a2) � �

� (4 � 1)a2 � (�3 � 6) � (4 � 1)

� or 3a2 � 9a � 3

Find each difference.a. (6a2

� 2) � (a2� 4)

b. (6x2� 2x � 2) � (x2

� 5x � 3)

c. (6x2� x) � (10 � 5x � x2)

Your Turn

REMEMBER ITBefore adding orsubtracting polynomials,be sure that the powersof the variable are indescending order.


Find each product.

x(x � 1)

x(x � 1) � x � x

� �

g(3g2 � 4)

g(3g2� 4) � g � g

� �

�y(2y � 6)

�y(2y � 6) � �y � (�y)

� �

b2(2b2 � 4b � 9)

b2(2b2� 4b � 9) � b2

� b2� b2

� � �

Find each product.a. �3x(x � 4) b. y(5y2

� 2)

c. �a(3a � 4) d. x3(3x3� 2x � 6)

Your Turn©

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Multiplying a Polynomial by a Monomial9–3

• Multiply a polynomialby a monomial.

WHAT YOU’LL LEARN

Multiplying a PolynomialBy a Monomial Tomultiply a polynomial by a monomial, use thedistributive property.

KEY CONCEPT



3(d � 4) � 8 � 5(5d � 1) � 3

3(d � 4) � 8 � 5(5d � 1) � 3

� 12 � 8 � � 5 � 3 DistributiveProperty

� 20 � � 2 Combine liketerms.

3d � 20 � � 25d � 2 � Subtract.

� 20 � 2

�22d � 20 � � 2 � Add to

each side.

�22d �

� Divide.

The solution is .

a(3 � a) � 2 � a(a � 1) � 6a(3 � a) � 2 � a(a � 1) � 6

3a � � 2 � � a � 6 DistributiveProperty

3a � a2� 2 � � a2

� a � 6 � Subtract.

� 2 � �a � 6

3a � 2 � a � �a � 6 � a Add a to eachside.

� 2 � 6

4a � 2 � � 6 � Add to

each side.4a � 8

� Divide.

The solution is .

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9–3

REVIEW ITExplain the procedurefor combiningexponents whenmultiplying like bases.(Lesson 8-2)


Solve each equation.a. 6(a � 3) � 2 � 5(a � 1) � 4 b. x(4 � x) � x(x � 10) � 6

Find the area of the shaded region in simplest form.

Subtract the area of the smaller rectangle from the area of the larger rectangle.

area of larger rectangle: (5x � 5) A � �w

area of smaller rectangle: (2x � 1)

area of shaded region: 4x(5x � 5) �

A � 4x(5x � 5) �

� 4x� � � 4x(5) � x� � � x(1) DistributiveProperty

� � 20x � � x

� Combine liketerms.

The area is square units.

Find the area of the shaded region in simplest form.

Your Turn

Your Turn

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9–3

x + 1

3x + 5

2x

3x

Page(s):Exercises:

HOMEWORKASSIGNMENT

x

2x + 1

5x + 5

4x


Multiplying Binomials

Find the product.

(x � 1)(x � 5)

(x � 1)(x � 5)

� x(x � 5) � (�1) Distributive Property

� x(x) � x � (�1)(x) � (�1) Distributive Property

� x2� � x � 5 Simplify.


Find each product.a. (x � 4)(x � 1) b. (p � 3)(2p � 4)

Find each product.

(d � 2)(d � 8)

F O I L

(d � 2)(d � 8) � (d) � (d)(8) � (2) � (2)(8)

� � 8d � � 16


Your Turn

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• Multiply two binomials.

WHAT YOU’LL LEARN

FOIL Method forMultiplying TwoBinomials To multiplytwo binomials, find thesum of the products of

F the First terms

O the Outer terms

I the Inner terms

L the Last terms

Write whatthe letters in FOIL standfor under the tab forMultiply Polynomials.

KEY CONCEPT


Find the product of a � 2 and a2 � 4.

(a � 2)(a2� 4) � (a)(a2) � (a) � (�2)(a2) � (�2)

� a3� � 2a2

� There are nolike terms.

�

Find each product.

a. ( y � 9)( y � 5) b. (k � 3)(3k � 5)

c. (2x � y)(x � 3y) d. (y � 2)(y2� 5)

Your Turn

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9–4

Page(s):Exercises:

HOMEWORKASSIGNMENT

(e � 4)(2e � 4)

F O I L

(e � 4)(2e � 4) � (e) � (e)(�4) � (4) � (4)(�4)

� � 4e � � 16


(5x � y)(4x � 2y)

F O I L

(5x � y)(4x � 2y) � (5x) � (5x)(�2y) � (y) � (y)(�2y)

� � 10xy � � 2y2



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Find each product.

(b � 5)2

(a � b)2� � 2ab � Square of a Sum

(b � 5)2� � 2(b)(5) � Replace a with

and b with .

�

(c � 3)2

(a � b)2� � 2ab � Square of a Difference

(c � 3)2� � 2(c)(3) � Replace a with

and b with .

�

(2d � 1)2

(a � b)2� � 2ab � Square of a Sum

(2d � 1)2� � 2(2d)(1) � Replace a with 2d

and b with 1.

�

(3e � 3f)2

(a � b)2� � 2ab � Square of a

Difference

(3e � 3f)2� � 2(3e)(3f) �

Replace a with 3eand b with 3f.

�

• Develop and use thepatterns for (a � b)2,(a � b)2, and(a � b)(a � b).

WHAT YOU’LL LEARN

Square of a Sum andSquare of a Difference

(a � b)2 � a2 � 2ab � b2

(a � b)2 � a2 � 2ab � b2

KEY CONCEPTS

9–5


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9–5

Find each product.

(3 � a)(3 � a)

(a � b)(a � b) � � b2 Product of a Sum and aDifference

(3 � a)(3 � a) � � a2 Replace a with and

b with .

�

(5b � 2)(5b � 2)

(a � b)(a � b) � � b2 Product of a Sum and aDifference

(5b � 2)(5b � 2) � � (2)2 Replace with 5b

and b with .

�

Find each product.

a. (y � 3)2 b. (k � 4)2

c. (2x � 5)2 d. (3x � 2y)2

e. (x � 6)(x � 6) f. (4x � 3y)(4x � 3y)

Your Turn

Product of a Sum and aDifference

(a � b)(a � b) � a2 � b2

KEY CONCEPT

Page(s):Exercises:

HOMEWORKASSIGNMENT

Write the three specialproduct models underthe tab for MultiplyPolynomials. Then, writean example of each andinclude a model.

Identify and

classify

polynomials

Add and

subtract

polynomials

Multiply

polynomials

ORGANIZE IT


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Match the expression on the left with the correct term at the right.

1. �5x2y3z � 2x

2. �3yx�

3. 12a2b3c4

4. 2k � 3m � 4mn

Find the degree of each polynomial.

5. 5ab � 2a2b2� 4a2b 6. �6xy3z � xy

7. 12 8. 3a2b � 2a2


9. (�3x3� 4x2

� 5x � 1) � (�5x3� 2x2

� 2x � 7)

10. (5k � 4) � (k � 1)



To make a crossword puzzle,word search, or jumble puzzle of the vocabulary wordsin Chapter 9, go to




9-1

Polynomials

a. monomial

b. binomial

c. trinomial

d. not a polynomial

9-2

Adding and Subtracting Polynomials


C H A P T E R

9STUDY GUIDE


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11. The property is used to multiply a

polynomial by a monomial.

Find each product.

12. 2y2(3y2� 2y � 7) 13. �3x3(x3

� 2x2� 3)

14. Complete. To multiply two binomials, find the sum of theproducts of

F � ,

O � ,

I � , and

L � .

Match each model to its special product name.

15. (a � b)2� a2

� 2ab � b2

16. (a � b)2� a2

� 2ab � b2

17. (a � b)(a � b) � a2� b2


9-3

Multiplying a Polynomial by a Monomial

9-4

Multiplying Binomials

9-5

Special Products

a. square of a sum

b. square of a difference

c. product of a sum and a difference

d. sum of a difference















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Teacher Signature

C H A P T E R

9Checklist



Factoring

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NOTE-TAKING TIP: When you start a new lesson,write down the objectives. Then look for sampleproblems and solutions that illustrate theobjectives.

C H A P T E R

10


FoldFold in half lengthwise.

FoldFold again in thirds.

OpenCut along the second foldto make three tabs.


Begin with a sheet of plain 8 " by 11" paper.1�2

FactoringGreatest

CommonFactor

TheDistributiveProperty

Trinomials

Ch

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C H A P T E R


difference of squares

factoring

greatest common factor(GCF)

perfect squaretrinomial

prime polynomial

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on page Example

Find the factors of each number. Then classify eachnumber as prime or composite.

47

There is only one pair of whole numbers whose product is 47.

1 �

The factors of 47 are and . Therefore, 47 is a

number.

35

To find the factors of 35, list all pairs of whole numbers whoseproduct is 35.

1 � 5 �

The factors of 35 are 1, 5, , and . Since 35 has more

than factors, it is a number.

Find the factors of each number. Thenclassify each number as prime or composite.a. 51 b. 18

Factor each monomial.

16b2c2

16b2c2� 2 � 2 � 2 � � b � � c � c

Your Turn

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10–1


Factors

• Find the greatestcommon factor of a set of numbers ormonomials.

WHAT YOU’LL LEARN

How can you tell thatthe expression listed in Example 3 is amonomial? (Lesson 9-1)

REVIEW IT

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10–1


�15xy2

To factor a negative integer, first express it as the product of awhole number and �1. Then find the prime factorization.

�15xy2� � 15xy2

�15xy2� �1 � 3 � � x � y �

Factor each monomial.

a. �36xy2 b. 6a3b2

Find the GCF of each set of numbers or monomials.

12, 20, and 24

12 � 2 � 2 � 2 � 3 Find the prime factorization of each number.

20 � 2 � 2 � 2 � 3 � 5 Line up as many factors as possible.

24 � 2 � 2 � 2 � 3 Circle the common factors.

The GCF of 12, 20, and 24 is � or .

21ab2 and 9a2b

21ab2� 3 � 3 � 7 � a � a � b � b

9a2b � 3 � 3 � 7 � a � a � b

The GCF of 21ab2 and 9a2b is � � or

.

Find the GCF of each set of numbers ormonomials.

a. 10, 20, and 50 b. 10 and 11 c. 9a2b and 12a3b2

Your Turn

Your Turn

Page(s):Exercises:

HOMEWORKASSIGNMENT

Greatest Common FactorThe greatest commonfactor of two or moreintegers is the productof the prime factorscommon to the integers.

Find thegreatest common factorfor a set of threenumbers and write itunder the tab forGreatest CommonFactor.

KEY CONCEPT

Factor each polynomial.

24y � 18y2

First, find the GCF of 24y and 18y2.

24y � 2 � 2 � 2 � 3 � 3 � y18y2

� 2 � 2 � 2 � 3 � 3 � y � y

The GCF of 24y and 18y2 is . Write each term as a

product of the GCF and its remaining factors.

24y � 18y2� (4) � (3y)

� 6y Distributive Property

18ƒg � 21gh2

First, find the GCF of 18ƒg and 21gh2.

18ƒg � 2 � 3 � 3 � 3 � ƒ � g21gh2

� 2 � 3 � 3 � 7 � g � h � h

The GCF is .

18ƒg � 21gh2� (6ƒ) � (7h2)

� 3g� � Distributive Property

Factor each polynomial.a. 10a2

� 5a b. 12mn2� 15n

Your Turn

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Factoring Using the Distributive Property

• Use the GCF and theDistributive Property tofactor polynomials.

WHAT YOU’LL LEARN

How can you use theDistributive Property tocheck your work?(Lesson 9-3)

REVIEW IT

The process of finding the factors of a product is factoring.

A polynomial that cannot be factored is called a primepolynomial.


Factor 5a � 20ab � 10a2.

5a � 2 � 2 � 5 � a

20ab � 2 � 2 � 5 � a � a � b

10a2� 2 � 2 � 5 � a � a

The GCF is . When 5a is factored from 5a, the

remaining factor is .

5a � 20ab � 10a2� (1) � (4b) � (2a)

� 5a Distributive Property

Factor each polynomial.

a. 6ab � 15ab2� 3a2b2 b. 15c � 11ab2

Divide (24a2 � 20a) by 4a.

(24a2� 20a) � 4a � � Divide each

term by .

� � �21

�

2�

1�

2�

1

�

2�

1�

5� a

�

1�

a1

�� Simplify.

� �

Therefore, (24a2� 20a) � 4a � .

Divide (36ab2� 9a2b) by 9ab.Your Turn

21

� � 21

� � 2 � 3 � a1

� � a��

21� � 2

1� � a�

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

Multiply a monomialand a trinomial togetherunder the tab forDistributive Property.

FactoringGreatest

CommonFactor


Trinomials

ORGANIZE IT 24a2 20a

1 1 1

REMEMBER ITThe product of apositive integer and a negative integer isnegative. The productof two negative integersis positive.

Factor each trinomial.

x2 � x � 12

x2� x � 12 � (x � � )(x � �)

Find integers whose product is and whose sum

is .

You can stop listing factors when you find a pair that works.

Therefore, x2� x � 12 � �x � ��x � �.

x2 � 9x � 8

x2� 9x � 8 � (x � � )(x � �)

Find integers whose product is and whose sum

is .

Therefore, x2� 9x � 8 � �x � ��x � �.

Factoring Trinomials: x2 � bx � c10–3


• Factor trinomials of theform x

2� bx � c.

WHAT YOU’LL LEARN

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Product Integers Sum

�12 �2, 6 �2 � 6 �

�12 2, �6 2 � (�6) �

�12 �3, 4 �3 � 4 �


8 �1, �8 �1 � (�8) �

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Page(s):Exercises:

HOMEWORKASSIGNMENT

REMEMBER ITSquare models canbe used to help find thecorrect combinationwhen factoringtrinomials.

x2 � x � 1

x2� x � 1 � (x � � )(x � � )

Find integers whose product is and whose sum is .

There are no factors of whose sum is .

Therefore, x2� x � 1 is a polynomial.

Factor each trinomial.

a. x2� 14x � 24 b. x2

� 7x � 18

c. x2� 2x � 1

Factor 4x2 � 8x � 60.First, check for a GCF.

4x2� 8x � 60 � (x2

� 2x � 15) The GCF is .

Now, factor x2� 2x � 15.

x2� 2x � 15 � �x � ��x � �

4x2� 8x � 60 �

Factor 5x2� 10x � 15.Your Turn

Your Turn


1 1, 1 1 � 1 �

1 �1, �1 �1 � (�1) �

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Factoring Trinomials: ax2 � bx � c10–4


Factor 3y2 � 7y � 6.

3y2 is the product of the first terms, and �6 is the product ofthe last terms.

3y2� 7y � 6 � (3y � �)(y � �)

The last term, , is negative. The sum of the inside and

outside terms, , is positive. So, one factor must be

and one must be . Try factor

pairs of until the sum of the products of the Outer

and Inner terms is .

Try 2 and �3.

�3y � ��y � � � 3y2� 9y � 2y � 6

� 3y2� 6 is

not the correct middle term.

�3y � ��y � � � 3y2� 6y � 3y � 6

� 3y2� 6 is not

the correct middle term.

Try �2 and 3.

�3y � ��y � � � 3y2� 9y � 2y � 6

� 3y2� � 6

Therefore, 3y2� 7y � 6 � .

• Factor trinomials of theform ax

2� bx � c.

WHAT YOU’LL LEARN

FactoringGreatest

CommonFactor


Trinomials

Choose two binomialsand use FOIL to multiplythem together underthe tab for Trinomials.Then use the factoringtechniques you’velearned to factor theresult back into twobinomials.

ORGANIZE IT

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Factor each trinomial.a. 2x2

� x � 3 b. 5z2� 22z � 8

Factor 4x2 � 4x � 15.

Try 4 and 1.

(4x � 3)(x � 5) � 4x2� � � 15

� 4x2� � 15 is not the

correct middle term.

(4x � 15)(x � 1) � 4x2� � � 15

� 4x2� � 15 is not the

correct middle term.

Try 2 and 2. (2x � 3)(2x � 5) � 4x2� � �15

� 4x2� � 15

Therefore, 4x2� 4x � 15 � .

Factor 4c2� 16c � 7.Your Turn

Your Turn

10–4

Page(s):Exercises:

HOMEWORKASSIGNMENT

Number Factor Pairs

4 4 and 1, 2 and 2�15 3 and �5, �3 and 5, 15 and �1, �15 and 1

Special Factors10–5


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• Recognize and factorthe differences ofsquares and perfectsquare trinomials.

WHAT YOU’LL LEARN Determine whether each trinomial is a perfect squaretrinomial. If so, factor it.

x2 � 14x � 49

To determine whether x2� 14x � 49 is a perfect square

trinomial, answer each question.

• Is the first term a perfect square?

x2 is the square of .

• Is the last term a perfect square?

49 is the square of .

• Is the middle term twice the product of x and ?

14x � (7x).

Therefore, x2� 14x � 49 is a perfect square trinomial.

x2� 14x � 49 � .

9a2 � 16a � 4

• Is the first term a perfect square?

9a2 is the square of .

• Is the last term a perfect square?

4 is the square of .

• Is the middle term twice the product of 3a and ?

2� � � 16a

Therefore, 9a2� 16a � 4 is not a perfect square trinomial.

REMEMBER ITIn a perfect squaretrinomial, the last termis always positive.

Factoring Perfect SquareTrinomials A perfectsquare trinomial is atrinomial that has twoequal binomial factors.For example, x2 � 6x � 9 � (x � 3)(x � 3) andx2 � 6x � 9 � (x � 3)(x � 3).

KEY CONCEPT

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Determine whether each trinomial is aperfect square trinomial. If so, factor it.

a. x2� 10x � 25 b. 4c2

� 20c � 100

c. 9m2� 12m � 4

Determine whether each binomial is the difference ofsquares. If so, factor it.

d2 � 81

d2 and 81 are both perfect and d2� 81 is a

.

d2� 81 � � �2

� � �2� d 2, � 81

� (d � 9)� � Difference of Squares

ƒ2 � 64

ƒ2 and 64 are both perfect squares. But ƒ2� 64 is a ,

not a difference. Therefore ƒ2� 64 is not a difference of

squares. It is a polynomial.

Determine whether each binomial is thedifference of squares. If so, factor it.

a. r2� 100 b. n2

� 4

c. 9x2� 121

Your Turn

Your Turn

10–5

Page(s):Exercises:

HOMEWORKASSIGNMENT

Factoring a Difference ofSquares A binomial thatcan be factored into twobinomials is a differenceof squares. For example,x2 � 9 � (x � 3)(x � 3).

KEY CONCEPT

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To make a crossword puzzle,word search, or jumble puzzle of the vocabulary wordsin Chapter 10, go to




C H A P T E R

10STUDY GUIDE

Find the prime factorization of each number or monomial.

1. 36 2. 14 3. 25a3b

4. 27x 5. 81 6. 42mn2

Refer to the polynomial 5ab � 25b2 � 5b.

7. What is the GCF of this polynomial?

8. Write this polynomial in factored form.

9. Explain how to check your answer.

10-1

Factors

10-2

Factoring Using the Distributive Property

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Chapter BRINGING IT ALL TOGETHER

Tell what sum and product you want m and n to have to usethe pattern (x � m)(x � n) to factor the given trinomial.

10. x2� 10x � 24 sum: product:




Match each trinomial with the correct factored form. If thetrinomial will not factor, choose answer F.

14. x2� 7x � 30

15. x2� 7x � 6

16. x2� 8x � 15

17. x2� 6x � 16

18. x2� 2x � 5

Refer to the trinomial 4x2 � 11x � 6.

19. What are the possibilities for the first term in each binomial?

20. What are the possibilities for the last term in each binomial?

10

10-3

Factoring Trinomials: x2 � bx � c

10-4

Factoring Trinomials: ax2 � bx � c

a. (x � 2)(x � 8)

b. (x � 1)(x � 6)

c. (x � 10)(x � 3)

d. (x � 3)(x � 5)

e. (x � 2)(x � 4)

f. prime polynomial

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Explain why each binomial is a difference of squares.

21. 4x2� 25

22. 49a2� 64b2

Match each polynomial from the first column with a factoringtechnique in the second column. Some of the techniques may be usedmore than once. If none of the techniques can be used to factor thepolynomial, choose prime polynomial.

23. 9x2� 64

24. 9x2� 12x � 4

25. x2� 5x � 6

26. 4x2� 13x � 9

27. x2� 25

28. x2� 4x � 4

29. 2x2� 16


10-5

Special Factors

a. factor as x2� bx � c

b. factor as ax2� bx � c

c. difference of squares

d. perfect square trinomial

e. factor out the GCF

f. prime polynomial


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Teacher Signature

C H A P T E R

10Checklist

Quadratic and Exponential Functions

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Use the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking note.

NOTE-TAKING TIP: When taking notes, listen forkey words your teacher may use to emphasizeimportant concepts.

Ch

apte

r 11

C H A P T E R

11


FoldFold each sheet in halfalong the width.

UnfoldUnfold each sheet and tapeto form one long piece.

LabelLabel each page with thelesson number as shown.

RefoldRefold to form a booklet.Label the front cover“Quadratic and ExponentialFunctions.”

Begin with four sheets of plain paper.

11-2 11-4 11-611-1 11-3 11-5 11-7Quadratic

and

Exponential

Functions

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C H A P T E R




on Page Example

axis of symmetry[SIH-muh-tree]

completing the square

discriminant[dis-KRIMH-uh-nunt]

exponential function[EKS-po-NEN-chul]

geometric sequence[JEE-uh-MET-rik]

initial value

maximum

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on Page Example

minimum

parabola[puh-RA-buh-la]

quadratic equation[kwad-RAT-ik]

Quadratic Formula

quadratic function

roots

vertex[VER-teks]

zeros

Graphing Quadratic Functions

Graph y � x2 � 2 by making a table of values.

First choose values for x. Evaluate the function

for each x-value.

Graph the points and connect them with a

curve.

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11–1

• Graph quadraticfunctions.

WHAT YOU’LL LEARNThe shape of a function is called a parabola.


x x2 � 2 y (x, y)

�2 (�2)2� 2

�1 (�1)2� 2

0 02� 2

1 12� 2

2 22� 2

O x

y

Quadratic Function Aquadratic function isa function that can be described by anequation of the form y � ax2 � bx � c, wherea � 0.

KEY CONCEPT

Graph each quadratic equation by making atable of values.

a. y � �x2� 3 b. y � x2

� 2

Use characteristics of quadratic functions to graphy � �x2

� 2x � 1.

a. Find the equation of the axis of symmetry.

y � ax2� bx � c

y � �x2� 2x � 1 �x2

� �1x2

So, a � , b � , and c � .

Now, find the equation of the axis of symmetry.

x � Equation of axis symmetry

a � , b �

x � Simplify.

b. Find the coordinates of the vertex of the parabola.

Since the equation of the axis of symmetry is ,

the x-coordinate of the vertex must be . Substitute

for x in the equation y � �x2� 2x � 1 to solve for y.

Your Turn

x � �

x � 2� �©

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11–1


Equation of the Axis ofSymmetry The equationof the axis of symmetryfor the graph ofy � ax2 � bx � c, wherea � 0, is x � ��

2ba�.

Draw aparabola under the tabfor Lesson 11-1. Labelthe axis of symmetry andthe vertex.

KEY CONCEPT

y � �x2� 2x � 1

� �2

� 2 � 1

� � � 1 or

The point at is the vertex.

c. Graph the function.Construct a table. Choose some values for x that are less than 1 and some that are greater than 1. This ensures that points on each side of the axis of symmetry are graphed.

Use characteristics of quadratic functionsto graph y � x2 � 4x � 3.a. Find the equation of the axis of symmetry.

b. Find the coordinates of the vertex of the parabola.

c. Graph the function.

Your Turn

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11–1


Page(s):Exercises:

HOMEWORKASSIGNMENT

x �x2 � 2x � 1 y (x, y)

�1 �(�1)2� 2(�1) � 1

0 �(0)2� 2(0) � 1

1 �(1)2� 2(1) � 1

2 �(2)2� 2(2) � 1

3 �(3)2� 2(3) � 1

O x

yREMEMBER ITThe axis ofsymmetry can help yougraph quadraticfunctions. If you knowthe axis of symmetry,additional points canbe found by reflectingacross the axis ofsymmetry.

Graph each group of equations on the same screen.Compare and contrast the graphs. What conclusions can be drawn?

y � �x2, y � �0.5x2, y � �2x2

Each graph opens

and has its vertex at the .

The graph of y � �0.5 x2 is than the graph of

y � �x2.

The graph of y � �2x2 is than the graph

of y � �x2.

The shape of the parabola narrows as the coefficient of �x2

becomes greater. The shape widens as the coefficient of �x2

becomes smaller.

y � �x2, y � �x2� 1, y � �x2

� 4

Each graph opens

and has the same shape as y � �x2

so they form a .

Each parabola has a different located along

the y-axis.

A constant greater than 0 shifts the graph along

the axis of .

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11–2


Families of Quadratic Functions

Families of Parabolas Infamilies of parabolas,graphs either share avertex or an axis ofsymmetry, or both. Also,a family can consist ofparabolas of the sameshape.

Draw anexample of a family ofparabolas that share thesame axis of symmetryunder the tab forLesson 11-2.

KEY CONCEPT

• Learn the characteristicsof families of parabolas.

WHAT YOU’LL LEARN

y � x2, y � (x � 3)2, y � (x � 1)2

Each graph opens and

has the same shape as y � x2.

However, each parabola has a different located

along the .

Find the number for x that results in inside the

parentheses. The graph shifts this number of to the

or .

y � x2, y � (x � 2)2� 1

The graph of y � (x � 2)2� 1 has

the same shape as the graph of

.

However, it shifts to the two units because a

negative will result in inside the parentheses.

It also shifts 1 unit because of the constant

outside the parentheses.

Each group of equations is graphed on thesame coordinate plane. Compare and contrast thegraphs. What conclusions can be drawn?

a. y � x2, y � 0.5x2, y � 3x2

O x

y

Your Turn

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Draw a graph of thelinear equations y � xand y � x � 1. How dothe shifting techniqueslearned in this lessonrelate to yourobservations of thesetwo linear graphs?

WRITE IT

b. y � 2x2, y � 2x2� 3, y � 2x2

� 2.

c. y � �x2, y � �(x � 2)2, y � �(x � 3)2

d. y � x2, y � (x � 1)2� 2

O x

y

x

y42

�2�4�2�6�8 6 842

�4�5�8

�10�12

O

O x

y

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11–2


Page(s):Exercises:

HOMEWORKASSIGNMENT

Find the roots of x2 � 2x � 15 � 0 by graphing therelated function.

Graph the function ƒ(x) � .

Before making a table of values, find the equation of the

of symmetry. This will make selecting

for your table easier.

x � ��2ba� Equation of the axis of symmetry

or a � , b �

The equation of the axis of symmetry is . Now,

make a table using x-values around . Graph each point

on a coordinate plane.

x � �

x � 2� �

Solving Quadratic Equations by Graphing©

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11–3

• Locate the roots ofquadratic equations bygraphing the relatedfunctions.

WHAT YOU’LL LEARN

Draw a parabola andlabel the zeros underthe tab for Lesson 11-3.

ORGANIZE IT

A quadratic equation is an equation of the form

, where a ≠ .

The of a quadratic equation are called the

roots of the equation.

The of a quadratic function are called

zeros.


11-2 11-4 11-611-1 11-3 11-5 11-7Quadratic

and

Exponential

Functions

Standard 21.0 Studentsgraph quadratic functionsand know that their rootsare the x–intercepts. (Key)

Standard 23.0 Studentsapply quadratic equationsto physical problems, suchas the motion of an objectunder the force of gravity.(Key)

x � �

x � 2� �

The zeros of the function appear to

be and .

So, the roots are and .

Find the roots of �x2� 4x � 12 � 0 by graphing

the related function.

Estimate the roots of �x2� 4x � 1 � 0.

Find the equation of the axis of symmetry.

x � ��2ba� Equation of the axis of symmetry

or a � , b �

The equation of the axis of symmetry is . Now, make

a table using x-values around . Graph each point on a

coordinate plane.

Your Turn

f(x)

O x

2

�2�2�1�3�4�5 321

�4�6�8

�10�12�14

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11–3


x x2 � 2x � 15 f(x)

�5 (�5)2� 2(�5) � 15

�3 (�3)2� 2(�3) � 15

�1 (�1)2� 2(�1) � 15

1 (1)2� 2(1) � 15

3 (3)2� 2(3) � 15

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11–3


Page(s):Exercises:

HOMEWORKASSIGNMENT

The x-intercepts of the graph are between 0 and 1 and between 3 and 4. So, one root of the equation

is between and the

other root is between .

Estimate the roots of x2� 3x � 2 � 0.Your Turn

x �x2 � 4x � 1 f(x)

0 �02� 4(0) � 1

1 �12� 4(1) � 1

2 �22� 4(2) � 1

3 �32� 4(3) � 1 2

4 �42� 4(4) � 1 �1

x

y

O

Solve �2x(x � 5) � 0. Check your solution.

If �2x(x � 5) � 0, then � 0 or � 0.

� 0 or � 0

x � or x �

Check: Substitute each value for x in the original equation.

�2x(x � 5) � 0 �2x(x � 5) � 0

�2 � �� 5� � 0 �2� �� 5� � 0

0� � � 0 (0) � 0

� 0 � 0

Solve �3x(x � 4) � 0. Check your solution.

A child throws a ball up in the air. The height h of theball t seconds after it has been thrown is given by theequation h � �16t2

� 8t � 4. Solve 4 � �16t2� 8t � 4 or

8t(�2t � 1) � 0 to find how long it would take the ball toreach the height from which it was thrown.

If 8t(�2t � 1) � 0, then � 0 or � 0.

� 0 or � 0

t � �2t �

t �

Your Turn

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11–4


Solving Quadratic Equations by Factoring

• Solve quadraticequations by factoringand by using the ZeroProduct Property.

WHAT YOU’LL LEARN

Zero Product PropertyFor all numbers a and b,if ab � 0, then a � 0,b � 0, or both a and bequal 0.

Summarizethe Zero ProductProperty in your ownwords under the tab forLesson 11-4.

KEY CONCEPT

The solutions are and . The solution 0 represents

the beginning of the throw. So, the ball would reach the height

from which it was thrown after of a second.

A child throws a ball up in the air. The height hof the ball t seconds after it has been thrown is given by theequation h � �16t2

� 4t � 3. Solve 3 � �16t2� 4t � 3 or

4t(�4t � 1) � 0 to find how long it would take the ball toreach the height from which it was thrown.

Solve x2 � 4x � 21 � 0. Check your solution.

x2� 4x � 21 � 0

�x � ��x � � � 0 Factor

� 0 or � 0 Zero ProductProperty

x � or x �

Check: x2� 4x � 21 � 0 x2

� 4x � 21 � 0

� �2� 4� � � 21 � 0 2

� 4� � � 21 � 0

� � 21 � 0 � � 21 � 0

� 0 � 0

Solve x2� 11x � 30 � 0. Check your solution.Your Turn

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11–4


Page(s):Exercises:

HOMEWORKASSIGNMENT

Find the value of c that makes x2 � 8x � c a perfectsquare.

Step 1 Find half of �8.

� �4

Step 2 Square the result of Step 1.

(�4)2�

Step 3 Add the result of Step 2 to x2� 8x.

x2� 8x �

Thus, c � .

Notice that x2� 8x � 16 � � �

2

.

Find the value of c that makes x2� 12x � c

a perfect square.

Your Turn©

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-Hill

11–5


Solving Quadratic Equations by Completing the Square

A method to make any expression a

square is called completing the square.


2

• Solve quadraticequations bycompleting the square

WHAT YOU’LL LEARN

Create your ownquadratic equation(choose any integersfor a, b, and c) andwrite it under the tabfor Lesson 11-5. Solvethe equation bycompleting the square.

ORGANIZE IT

REMEMBER ITWhen the squareroot is taken on bothsides of an equation,it is necessary to add a� sign.

11-2 11-4 11-611-1 11-3 11-5 11-7Quadratic

and

Exponential

Functions

Solve x2 � 4x � 5 � 0 by completing the square.

x2� 4x � 5 � 0 x 2

� 4x � 5 is not a perfectsquare.

x2� 4x � Add to each side.

x2� 4x � � 5 � Since � �

2

� , add

to each side.

�x � �2� Factor x2 � 4x � 4.

�x � �2� �222 Take the square root of each

side.

x � � �

x � 2 � � �3 � Subtract from each

side.

x � or x � Simplify each equation.

x � or x � Check the solution.

The solutions are and .

Solve x2� 4x � 21 � 0 by completing the

square.

Your Turn

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11–5


Page(s):Exercises:

HOMEWORKASSIGNMENT

Use the Quadratic Formula to solve each equation.

2x2 � 5x � 3 � 0

x � , a � , b � , and c �

x ��

x�

�

�

5 � or

5 �

x �

5 �or x �

5 �

4 4

x � or x � or

�x2� 6x � 9 � 0

x � , a � , b � , and c �

x�

� � � �

x

�

�

x�

�6 �� or

�2 �2

�b��b2� 4�ac�

��2a

�b ��b2� 4�ac�

��2a

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11–6


The Quadratic Formula

• Solve quadraticequations by using theQuadratic Formula.

WHAT YOU’LL LEARN


For ax2 � bx � c � 0,

x � ,

a � 0.

�b��b2 ��4ac��

2a

Write theQuadratic Formula underthe tab for Lesson 11-6.Use the formula to solvethe quadratic equationthat you created underthe tab for Lesson 11-5.

KEY CONCEPT

2� �

4 4

� �2� 4� ��

2� �� 2

� 4� ��

�

4 4

�

Standard 19.0 Studentsknow the quadraticformula and are familiarwith its proof bycompleting the square.(Key)

Standard 20.0 Studentsuse the quadratic formulato find the roots of asecond-degree polynomialand to solve quadraticequations. (Key)

Use the Quadratic Formula to solve eachequation.a. 2x2

� 3x � 2 � 0 b. 6n2� 7n � 3 � 0

“Hang time” is the total amount of time a ball stays inthe air. A punter kicks the football with an upwardvelocity of 58 ft/s and his foot meets the ball 1 foot offthe ground. His formula is h(t) � �16t2 � 58t � 1, whereh(t) is the ball’s height for any time t after the ball waskicked. What is the hang time?

t � , a � , b � , and c �

t

��

t

�

�58 �

t ��58 �

or t ��58 �

t � or �

Since time cannot be negative, the only approximate

solution is . The football has a hang time of

about .

A baseball player hits a baseball with anupward velocity of 50 feet per second from a height of 3 feet.The equation y � �16t2

� 50t � 3 gives the height y of theball after t seconds. How long will it take the ball to be 5 feetabove the ground on the way down?

Your Turn

�b ��b2� 4�ac�

��2a

Your Turn©

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11–6


Page(s):Exercises:

HOMEWORKASSIGNMENT

REMEMBER ITThe square root ofa negative number isnot a real number. Thismeans that there is noreal solution for x.

2(�16)

�32 �32

� �2� 4� �(1)

�

Graph y � 1.5x.

Graph y � 3x.Your Turn©

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11–7


Exponential Functions

A function in which the is a variable is an

exponential function.

The initial value of an exponential function is the value of

the function when .


x 1.5x y

�1 1.5�1

0 1.50

1 1.51

2 1.52

3 1.53

O x

y

• Graph exponentialfunctions.

WHAT YOU’LL LEARN

Sketch an exponentialfunction under the tabfor Lesson 11-7. Labelthe initial value.

ORGANIZE IT

11-2 11-4 11-611-1 11-3 11-5 11-7Quadratic

and

Exponential

Functions

Graph the exponential function. Then state they-intercept.

y � 3x � 1

To find the y-intercept, let x � and solve for y.

y � 30� 1 or

So, the y-intercept is .

Graph each exponential function. Thenstate the y-intercept.a. y � 2x

� 1 b. y � 3x� 2

Your Turn

x 3x� 1 y

�1 3�1� 1

0 30� 1

1 31� 1

2 32� 1

3 33� 1

O x

y

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11–7


Page(s):Exercises:

HOMEWORKASSIGNMENT



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Complete each statement about the graph at the right.

1. The graph is a curve called a .

2. The highest point of the graph is located

at .

3. This point is the (maximum/minimum) point

of the graph.

4. If you fold a parabola along a line to get two halves that match

exactly, the line where you fold the parabola is the

of the parabola. This line goes through the of the parabola.

Refer to these parabolas that were graphed on a calculator.

5. Do the parabolas graphed form afamily of parabolas? Explain whyor why not.

y � x 2

y � (x � 4)2

y � (x � 3)2

x

y

O







C H A P T E R

11STUDY GUIDE

11-1

Graphing Quadratic Functions

11-2

Families of Quadratic Functions


Glencoe/M

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-Hill

The graphs of two functions are shown. Use the graphs to providethe requested information about the related quadratic equations.

a. ƒ(x) � x2� 2x � 3 b. ƒ(x) � x2

� x � 2

6. For Graph a, the related quadratic equation is .

7. How many solutions are there?

8. Name any solutions.

9. For Graph b, the related quadratic equation is .

10. How many solutions are there?

11. Name any solutions.

Provide a reason for each step in the solution of the equation.

2x2 � 8x � 42

12. 2x2� 8x � 42 � 0

13. 2(x2� 4x � 21) � 0

14. 2(x � 3)(x � 7) � 0

15. x � 3 � 0 or x � 7 � 0

16. x � 3 or x � �7

y

xO

y

xO

11


11-3

Solving Quadratic Equations by Graphing

11-4

Solving Quadratic Equations by Factoring

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Find the value of c that makes each trinomial a square.

17. x2� 2x � c 18. x2

� 9x � c

19. Solve x2� 6x � 72 � 0 by

completing the square.

20. Solve 12x2� 7x � 15

using the Quadratic Formula. Round to the nearest hundredth.

21. Find the length and widthof the rectangle shown.

Match the correct equation with each graph shown.

22. 23. 24.

x

y

x

y

Ox

y

O

w � 3 in.

w � 2 in.

Area � 84 in.2



11-5

Solving Quadratic Equations by Completing the Square

11-6


11-7

Exponential Functions

a. y � 3x b. y � 4�x c. y � 3x� 3 d. y � �4x

Inequalities

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NOTE-TAKING TIP: When you take notes, definenew terms and write about the new concepts inyour own words. Write your own examples thatuse the new terms and concepts.

C H A P T E R

12


Ch

apte

r 12

FoldFold each sheet in halffrom top to bottom.

CutCut along fold. Staplethe eight half-sheetstogether to form a booklet.



12–1Inequalities

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C H A P T E R




on Page Example

boundary

compound inequality

half-plane

intersection

quadratic inequalities

set-builder notation

union


Many movie theaters give a senior-citizen discount topeople who are 65 or over. Write an inequality thatdescribes those who are eligible to receive the discount.Let a represent the ages of people who are eligible to receive the discount.The ages of all those eligible are greater than or equal to 65 years.

is the same as 65 � a.

Then a is 65.

In Colorado, the speed limit on an interstatehighway is 75 miles per hour. Write an inequality thatdescribes the speed cars are allowed to travel.

Graph each inequality on a number line.

x ∞ �1

Since x can equal , graph a at and

shade to the .

k � �34

�

Since x cannot equal , graph a at and

shade to the .

11–4

1–2

3–4

0

0 1–2 –1–3

Your Turn

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12–1 Inequalities and Their Graphs


• Graph inequalities on anumber line.

WHAT YOU’LL LEARN

REMEMBER ITA bullet, or closedcircle, is used when theinequality includes theendpoint. An open circleis used when theinequality does notinclude the endpoint.

GLE 14. Graph and interpret linear inequalities in one or two variables and systemsof linear inequalities (A-2-H, A-4-H)

Graph each inequality on a number line.a. b � �1.8 b. w � 5

Write an inequality for the graph.

Locate where the graph begins. This graph begins at ,

and 2 is included. Also note that the arrow points

to the . The graph describes values that are

2. So, .

Write an inequality for the graph.

Locate where the graph begins. This graph begins at

, and ��23

� is not included. Note that the

arrow points to the . The graph describes

values that are . So, .

Write an inequality for each graph.

a. b.0– – – 1–4

1–41–2

3–4–6 –5–8 –7–9

Your Turn

– 0–1 – – 1–3

2–3

4–3

3 41 20

Your Turn

12–1


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Summarize the meaningof the following signs:�, �, �, and � underthe tab for Lesson 12–1.

12–1Inequalities

ORGANIZE IT

No more than 12 students can fit in a large school van.Write an inequality to express this information. Thengraph the inequality on a number line.

Let s represent the who can fit in

a large school van. Then write an inequality using the

symbol since the number

12 students.

To graph the inequality, first graph a

at . Then include the numbers than

by drawing and shading an to the .

A classroom can seat no more than 30 people.Write an inequality to represent this situation. Then graph theinequality on a number line.

Your Turn

13 1411 1210

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12–1


Page(s):Exercises:

HOMEWORKASSIGNMENT


Solve y � 5 � �2. Check your solution.y � 5 � �2

y � 5 � � �2 � Add to each side.

y �

Check: Substitute a number less than , the

number , and a number greater than into

the inequality.

The solution is {all numbers }.

Solve r � 12 � �2. Check your solution.Your Turn

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12–2

• Solve inequalitiesinvolving addition andsubtraction.

WHAT YOU’LL LEARN

Addition andSubtraction Propertiesfor Inequalities For anyinequality, if the samequantity is added orsubtracted to each side,the resulting inequalityis true.

KEY CONCEPT

Let x � 0. Let x � 3. Let x � 5.

y � 5 � �2 y � 5 � �2 y � 5 � �2

� 5 / �2 � 5 / �2 � 5 / �2

/ �2 / �2 / �2

Set-builder notation is a method of writing the solution

set for an ; {x � x � 3}.



Suppose that you plan to run at least 7.5 miles per week during the summer to train for the cross-countryseason in the fall. During one week, you run 2 miles on Sunday and 2.5 miles on Wednesday. Solve 2 � 2.5 �m § 7.5 to find out how many more miles need to be runbefore the week is over.

2 � 2.5 � m � 7.5

� m � 7.5

4.5 � m � � 7.5 � Subtract from

each side.

m �

The solution can be written as {m � }. So, at least

more miles need to be run before the week is over.

Suppose that your teacher recommends that youread 3 hours each week. During one week, you read 35 minuteson Monday and 25 minutes on Tuesday. Solve 35 � 25 � m � 180to find how many more minutes m you must read this week foryour weekly total to be at least 3 hours.

Your Turn

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12–2


Page(s):Exercises:

HOMEWORKASSIGNMENT

Write the solution setfor Example 3 usingwords under the tab forLesson 12-2.

12–1Inequalities

ORGANIZE IT

REMEMBER ITThe vertical line, �,used in set buildernotation can be read“such that”.

Solving Multiplication and Division Inequalities

An electric car that needs to be recharged every 260 miles should travel no more than 130 miles from thecharger. If you drive at an average speed of 50 miles perhour, what are the lengths of time t you can drive awayfrom the charger and then still make it back withoutrunning out of energy?

Recall that rt � d.

t �


t � The inequality symbol is facing thesame direction.

You should travel no more than hours or 2 hours and

minutes away from the charger.

Two friends walk at least 3 miles every day. Ifthey walk 2 miles per hour, how long do they walk every day?

Solve �2.5x � 10. Check your solution.

�2.5x � 10

� Divide each side by and

reverse the inequality symbol.

x �

Your Turn

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• Solve inequalitiesinvolving multiplicationand division.

WHAT YOU’LL LEARN

Division Propertyfor Inequalities If youdivide each side of aninequality by a positivenumber, the inequalityremains true. If youdivide each side of aninequality by a negativenumber, the inequalitysymbol must by reversedfor the inequality toremain true.

KEY CONCEPT

50t 130

�2.5x 10


12–3

Solve ��12

�x � �5. Check your solution.

��12

�x � �5

(��12

�x) � (�5) Multiply each side by �2 and

reverse the inequality symbol.

x �

Check: Substitute and a number less than ,

such as 8, into the original inequality.

Let x � 10. Let x � 8.

��12

�x � �5 ��12

�x � �5

��12

� ( ) 2 �5 ��12

� ( ) 2 �5

� �5 � �5

The solution is {x � }.

Solve each inequality. Check your solution.

a. �3r � 21 b. ��43

�y � 12

Your Turn

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12–3


Page(s):Exercises:

HOMEWORKASSIGNMENT

Check: Substitute and a number less than ,

such as �5, into the original inequality.

Let x � �4. Let x � �5.

�2.5x � 10 �2.5x � 10

�2.5( ) 0 10 �2.5( ) 0 10

� 10 � 10

The solution set is {x � }.

Multiplication Propertyfor Inequalities If youmultiply each side of aninequality by a positivenumber, the inequalityremains true. If youmultiply each side of aninequality by a negativenumber, the inequalitysymbol must be reversedfor the inequality toremain true.

Write theMultiplication andDivision Properties forInequalities under thetab for Lesson 12–3.

KEY CONCEPT

Solving Multi-Step Inequalities

Solve 5x � 9 � 21. Check your solution.

5x � 9 � 21

5x � 9 � � 21 � Add to each side.

5x �

�55x� � �

350� Divide each side by 5.

x �

Check: Substitute 6 and 7 into the original inequality.

5x � 9 � 21 5x � 9 � 21

5(6) � 9 0 21 5(7) � 9 0 21

30 � 9 0 21 35 � 9 0 21

� 21 false � 21 true


Solve 16 � 2x � 3x � 1. Check your solution.

16 � 2x � 3x � 1

16 � 2x � � 3x � 1 �

16 � 5x � 1

16 � 5x � � 1 �

�5x � �15

��

55x

� � ��155

� Divide and reverse theinequality symbol.

x �

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12–4

• Solve inequalitiesinvolving more thanone operation.

WHAT YOU’LL LEARN

Write and solve aninequality requiringmore than one step tosolve under the tab forLesson 12–4. Then, namethe operations used.

12–1Inequalities

ORGANIZE IT


Check: Substitute 3 and 4 into the original inequality.Let x � 3. Let x � 4.

16 � 2x � 3x � 1 16 � 2x � 3x � 1

16 � 2(3) 1 3(3) � 1 16 � 2(4) 1 3(4) � 1

10 � true 8 � true


Solve 3(x � 2) � �75. Check your solution.3(x � 2) � �75

3x � � �75 Distributive Property

3x � 6 � � �75 � Subtract from each side.

3x �


x �

The solution is {x � }.Check your solution.

Solve each inequality. Check your solution.a. �3t � 2 � �4 b. 10 � 3x � 5x � 4

c. 15 � �5(x � 6)

Your Turn

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12–4


3x �81

Karl’s point totals in the first four of five basketballgames were 15, 12, 19, and 18. How many points t musthe score in the fifth game to have a mean point total ofmore than 16?

The sum of Karl’s points, divided by , will give the

mean score. The mean must be more than .

�

( ) � (16)

15 � 12 � 19 � 18 � t �

� t � 80

64 � t � � 80 �

t �

Karl must score more than points in the fifth game to

have a mean point total of more than .

Lien’s score on the first four of five 100-pointtests were 82, 85, 95, and 91. What score s on the fifth test willgive her a mean score of at least 90 for all five tests?

Your Turn

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12–4


Page(s):Exercises:

HOMEWORKASSIGNMENT

15 � 12 � 19 � 18 � t

15 � 12 � 19 � 18 � t Multiply each

side by .

Subtract.

Write x � 0 and x � �3 as a compound inequalitywithout using and.

x � 0 and x � �3 can be written as � x � or

� x � .

Write x � �2 and x � 5 as a compoundinequality without using and.

A veterinarian has a scale for weighing dogs and catsthat weigh more than 10 pounds but no more than 65 pounds. The weights w that can be measured on thisscale can be written as 10 � w � 65. Graph the solutionof this inequality.

Rewrite the compound inequality using . 10 � w � 65

is the same as w � 10 and .

Your Turn

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12–5



Two or more inequalities that are connected by the

words or form a compound inequality.

An intersection is the set of elements common

to inequalities.

A union is the set of elements in each of

inequalities.


Summarize thedifference between“intersection” and“union” under the tabfor Lesson 12–5. Then,give examples of whento use each one.

12–1Inequalities

ORGANIZE IT

• Solve compoundinequalities.

WHAT YOU’LL LEARN


Solve �6 � x � 3 � �1. Graph the solution.STEP 1 Rewrite the compound inequality using and.

�6 � x � 3 � �1

� �6 and � �1

STEP 2 Solve each inequality.

x � 3 � �6 and x � 3 � �1

x � 3 � � �6 � x � 3 � � �1 �

x � x �

STEP 3 Rewrite the inequality as � x � .


Solve 4 � �2x � 12. Graph the solution.Your Turn

2 3 4–1–2–3–4 0 1

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12–5


STEP 1 Graph w � .

STEP 2 Graph w � .

STEP 3 Find their .

The solution is {w � � w � }.

An amusement park charges $15 admission forchildren between the ages of 5 and 13. The age a of peoplecharged $15 for admission can be written as 5 � a � 13.Graph the solution of this inequality.

Your Turn

30 40 50 60 700 10 20

30 40 50 60 700 10 20

30 40 50 60 700 10 20

Graph the solution of x � 7 or x � 3.

STEP 1 Graph x � .


STEP 3 Find the union of the graphs.

Graph the solution of x � �3 or x � 2.

Solve �23

�x � 4 or �5x � 20. Graph the solution.

�23

�x � 4 or �5x � 20

(�23

�x) � (4) �

x � x �

Now graph the solution.


STEP 2 Graph x � �4.

STEP 3 Find the union of the graphs.

The last graph shows the solution {x � }.

Solve �14

�x � 3 or �3x � �48. Graph the solution.Your Turn

Your Turn

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12–5


Page(s):Exercises:

HOMEWORKASSIGNMENT

�5x 20

2 4 6–2 0–4

2 4 6–2 0–4

2 4 6–2 0–4

6 7 84 532

6 7 84 532

6 7 84 532

Solving Inequalities Involving Absolute Value

Solve � x � 3� � 4. Graph the solution.CASE 1 CASE 2

x � 3 is positive. x � 3 is negative.

x � 3 � �(x � 3) �

x � 3 � � 4 � x � 3 �

x � x � 3 � � �4 �

x �

The solution is {x � � x � }.

The solution makes sense since and are at most

units from .

Solve � t � 3� � 10. Graph the solution.

Solve �4x� � 16. Graph the solution.CASE 1 CASE 2

4x is positive. 4x is negative.

4x � �4x �

�44x� � �

146� 4x �

x � �44x� � �

�416�

x �

The solution is {xx � or x � }.

Your Turn

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12–6

• Solve inequalitiesinvolving absolutevalue.

WHAT YOU’LL LEARN

–4 –3 –2 –1 0 1–7 –6 –5

2 3 4–1–2–3–4 0 1

Summarize thedifference between thesolution of � x � � 4 and� x � � 4 under the tabfor Lesson 12–6.

12–1Inequalities

ORGANIZE IT


Solve �5x� � 15. Graph the solution.

A lumber company makes 3�14

�-foot railing posts to use in making decks. The tolerance for the posts is 0.025 feet.What is the range of acceptable post lengths?

Let p � the actual measure of the posts.

Then, �p � 3�14

�� .

� p � 3.25� � 0.025 Write �14

� as 0.25.

CASE 1

p � 3.25 is positive.

p � 3.25 �

p � 3.25 � � 0.025 �

p �

CASE 2

p � 3.25 is negative.

�(p � 3.25) �

p � 3.25 �

p � 3.25 � � �0.025 �

p �

The solution is {p � � p � }.

A company makes 1�12

�-inch washers. The tolerance for the washers is 0.05 inch. What is the range of acceptable washer sizes?

Your Turn

Your Turn

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12–6


Page(s):Exercises:

HOMEWORKASSIGNMENT

REMEMBER ITThe word “and”refers to an intersectionof solutions and theword “or” refers to aunion of solutions.

Graph y � x � 2.

STEP 1 Determine the boundary by graphing the

related equation, y � x � 2.

STEP 2 Draw a line since

the boundary is included.

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Graphing Inequalities in Two Variables

A line that the coordinate plane into

half-planes is a boundary.

The region of the graph of an inequality on side

of a boundary is a half-plane.


• Graph inequalities onthe coordinate plane.

WHAT YOU’LL LEARN

Describe the term “half-plane” using the words“half” and “plane”.

WRITE IT

x x � 2 y

�2 �2 � 2

�1 �1 � 2

0 0 � 2

1 1 � 2

2 2 � 2

O x

y


Graph y � �x � 2.

Graph �3x � y � 1.

To make a table or graph for the line, solve the

inequality for y in terms of .

�3x � y � 1

�3x � y � � 1 � Add to each side.

y � � 1 Rewrite as

3x � 1.

Your Turn

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Explain when to use adashed line and when touse a solid line whengraphing inequalities intwo variables under thetab for Lesson 12–7.Then explain how todetermine which half-plane should be shaded.

12–1Inequalities

ORGANIZE IT

STEP 3 Test any point to find which half-plane is thesolution. Use (0, 0) since it is the easiest point to usein calculations.

y � x � 2

1 � 2 x � , y �

0 � false

Since (0, 0) does not result in a

inequality, the half-plane

containing (0, 0) is not the solution. O x

y

STEP 1 Determine the boundry by graphing the relatedequation, y � 3x � 1

STEP 2 Draw a line

since the boundary is not included.

STEP 3 Test (0, 0) to find which half-plane contains the solution.

y � 3x � 1

� 3( ) � 1 x � , y �

0 � 1 false

Since (0, 0) does not result in a true inequality, the half-plane containing (0, 0) is not the solution. Thus, shade the other half-plane.

Graph y � 2x � 1.Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

O x

y

x 3x � 1 y

�2 3(�2) � 1

�1 3(�1) � 1

0 3(0) � 1

1 3(1) � 1

2 3(2) � 1

O x

y

12-2



C H A P T E R

12STUDY GUIDE


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12-1

Inequalities and Their Graphs

Write the letter of the graph that matches each inequality.

1. x � �1 a.

2. x � �1 b.

3. x � �1 c.

4. x � �1 d.

Write an inequality for each statement. Then solve.

5. A number subtracted from 16. A number added to 12 is a 21 is no less than �2. minimum of �1.

7. 5 more than a number is 18. 18 less than a number isat least 15. at most 45.

1 2 3�1 0�2�3

1 2 3�1 0�2�3

1 2 3�1 0�2�3

1 2 3�1 0�2�3


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12



19. 12 � 6n

10. ��t3� � 14

11. 12x � 32


12. 2x � 8 � �16

13. 3y � 5 � 16

14. n � 3(n � 1) � 1

Write the letter of the graph that matches each compound inequality.

15. x � 3 or x � �3 a.

16. �3 � x � 3 b.

17. x � �3 or x � 3 c.

Solve each compound inequality. Graph the solution.

18. �13 � 2x � 1 � �5

19. 3a � 21 or �2a � �24

1 2 3�1 0�2�3

1 2 3�1 0�2�3

1 2 3�1 0�2�3

12-3

Solving Multiplication and Division Inequalities

12-4

Solving Multi-Step Inequalities

12-5


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Solve each inequality. Graph the solution.

20. �2x � 2� � 8

2x � 2 � 8

2x � 2 � �8

21. �x � 5� � 4

x � 5 � 4

x � 5 � �4

22. �2x � 3� � 5

2x � 3 � 5

2x � 3 � �5

23. �2x � 2� � 6

2x � 2 � 6

2x � 2 � �6

24. BUSINESS A small business is charged 5 cents per minute forin-state calls and 10 cents per minute for out of state calls.The inequality 5x � 10y � 2500 represents how many minutesof each type of call can be made for under $25 per month.Graph the inequality. List three solutions of the inequality.



12-6

Solving Inequalities Involving Absolute Value

12-7

Graphing Inequalities in Two Variables

Systems of Equations and Inequalities

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NOTE-TAKING TIP: When taking notes, thinkabout the order in which concepts are beingpresented. Write why you think the concepts were presented in this sequence.

Ch

apte

r 13

C H A P T E R

13


Stack Stack sheets of paper with edges four grids apart to create tabs.

FoldFold up bottom edges.All tabs should be thesame size.

Staple Staple along the fold.

LabelLabel the tabs using lessonnumbers and titles.




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C H A P T E R




on Page Example

augmented matrix

consistent[kun-SIS-tunt]

dependent

digit problems

elimination[ee-LIM-in-AY-shun]

identity matrix


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on Page Example

inconsistent[in-kun-SIS-tunt]

independent

matrices[MAY-tra-seez]

quadratic-linear system of equations

row operations

substitution[SUB-sti-TOO-shun]

system of equations

system of inequalities

Graphing Systems of Equations©

Glencoe/M

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13–1

• Solve systems ofequations by graphing.

WHAT YOU’LL LEARN Solve each system of equations by graphing.

y � x � 6y � 3x

The graphs appear to

intersect at .

You can check this estimate by substituting the coordinates into each equation. The solution of the system of equations

is .

x � y � 3x � y � 1

The graphs appear to

intersect at .

Check this estimate.The solution of the system of

equations is .

Solve each system of equations bygraphing.a. y � x b. y � x � 4

x � y � 6 y � x � 6

Your Turn

y = x + 6y = 3x

(3, 9)

O x

y

System of Equations A system of equations is a set of two or moreequations with the samevariables. The solution is the ordered pair thatsatisfies all of theequations.

KEY CONCEPT

GLE 16. Interpret and solve systems of linear equations using graphing, substitution,elimination, with and without technology, and matrices using technology (A-4-H)

x – y = 3

x + y = 1

(2, –1)(2, –1)x

y

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Page(s):Exercises:

HOMEWORKASSIGNMENT

The Math Club is selling T-shirts for a profit of $4 eachand caps for a profit of $5 each. The club wants to sell50 items and make a profit of $240. How many of eachitem should the club try to sell?

Let x � the number of T-shirts and y � the number of caps.You can write two equations to represent this situation.

x � y � the number of items

4x � 5y � the total profit

Graph x � y � and 4x � 5y � . The graphs

appear to intersect at ( , 40). Check this estimate.

x � y � 50 4x � 5y � 240

� � 50 4( ) � 5( ) � 240

� 50 ✓ � � 240

� 240 ✓

They should try to sell T-shirts and caps.

A service organization is selling flowers for aprofit of $2 each and vegetable plants for a profit of $3 each.The club wants to sell 100 items and make a profit of $230.How many of each item should the organization try to sell?

Your Turn

x + y = 50

4x + 5y = 240

(10, 40)

50

40

30

20

10

10 20 30 40 50

Tell what a system ofequations is under thetab for Lesson 13-1.Then describe thesolution to a system ofequations.



ORGANIZE IT

Solutions of Systems of Equations©

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13–2

• Determine whether a system of equationshas one solution, nosolution, or infinitelymany solutions bygraphing.

WHAT YOU’LL LEARN

How do you know if twolines are parallel withoutdrawing their graphs?How do you know,without graphing, whentwo equations representthe same line?

WRITE IT

A system of equations with solution

is consistent.

A system of equations with solutions

solutions is dependent.

A system of equations with

is inconsistent.

A system of equations with

solution is independent.


State whether each system is consistent andindependent, consistent and dependent, or inconsistent.

Both equations have the same graph. Because any ordered pair on the graph will satisfy both equations,

there are

solutions. The system is

and

.

The graphs appear to intersect at

the point at . Because there

is solution, this system of

equations is and

.

O x

y

y = 3

y = 1–2x – 1

x – 2 = yy = x – 2

O x

y


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State whether each system is consistentand independent, consistent and dependent, orinconsistent.a. b.

Determine whether the system of equations has onesolution, no solution, or infinitely many solutions bygraphing. If the system has one solution, name it.

3x � y � 16x � 2y � 2

Write each equation in slope-intercept form.

3x � y � 1 y �

6x � 2y � 2 y �

The graphs have the same and

the same . The system

of equations has

solutions.

y � �xy � x � 2


. Therefore, this system of

equations has one solution .

Remember to check by substituting the values for x and y into each of the originalequations.

y = x – 2

y = –x

(1, –1)O x

y

y = 3x – 1

6x – 2y = 2x

y

O

y

xO

y � �2x � 4y � �2x � 4

y � �2x y � �2x

y

xO

y � x � 1

y � �x � 3

Your Turn

Describe the possiblegraphs of a system oflinear equations underthe tab for Lesson 13-2.Then name the type ofsystem that each graphrepresents.



ORGANIZE IT

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Page(s):Exercises:

HOMEWORKASSIGNMENT

Determine whether the system ofequations has one solution, no solution, or infinitelymany solutions by graphing. If the system has onesolution, name it.a. y � 3x � 1 b. 2x � 3y � 1

y � 2 4x � 6y � 5

The system of equations 4x � 2y � 6 and 8x � 4y � 20represents the tracks of two trains. Do the tracksintersect, run parallel, or are the trains running on thesame track? Explain.

Write each equation in slope-intercept form.

4x � 2y � 6 y �

8x � 4y � 20 y �

The graphs have the same

and

y-intercepts, so the lines are

and the tracks do

not intersect.

The system of equations 3x � y � 2 and 9x � 3y � 6 represents the tracks of two trains. Do the tracks intersect, run parallel, or are the trains on the same track? Explain.

Your Turn

y

xO

Your Turn

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Substitution

Substitution is an algebraic method to solve a of equations.


• Solve systems ofequations by thesubstitution method.

WHAT YOU’LL LEARN

Summarize thesubstitution method and explain when it is preferred over thegraphing method underthe tab for Lesson 13-3.



ORGANIZE IT

Use substitution to solve x � 2y and 2x � 3y � 5.

The first equation tells you that x is equal to 2y. So, substitute2y for x in the second equation. Then solve for y.

2x � 3y � 5

2( ) � 3y � 5 Replace x with .

� 3y � 5

� 5

Now substitute 5 for y in either equation and solve for x.Choose the equation that is easier to solve.

x � 2y

x � 2( ) or Replace y with .

The solution of this system of equations is .

Use substitution to solve y � 3x � 1 and x � y � �3.

Your Turn

y

xO

�4�6�8

2 4 6 8�6�8 �4�2

8642


Use substitution to solve �2x � y � �7 and 3x � 2y � 12.

Solve the first equation for y since the coefficient of y is .

�2x � y � �7 y � 2x �7

Next, find the value of x by Now substitute for

substituting 2x �7 for y in the x in either equation and second equation. solve for y.

3x � 2y � 12 �2( ) � y � �7

3x � 2( ) � 12 � y � �7

3x � � � 12 �4 � y � � �7 �

� 14 � 12 y �

�x � 14 � � 12 �

�x �

�

x �

The solution is .

Use substitution to solve each system ofequations.a. x � 2y � 0 b. 2x � y � 3

3x � 2y � 12 x � 2y � 6

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�x �2

Use substitution to solve each system of equations.

3x � 1 � y9x � 3y � �3

Find the value of x by substituting for y in the

second equation.

9 � 3y � �3

9x � 3( ) � �3 Replace y with .

9x � � � �3 Distributive Property

� �3

The statement �3 � �3 is . This means that an

ordered pair for any point on either line is a solution to both

equations. The system has solutions.

y � 3x � 56x � 2y � 4

6x � 2( ) � 4 Replace y with .

6x � � � 4 Distributive Property

� 4

The statement 10 � 4 is . This means that there are

no ordered pairs that are solutions to both equations. Thus, the

lines are , and the system has .

Use substitution to solve each system ofequations.a. x � 2y � 2 b. y � 2x � 2

3x � 6y � 12 2x � y � 2

Your Turn©

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Page(s):Exercises:

HOMEWORKASSIGNMENT

REMEMBER ITWhen using analgebraic method tosolve a system ofequations, a truestatement with novariables represents an infinite number of solutions. A falsestatement with novariables represents no solution.

Elimination Using Addition and Subtraction©

Glencoe/M

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13–4

• Solve systems ofequations by theelimination methodusing addition andsubtraction.

WHAT YOU’LL LEARN Elimination is an algebraic method to solve a system ofequations by adding or subtracting the equations.


Use elimination to solve 3x � 2y � 21 and 3x � 4y � 3.

3x � 2y � 21 Write the equations in column form.(�)3x � 4y � 3 Subtract to eliminate the x terms.

� � 18

� 18

��

�

66y

� � ��

186�

y � The value of y is .

Now substitute in either equation to find the value of x.Choose the equation that is easier for you to solve.

3x � 4( ) � 3

3x � � 3

3x � 12 � � 3 �

3x � 15

�33x� � �

135�

x � The value of x is .

The solution of the system of equations is .

Use elimination to solve x � 2y � 4 and x � y � 1.Your Turn

How can you tell whena linear equation is in standard form?(Lesson 6-3)

REVIEW IT


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For a special event at the Kartchner Caverns, the costof tickets for 4 adults and 3 students was $84 and thecost of tickets for 2 adults and 3 students was $54. Findthe admission price for an adult and for a student at thespecial event.

Let a � the cost for an and s � the cost for a

.

4a � 3s � 84 total cost for the group

2a � 3s � 54 total cost for the group

4a � 3s � 84 Write the equations in column form.

(�) 2a � 3s � 54 Subtract the equations to eliminatethe s terms.

� � 30

� 30

�22a� � �

320�

a � The value of a is .

Now substitute in either equation to find the value of s.4a � 3s � 84

4( ) � 3s � 84

� 3s � 84

60 � 3s � � 84 �

3s � 24

�33s� � �

234�

s � The value of s is .

The solution of the system of equations is . This

means that the cost for adults was and the cost for

students was .

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Describe the types ofsystems where theelimination method isthe preferred methodused to solve under thetab for Lesson 13-4.



ORGANIZE IT

The group admission cost for 2 adults and 6 children to enter a museum is $60. The group admission costfor 2 adults and 10 children to enter the same museum is$84. Find the group admission price for an adult and a child.

Use elimination to solve 3x � 4y � 6 and 5x � 4y � �22.

3x � 4y � 6 4y and �4y are additive inverses.(�)5x � 4y � �22 Add to eliminate the y terms.

� � �16

� �16

�88x� � �

�81x6

�

x � The value of x is .

Now substitute in either equation to find the value of y.3x � 4y � 6

3( ) � 4y � 6

� 4y � 6

�6 � 4y � � 6 �

4y � 12

�44y� � �

142�

y � The value of y is .

The solution of the system is .

Use elimination to solve 2x � 3y � �6 and 3x � 3y � 21.

Your Turn

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

Digit problems explore the relationship between digits of a number.


The sum of the digits of a two-digit number is 14. If the units digit is 2 more than the tens digit, what is the number?

Let t represent the tens digit and let u represent the units digit.t � u � 14 the sum of the digitsu � t � 2 the relationship between the digits

Rewrite the second equation so that the t and u are on thesame side of the equation.

u � t � 2 �t � u � 2

Then use elimination to solve.

t � u � 14 Write the equations in column form.(�)�t � u � 2 Add the equations to eliminate the t terms.

� � 16

� 16

�22u� � �

126�

u � The units digit is .

Now substitute to find the tens digit.

t � u � 14

t � � 14

t � 8 � 8 � 14 � 8

t � The tens digit is .

Since t is and u is , the number is .

The sum of thedigits of a two-digit number is 7.If the units digit is 7 less thanthe tens digit, what is thenumber?

Your Turn

Elimination Using Multiplication©

Glencoe/M

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13–5

• Solve systems ofequations by theelimination methodusing multiplication and addition.

WHAT YOU’LL LEARN

Use elimination to solve 5x � 2y � �2 and 3x � 6y � �30.

Multiply the first equation by 3 so that the y terms areadditive inverses.

5x � 2y � �2 x � y � �6

3x � 6y � �30 (�) 3x � 6y � �30

� 0 �

� �36

�

x �

Now find the value of y by replacing x in either equation.

5x � 2y � �2

5( ) � 2y � �2

�10 � 2y � �2

�10 � 2y � � �2 �

2y �

�22y� � �

82

�

y �

The solution of this system of equations is .


Your Turn

18x �36

Summarize thedifference between theelimination methodtaught in Lesson 13-4and the eliminationmethod taught inLesson 13-5 under thetab for Lesson 13-5.Then give examples ofwhen to use eachmethod.



ORGANIZE IT

GLE 16. Interpret and solve systems of linear equations using graphing, substitution, elimination, with and without technology, and matrices using technology (A-4-H)

Multiply by 3.

Use elimination to solve 3x � 2y � 4 and 9x � 4y � 7.

Multiply the first equation by �3 so that the x terms areadditive inverses.

3x � 2y � 4 x � y �

9x � 4y � 7 (�) 9x � 4y � 7

0 � �

� �5

�

y �

Now find the value of x by replacing y in either equation.3x � 2y � 4

3x � 2� � � 4

3x � � 4

3x � 1 � � 4 �

3x �

�

x �

The solution of the system of equations is .

Use elimination to solve x � 2y � 3 and 4x � y � 12.

Your Turn

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�10y �5

Multiply by 3.

REMEMBER ITAnother methodcan be used to checkyour answers. Forexample, if eliminationis used to solve thesystem, the system canthen be graphed orsubstitution can be usedto check the answer.

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A music store has one price for all CDs except for theCDs in the Sale section. One customer bought 8 regularCDs and 2 sale CDs and paid $79.50. Another customerbought 4 regular CDs and 5 sale CDs and paid $62.75.What are the costs of regular and sale CDs?

Let r represent the cost of the regular CDs and let s representthe cost of the sale CDs.

8r � 2s � 79.50 first customer’s expense

4r � 5s � 62.75 second customer’s expense

Multiply the second equation by �2 to eliminate the r terms.

8r � 2s � 79.50 8r � 2s � 79.50

4r � 5s � 62.75 (�) r � s � �125.50

0 � �

� �46

�

s �

Now find the value of r by replacing s with in either equation.

8r � 2s � 79.50

8r � 2� � � 79.50

8r � � 79.50

8r �

�

r �

The solution is . This means that the regular

CDs sell for and the sale CDs sell for .

�8s �46

Multiply by �2.

8r 68

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Morgan and Amy found a special on shorts andshirts. Morgan bought 5 pairs of shorts and 5 shirts for $125,not including tax. Amy bought 3 pairs of shorts and 6 shirtsfor $105, not including tax. Find the cost of each pair of shortsand the cost of each shirt.


Multiply the first equation by 5 and the second equation by 3so that the y terms are additive inverses.

4x � 3y � �8 x � y �

7x � 5y � 27 (�) x � y �

41x �

�4411x

� � �4411�

x �

Now find the value of y by replacing x in either equation.

7x � 5y � 27

7( ) � 5y � 27

� 5y � 27

7 � 5y � 7 � 27 � 7 5y � 20

�55y� � �

250�

y � The solution is .

Use elimination to solve 2x � 3y � 13 and3x � 4y � �6.

Your Turn

Your Turn

Multiply by 5.

Multiply by 3.

REMEMBER ITThere may be morethan one way to solve aproblem. In Example 4,you can also solve thissystem of equations bymultiplying the firstequation by 7 and thesecond equation by �4.

Page(s):Exercises:

HOMEWORKASSIGNMENT

Solving Quadratic-Linear Systems of Equations

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13–6

• Solve systems ofquadratic and linearequations.

WHAT YOU’LL LEARN

Quadratic-Linear Systems

no solution graphs donot intersect

one solution graphsintersect at one point

two solutions graphsintersect at two points

y

xO(�2, �2)

(1, 1)

y

xO

32(�1, )

y

xO

KEY CONCEPT

Determine whether each system of equations has onesolution, two solutions, or no solution by graphing. If thesystem has one solution or two solutions, name them.

y � x2

y � �x


and . Check this estimate by

substituting the coordinates into each equation.

Check:y � x2 y � �x

�

2

(x, y) � � � (x, y) �

� 0 ✓ � 0 ✓

y � x2 y � �x

� � �2 (x, y) � � �� (x, y) �

� 1 ✓ � 1 ✓

y � �x2 � 1y � 1

The graphs appear to intersect

at .

Check:y � �x2 � 1 y � 1

� �� 2 � 1 (x, y) � � 1 ✓

1 � ✓

y = –x2 + 1

y = 1(0, 1)x

y

O

y

x

y = x2

y = –x

(–1, 1)(0, 0)

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Determine whether each system ofequations has one solution, two solutions, or no solutionby graphing. If the system has one solution or twosolutions, name them.a. y � x2 � 1 b. y � �x2 � 3

y � 5 2x � y � 6

Use substitution to solve y � 2x2 � 3 and y � �1.

Substitute �1 for y in the first equation. Then solve for x.

y � 2x2 � 3

� 2x2 � 3 Replace y with .

�1 � � 2x2 � 3 � Add to each side.

� 2x2

�

1 � x2

� x Take the square root of each side.

The solutions of the system of equations are and

.

Your Turn

2 2x2

Use substitution to solve y � 2x2 � 3 and y � 1.

Substitute 1 for y in the first equation.

y � 2x2 � 3

� 2x2 � 3 Replace y with .

1 � � 2x2 � 3 � Subtract from each side.

� 2x2

�

�1 � x2

� x Take the square root of each side.

There is no real solution because the square root of a negativenumber is not a real number.

Use substitution to solve the system ofequations.a. y � �3x2 � 4

y � �1

b. y � x2 � 3y � 7

c. y � x2 � 6y � x

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

List the differentmethods of solvingquadratic-linear systemsunder the tab for Lesson13-6. Then describe thesituations in which eachmethod is preferred.



ORGANIZE IT�2 2x2

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Graphing Systems of Inequalities

• Solve systems ofinequalities bygraphing.

WHAT YOU’LL LEARN

Compare the solution of a linear system ofequations with a linearsystem of inequalitiesunder the tab for Lesson13-7. Then give anexample of each.



ORGANIZE IT

A set of two or more inequalities is a system of inequalities.


Solve each system of inequalities by graphing. If thesystem does not have a solution, write no solution.

x � 3y � x � 4

The solution is the ordered pairs in

the of the graphs

of x � 3 and y � x � 4. Shade the region darkest on the graph. The graphs of x � 3 and y � x � 4 are

the of this region.

The graph of y � x � 4 is a

line and is

in the solution of the system. Choose a point and check the solution.

x � y � 22y � �2x � 2

The graphs of x � y � 2 and

2y � �2x � 2 are

lines. Check this by graphing or by

comparing the .

Because the regions in the solution of x � y � 2 and 2y � �2x � 2 have

in common, the system

of inequalities has solution.

y = x + 4

x = 3

O x

y

x + y = 2

2y = –2x – 2O x

y

GLE 14. Graph and interpret linear inequalities in one or two variables and systems of linear inequalities (A-2-H, A-4-H)

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Page(s):Exercises:

HOMEWORKASSIGNMENT

Solve each system of inequalities bygraphing. If the system does not have a solution, writeno solution.a. y � 2x � 1

3x � 2y � 6

b. y � 5y � x � 2

Your Turn



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1. Solve the system of equations by graphing.

y � 2x � 1 y � �3x � 6

Describe the solution of each system of equations.

2. 3. y

xO

y

xO







C H A P T E R

13STUDY GUIDE

13-1

Graphing Systems of Equations

13-2

Solutions of Systems of Equations


Glencoe/M

cGraw

-Hill

4. Describe how you would use substitution to solve the systemof equations. y � �2x

x � 3y � 15

5. Amy solved a system of equations and her result was �8 � �8.All of her work was correct. Describe the graph of the system.Explain.

6. Luis solved a system of equations and his result was 5 � �2.All of his work was correct. Describe the graph of the system.Explain.

7. The sum of the digits of a two-digit number is 8. If the tens digit is 6 more than the units digit, what is the number?

Use elimination to solve each system of equations.

8. 3x � 5y � 15 9. �x � 4y � �4�3x � 2y � 6 4x � 4y � 6

10. 3x � 5y � 7 11. 5x � 7y � 173x � 2y � 14 8x � 7y � 9

13


13-3

Substitution

13-4

Elimination Using Addition and Subtraction


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Match each system with its solution.

12. x � y � 4 13. 4x � 3y � �1

�3x � 2y � 8 2x � y � 2

14. 2x � y � 4 15. 5x � 3y � �5

4x � y � 2 2x � 7y � �2

Match each graph with the correct system of equations.

16. 17. 18.

a. y � x2 b. y � x2 � 6 c. y � �12

� x � 4 d. y � 2x � 3

y � 2x � 3 y � 3 y � �3x2 � 5 y � �x2

Write the inequality symbols that you need to get a system whosegraph looks like the one shown. Use �, �, �, or �.

19. 20.

y x � 2 y x � 2

y �2x � 1 y �2x � 1

y

xO

y � �2x � 1y � �2x � 1

y � x � 2y � x � 2

y

xO

y � �2x � 1y � �2x � 1

y � x � 2y � x � 2

y

xO

y

xO

y

xO


13-5

Elimination Using Multiplication

13-6

Solving Quadratic-Linear Systems of Equations

13-7

Graphing Systems of Inequalities

a. (4, �2)b. (1, 2)c. (0, 4)

d. (�12

�, 1)e. (�1, 0)

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Teacher Signature

C H A P T E R

13Checklist







• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).

• You may also want to take the Chapter 13 Practice Test onpage 595.




• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).

• You may also want to take the Chapter 13 Practice Test onpage 595.

Radical Expressions

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NOTE-TAKING TIP: As you study a lesson, writedown questions you have, comments, reactions,and/or short summaries of the main ideas of thelesson that are highlighted or underlined.

Ch

apte

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C H A P T E R

14


FoldFold the short sides tomeet in the middle.

FoldFold the top to thebottom.

OpenCut along the second foldto make four tabs.


Begin with a sheet of 11" � 17" paper.

Describe the relationships among sets of numbers

Solvesimple radical

equations

Simplify, add, and subtract radical

expressions

Find the distance between two points in the coordinate

plane

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C H A P T E R




on Page Example

conjugates[CON-ja-guts]

Distance Formula

radical equations

radicand[RA-di-KAND]

rationalizing thedenominator[RA-shun-ul-eyes-ing]

This is an alphabetical list of new vocabulary terms you will learn inChapter 14. As you complete the study notes for the chapter, you willsee Build Your Vocabulary reminders to complete each term’s definitionor description on these pages. Remember to add the textbook pagenumber in the second column for reference when you study.

• Describe therelationships amongsets of numbers.

WHAT YOU’LL LEARN Name the set or sets of numbers to which each realnumber belongs.

10

This number is a natural number, a number,

an , and a number.

��100�

Since ��100� � , this number is an and a

rational number.

52

Since 52� , this number is a natural number, a

number, an , and a rational number.

�0.666 . . .

This repeating decimal is a number since it is

equivalent to ��23

�.

�17�

�17� � 4.123105626 . . . It is not the square root of a perfect

square. So, it is an number.

Name the set or sets of numbers to whicheach real number belongs.

a. �36�

b. ��138�

Your Turn

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The Real Numbers

Explain the similaritiesand differences betweenthe sets of naturalnumbers, wholenumbers, integers,rational numbers,irrational numbers, andreal numbers under thetab for “Describe therelationships among setsof numbers.”


Solvesimple radical

equations


expressions


plane

ORGANIZE IT

GLE 1. Identify and describe differences among natural numbers, whole numbers,integers, rational numbers, and irrational numbers (N-1-H, N-2-H, N-3-H)

c. ��2�

d. �23

e. 0.5�

Find an approximation, to the nearest tenth, for eachsquare root. Then graph the square root on a numberline.

�10�Enter: [��] 3.16227766

An approximate value for �10� is .

��8�

Enter: [��] 8 –2.828427125

An approximate value for ��8� is � .

Find an approximation, to the nearesttenth, for each square root. Then graph the square rooton a number line.a. �7�

b. ��19�

Your Turn

–5 –3–4 –2 –1

8��

0 1 2

ENTER2nd(–)

–1 10 2 3

10��

4 5 6

ENTER2nd

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Completeness Propertyfor Points on theNumber Line Each realnumber corresponds toexactly one point on thenumber line. Each pointon the number linecorresponds to exactlyone real number.

KEY CONCEPT

Determine whether each number is rational or irrational.If it is irrational, find two consecutive integers betweenwhich its graph lies on the number line.

��59�

� ��59� �

� ��59� �

59 is not a . So, its square root is

. The graph of ��59� lies between

and .

�121�

Since �121� � , it is a .

Determine whether each number isrational or irrational. If it is irrational, find twoconsecutive integers between which its graph lies on the number line.

a. ��64�

b. �23�

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

REMEMBER ITTo find the twoconsecutive integersbetween which anirrational square rootlies, make a list ofperfect square numbers:1, 4, 9, 16, 25, 36, 49, 64,81, . . . Then find thetwo numbers betweenwhich the numberunder the square rootsymbol lies.

The Distance Formula

Find the distance between points E(�3, 8) and F(5, 1).

d � �(x2 ��x1)2�� (y2 �� y1)2� Distance Formula

d � � � �2� � � �2

(x1, y1) � (�3, 8), and(x2, y2) � (5, 1)

d �2

� � �2

d � �

d �

The distance between points E(�3, 8) and F(5, 1) is

or about units.

A coordinate system issuperimposed over a map of Washington, D.C.Two taxis leave the intersection of 15th Street and G Street.One taxi travels 7 blocks north, and the other taxi travels 11 blocks east. How far apart are the taxis when they stop?

x

y

O

Sales St. Green Ct.L St.

Mount VernonSquare15th S

t.

FarragutSquare

McPhersonSquare

New York Ave.

G St. Metro Center

F St. White House

E St.

8th St.

9th St.

D St.

10th St.

11th St.

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• Find the distancebetween two points inthe coordinate plane.

WHAT YOU’LL LEARN

The Distance FormulaThe distance d betweenany two points withcoordinates (x1, y1) and(x2, y2) is given by d �

�(x2 ��x1)2 �� (y2 �� y1)�2.

KEY CONCEPT

The Distance Formula is a formula derived from the

Theorem to find the distance between

two on the coordinate plane.


Let the first taxi’s location be represented by (0, 7)

and the other taxi’s location by (11, 0). So, x1 � ,

y1 � , x2 � , and y2 � .


d � � �2� � �2

(x1, y1) � (0, 7), and(x2, y2) � (11, 0)

d � 2

� � �2

d � �

d � or about units

The taxis are about blocks apart when they stop.

a. Find the distance between points C(2, 7) and D(6, �4).

b. A coordinate system is superimposed over a map of Jake’sneighborhood. Jake leaves his school and travels 3 blockswest to a friend’s house. Then he turns and travels 5 blockssouth to his home. How far is Jake’s school from his house?

School

Home

Your Turn

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Write the distanceformula and twoexamples to help youremember how to use itunder the tab for “Findthe distance betweentwo points in thecoordinate plane.”


Solvesimple radical

equations


expressions


plane

ORGANIZE IT

Find the value of a if M(a, 6) and N(2, 3) are �18� unitsapart.


� � �2� � �2

(x1, y1) � (a, 6), and(x2, y2) � (2, 3)

� � � � � �2

�18� � �4 � 4a � a2�� 9 (2 � a)2 �

4 � 4a � a2

and (�3)2 � 9

�18� � �a2� 4�a � 1�3�

��18��2� ��a2

� 4�a � 1�3��2Square each side.

� a2� 4a � 13

0 � Subtract.

0 � (a � 1)(a � 5) Factor.

� 0 or � 0 Zero Product Property

a � a � .

The value of a is or .

Find the value of a if P(�3, 5) and Q(2, a) are

�194� units apart.

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

The Zero ProductProperty says that for allnumbers a and b, if ab� 0, then b � 0, b � 0,or both a and b equal 0.(Lesson 11-4)

REVIEW IT

Simplify �48�.

�48� � Prime factorization

� �16 � 3� 2 � 2 � 2 � 2 � 16

� �16� � Product Property ofSquare Roots

� Simplify �16�.

Simplify �15� � �75�.

�15� � �75� � �3 � 5� � �3 � 5 �� 5� Prime factorization

� �3� � �5� � �3� � �5� � �5� Product Property ofSquare Roots

� � �5� � �5� � �5� Commutative Property

� �32� � �52� � �3� � �3� � �32� and

�5� � �5� � �52�� 3 � 5 �

� Simplify.

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Simplifying Radical Expressions

• Simplify radicalexpressions.

WHAT YOU’LL LEARNThe radicand is the number under the sign.

Rationalizing the denominator is a method used to remove

from the of a fraction.

Conjugates are two expressions of the form a�b� � c�d�and a�b� � c�d�.


The Product Property ofSquare Roots says thatthe square root of aproduct is equal to theproduct of each squareroot. In symbols,

�ab� � �a� � �b� if a � 0and b � 0. (Lesson 8-5)

REVIEW IT

GLE 6. Simplify and perform basic operations on numerical expressions involving radicals (e.g., 2�3� + 5�3� = 7�3�) (N-5-H)

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a. �72� b. �6� � �12�

Simplify .

� ��31050

�� Quotient Property of Square Roots

� �20� Divide 300 by 15.

� �2 � 2 �� 5� Prime factorization

� 2� 5 2 � 2 � 22

� �22� � Product Property of Square Roots

� Simplify.

Simplify .

� � � 1

�

�

Simplify.

�900�

�

� � � Product Property ofSquare Roots

�

� Simplify.

3 � 2 � �5��

30

�9 � 4 �� 5��

30

�6 � 30��30 � 3�0�

�30��30�

�30��30�

�6��30�

�6��30�

�6��30�

�300��15�

�300��15�

Your Turn

The Quotient Propertyof Square Roots saysthat the square root ofa quotient is equal tothe quotient of eachsquare root. In symbols,

��ba

�� if a � 0 and

b � 0. (Lesson 8-5)

�a��b�

REVIEW IT

30

Simplified Form forRadicals A radicalexpression isin simplest form when:

• No radicands haveperfect square factorsother than 1.

• No radicands containfractions.

• No radicals appear inthe denominator of afraction.

KEY CONCEPT


a. b.

Simplify .

To rationalize the denominator, multiply both the numeratorand denominator by the conjugate of 4 � �3�.

�4 �

5�3��

4 �

5�3��

4

4

�

�

��

3

3

��

� � 1

� 5(4) � 5�3� Distributive Property;(a � b)(a � b) � a2 � b2

2�

� Simplify.16 � 3

� �20 �

135�3��

Simplify �48y3z�6�.

�48y3z6�� 2 � 2 �� 2 � 2�� 3 � y�3

� z6� Prime factorization

� 16 � 3 � � y2� z6 2 � 2 � 2 � 2 � 16

� � �3� � �y� � �y2� � �z6� Product Property ofSquare Roots

� 4 � �3� � �y� � � z3 Simplify.

� 4�y�z3�3y� The absolute value of yensures a nonnegativeresult.


a. �2 �

7�5�� b. �108a4�b3�

Your Turn

4 � �3��4 � �3�

5��4 � �3�

�12��72�

�16��48�

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

Adding and Subtracting Radical Expressions


12�10� � 15�10�12�10� � 15�10�

� � ��10� Distributive Property

�

3�2� � 9�2�3�2� � 9�2� � � ��2� Distributive Property

�

The lengths of the three sides of a triangle are 10�3� centimeters, 5�5� centimeters, and9�3� centimeters. Find the exact perimeter ofthe triangle.

P � � � Like terms: 10�3�

and 9�3�� 10�3� � 9�3� � 5�5� Commutative

Property

� (10 � 9)�3� � DistributiveProperty

� �

The exact perimeter of the triangle is �centimeters.


a. �8�5� � 12�5�

Your Turn

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14–4

• Add and subtractradical expressions.

WHAT YOU’LL LEARN

REMEMBER ITTo add or subtractradicals, the radicandsmust be the same.

GLE 6. Simplify and perform basic operations on numerical expressions involving radicals (e.g., 2�3� + 5�3� = 7�3�) (N-5-H)

b. 2�11� � 4�11� � 9�11�

c. The lengths of the sides of a quadrilateral are 8�2� inches,3�7� inches, 5�7� inches, and 6�2� inches. Find the exactperimeter of the quadrilateral.

Simplify 2�48� � 3�75�.

2�48� � 3�75�

� 2 4

� 3 � 3 2

� 3 Prime factorization

� 2 � � � �3�� 3� � �3�� Simplify.

� 2 � 4 � �3� � 15�3�

� � 15�3�

� (8 � 15)�3� Distributive Property

�

Simplify 3�72� � 5�18�.Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT

Write a note to explainthe process forsimplifying, adding, andsubtracting radicalexpressions under thetab for “Simplify, addand subtract radicalexpressions.” Then showone example of each.


Solvesimple radical

equations


expressions


plane

ORGANIZE IT

Solving Radical Equations


3 � �x� � 12

3 � �x� � 12

3 � �x� � � 12 � Subtract from each side.

�x� � 9

(�x�)2�

2Square each side.

x �

Check: 3 � �x� � 12

3 � � 12 Replace x with .

3 � � 12

12 � 12 ✓

�6 � �y � 2� � 1

�6 � �y � 2� � 1

�6 � �y � 2� � � 1 � Add to each side.

�y � 2� �

(�y � 2� )2�

2Square each side.

y � 2 � 49

y � 2 � 2 � 49 � 2 Add 2 to each side.

y � Check this result.

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14–5

• Solve simple radicalequations in which onlyone radical contains avariable.

WHAT YOU’LL LEARN A radical equation is an equation that contains a

expression.


Explain how toeliminate the radicalfrom an equationunder the tab for“Solve simple radicalequations.” Then showtwo examples.


Solvesimple radical

equations


expressions


plane

ORGANIZE IT


a. �2 � �x� � 9 b. �z � 5� � 1 � 8


�3m �� 4� � m

�3m �� 4� � m

(�3m �� 4�)2� m2 Square each side.

� m2

3m � 3m � 4 � 4 � m2� 3m � 4 Subtract 3m and 4

from each side.

0 �

0 � �m � ��m � � Factor.

m � � 0 or m � � 0 Zero ProductProperty

m � m �

Check: �3m �� 4� � m �3m �� 4� � m

3� � � 4 � 4 3� � � 4 � �1

� 4 � 4 � 4 � �1

� 4 � �1

4 � 4 1 �1

Since does not satisfy the original equation, is

the only solution.

Your Turn

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Why is it important tocheck your solutionswhen solving radicalequations?

WRITE IT

REMEMBER ITWhen solving aradical equation, alwaysisolate the radical beforesquaring each side.

�n � 5� � n � 5

�n � 5� � n � 5

(�n � 5� )2� (n � 5)2 Square each side.

n � 5 � n2� 10n � 25

n � n � 5 � 5 � n2� 10n � 25 � n � 5 Subtract.

0 � n2� � 20

0 � �n � ��n � � Factor.

n � � 0 or n � � 0 Zero Product Property

n � n �

Check: �n � 5� � n � 5 �n � 5� � n � 5

� � � 5 � � � � 5 � � � 5 � � � � 5

�0� � 0 �1� � 1

0 � 0✓ 1 � 1✓

Since both and satisfy the original equation,

they are both solutions.

Solve each equation. Check your solution.a. y � �3y � 1�0�

b. �x � 3� � x � 3

Your Turn

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Page(s):Exercises:

HOMEWORKASSIGNMENT


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For each of the following, choose the letter of each set ofnumbers to which each real number belongs. Each realnumber may belong to more than one set of numbers.

1. 3.6

2. �5

3. �41�

4. 0

5. 120

6. Suppose you want to use the Distance Formula to find thedistance between M(3, 7) and N(9, �2). Use (x1, y1) � (3, 7) and(x2, y2) � (9, �2). Complete the equation by writing the correctnumbers in the boxes.

d � � � �2� � � �2

7. What is the value of b if S(7, b) and T(13, 1) are 10 units apart?





You can use your completedVocabulary Builder(page 298) to help you solvethe puzzle.


C H A P T E R

14STUDY GUIDE


14-1

The Real Numbers

a. Natural numbers

b. Whole numbers

c. Integers

d. Rational numbers

e. Irrational numbers

14-2

The Distance Formula


Glencoe/M

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8. Of 5�3�, 5�6�, and 3�3�, which two radical expressions havethe same radicand?

Simplify each expression. Leave in radical form.

9. �32�

10. �6� � �18�

11.

12. a. How can you tell that the radical expression �28x2y�4� is not insimplest form?

b. To simplify �28x2y�4�, you first find the

of 28x2y4. Then apply the Property of Square Roots.

In this case, 4 � 7 � x2� y4 is equal to the product

. Simplify again to get a final

answer of 2 �x� y2�7�.

13. What method would you use to simplify ?�12t��15�

2��1 � �7�

14


14-3

Simplifying Radical Expressions

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14. Radical expressions can be added or subtracted if they have

the same .

15. Indicate whether the following expressions are in simplestform. Explain your answer.

a. 6�3� � �12�

b. 12�6� � 7�10�

c. 4�15� � 3�60�

d. 3�20� � 5�30�

16. Provide the reason for each step in the solution of the givenradical equation.

�5x � 1� � 4 � x � 3 Original equation

�5x � 1� � x � 1

(�5x � 1� )2� (x � 1)2

5x � 1 � x2� 2x � 1

0 � x2� 3x � 2

0 � (x � 1)(x � 2)

x � 1 � 0 or x � 2 � 0

x � 1 or x � 2



14-4

Adding and Subtracting Radical Expressions

14-5

Solving Radical Equations

Rational Expressions and Equations

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NOTE-TAKING TIP: When taking notes, alwayswrite definitions and examples of each of theterms learned.

Ch

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C H A P T E R

15


Fold Fold lengthwise to the holes.

CutCut along the top line and then cut ten tabs.

LabelLable the tabs using thevocabulary words in the chapter.


Rational Expression

Excluded Values

Rational Function

Multiplying Rational Expression

Dividing Rational Expression

Dividing polynomials

Least Common Multiple

Least Common Denominator

Rational Equation

Uniform Motion Porblems

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C H A P T E R




on Page Example

excluded value

least common denominator (LCD)

least common multiple (LCM)

rational equation [RA-shun-ul]

rational expression

rational function

uniform motion problem

work problems

This is an alphabetical list of new vocabulary terms you will learn in Chapter 15.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description on these pages. Remember to add the textbook page number in the secondcolumn for reference when you study.

Find the excluded value(s) for each rational expression.

�(x �

203)x�

Exclude the values for which (x � 3)x � .

(x � 3)x �

� 0 or � 0 Product Property

So, x cannot equal or .

�y2

10�

y4

�

Exclude the values for which y2� 4 � .

y2� 4 �

� 0 Factor .

y � or y � Zero Product Property

So, y cannot equal or .

Find the excluded value(s) for eachrational expression.

a. �m(7

3� m)� b. �

y2�

�

y16

�

Your Turn

©G

lencoe/McG

raw-H

ill

Simplify Rational Expressions15–1


• Simplify rationalexpressions.

WHAT YOU’LL LEARN An excluded value is any value assigned to a

that results in a denominator of .


Rational Expression A rational expression is an algebraic fractionwhose numerator and denominator are polynomials.

KEY CONCEPT

Simplify each rational expression.

�2250xx2

5

yy3�

�2250xx2

5

yy3� �

Note that x � 0 and y � 0.

� or

The GCF is 5x2y.

�47xx

��

1221

�

�47xx

��

1221

� � Factor 4x � 12 and 7x � 21.

� or The GCF is

.


a. �2118aa5

2

bb4�

b. �35xx

�

�

1220

�

Your Turn

©G

lencoe/McG

raw-H

ill

15–1


2 � 2 � 5 � x � x � x � x � x � y

2 � 2 � 5 � x � x � x � x � x � y1 1 1 1

REMEMBER ITDivision by zero is undefined.

Write an example of arational expressionunder the tab forRational Expression.Then find all theexcluded values for theexpression and writethose under the tab forExcluded Values. Writethe rational expressionas a function and drawits graph on the tab forRational Function.

Rational Expression

Excluded Values

Rational Function






Rational Equation


ORGANIZE IT

4x3

(x � 3)

(x � 3)

(x � 3)

(x � 3)

1

1


�a2

a�

2

2�

a2�

515

�

�a2

�

a2

2�

a2�

515

� � Factor a2� 25

and a2� 2a � 15.

� or

The GCF is .

�x2

6�

�

x2�

x12

�

�x2

6�

�

x �

2x12

� � Factor 6 � 2x

and x2� x � 12.

� Factor �1 from .

� or � The GCF

is x � 3.


a. �x2

1�

52�

x �

5x15

� b. �b2 �

b212

�

b9� 27

�

Your Turn

©G

lencoe/McG

raw-H

ill

15–1


Page(s):Exercises:

HOMEWORKASSIGNMENT

(a � )(a � )

(a � )(a � )

2(3 � x)

(x � )(x � )

2(� 1)(x � 3)

(x � 4)� �

(a � 5)� �(a � 3)� �

1

1

� �a � 3

2(�1)� �(x � 4)� �

1

1

x � 4

Explain how to factor a trinomial. How do you find the correctcombination of innerand outer terms?(Lesson 10-3)

REVIEW IT

Multiplying and Dividing Rational Expressions

Find each product.

�180aab

2� � �

140aab5�

�180aab

2� � �

140aab5� � �

180aab

2� � �

140aab5� Simplify.

� Multiply.

�zz

�

�

35

� � �34zz

2 �

�

1152z

�

�zz

�

�

35

� � �34zz

2 �

�

1152z

� � �zz

�

�

35

� � Factor 3z 2� 15z

and 4z � 12.

� � (z � 3) and (z � 5) are common factors.

�

�m

m�

24

� � �m2

m�

2

3�

m1� 4

�

�m

m�

24

� � �m2

m�

2

3�

m1� 4

� � �m

m�

24

� �

� �m

m�

24

� � or

©G

lencoe/McG

raw-H

ill


• Multiply and dividerational expressions.

WHAT YOU’LL LEARN

1 1 1

1 a32

4

Summarize the steps inmultiplying two rationalexpressions under thetab for MultiplyingRational Expressions.Then, give an example.

Rational Expression

Excluded Values

Rational Function






Rational Equation


ORGANIZE IT

(z � 5)

(z � 3)

z � 5

1

4

1

1

1

3z(z � 5)

(m � )(m � )

(m � )(m � )

(m � 1)

(m � 4)

1

1 1

m � 11

15–2

Find each product.

a. �1152nm

2� � �

48mnn5

2� b. �

2dd2

�

�

71d0

� � �3dd

�

�

251

� c. �y2 �

y2

1�

0y6�

416

� � �y2

2�

y32y

�

Find each quotient.

�15

ba3

3� � �

52ab

2�

�15

ba3

3� � �

52ab

2�

� �15

ba3

3� � The reciprocal of �5

2ab

2� is .

� �15

ba3

3� � �

52ab2� 5, a and b are common factors.

�

�6xx

��

36

� � (x � 1)

�6xx

��

36

� � (x � 1)

� �6xx

��

36

� � The reciprocal of

(x � 1) is .

� � Factor 6x � 6.

� � is a

common factor.

�

Your Turn

©G

lencoe/McG

raw-H

ill

15–2


1 1

1

1

6a

1

x � 3

x � 3

1

1

x � 3

Summarize the steps individing two rationalexpressions under thetab for Dividing RationalExpressions. Then, givean example.

Rational Expression

Excluded Values

Rational Function






Rational Equation


ORGANIZE IT

�x2

4�

x2y36

� � �64x

�2y

x5�

�x2

4�

x2y36

� � �64x

�2y

x5�

� �x2

4�

x2y36

� � The reciprocal of �46x�2y

x5� is .

� � Factor x2� 36.

� � Factor

from 6 � x .

� x � 6, x2, y, and are commonfactors.

�

Find each quotient.

a. �4st3

3� � �

122st4�

b. �1p0

2

q�2p

43� � �

25q

�2p

p�

c. �2x

x�

� 510

� � (x � 5)

Your Turn

©G

lencoe/McG

raw-H

ill

15–2


Page(s):Exercises:

HOMEWORKASSIGNMENT

4x2y

4x2y5

(4x2y)

1

1 1 1

1

4x2y51 1

� (x � 6)

1

Find each quotient.

(15x � 10) � (3x � 2)5

3x � 2�1�5�x� �� 1�0� 15x � 3x �

(�) 15x � Multiply 5 and .

.

Therefore, (15x � 10) � (3x � 2) � .

(x2� x � 12) � (x � 3)

xx � 3�� x 2

� x � x

(�) x2� Multiply x and .

�4x Subtract: �x � 3x � � 4x;

bring down .

(�) �4x Multiply �4 and .

Subtract.

Therefore, (x2� x � 12) � (x � 3) � .

Find each quotient.a. (14x � 7) � (2x � 1) b. (x2

� 8x �9) � (x � 1)

Your Turn

©G

lencoe/McG

raw-H

ill

Dividing Polynomials15–3


• Divide polynomials by binomials.

WHAT YOU’LL LEARN

Give an example of adivision problem using two polynomials under the tab for DividingPolynomials. Thenidentify the dividend,divisor, quotient and remainder.

Rational Expression

Excluded Values

Rational Function






Rational Equation


ORGANIZE IT

x2� x � 12

(8a2� 14a � 9) � (2a � 3)

4a � 12a � 3�� 8a2

� 2a �

(�) 8a2� Multiply 4a and .

2a � Subtract. Then bring

down .

(�) 2a � 3 Multiply 1 and .

Subtract. The remainder

is .

The quotient is 4a � 1 with remainder 6.

So, (8a2� 14a � 9) � (2a � 3) � 4a � 1 � .

(x2� 20) � (x � 5)

x � 5

x � 5 Rename x2� 20 as

x2� � 20

(�) x2� Multiply x and .

5x Subtract. Then bring

down .

(�) 5x Multiply 5 and .

Subtract. The remainder

is .

Therefore, (x2� 20) � (x � 5) � � .

©G

lencoe/McG

raw-H

ill

15–3


x � 5

2a � 3

REMEMBER ITAfter the divisor ismultiplied by the lastterm in the quotient,the result is subtracted.If you prefer, you mayadd the opposite of theentire expression.

8a2� 14a � 9

) x2� � 20

Find each quotient.a. (9y2

� 9y � 4) � (3y � 1) b. (x2� 32) � (x � 6)

Find the length of a rectangle if itsarea is 12x2

� 13x � 3 square unitsand its width is 3x � 1 units.

To find the length, divide the area by

the length .

4x � 33x � 1��

(�) 12x2� Multiply 4x and .

9x � Subtract. Then bring down .

(�) 9x � 3 Multiply 3 and .

The remainder is .

Therefore, the length of the rectangle is units.

You can check your answer by multiplying

and .

Find the length of a rectangle if its area is 5a2

� 8a � 4 square units and its width is 5a � 2 units.

Your Turn

Your Turn

©G

lencoe/McG

raw-H

ill

15–3


Page(s):Exercises:

HOMEWORKASSIGNMENT

12x2 � 13x � 33x � 1

5a � 2 5a2 � 8a � 4

12x2� 13x � 3

Combining Rational Expressions with Like Denominators


�7y

� � �1y0�

�7y

� � �1y0� �

The common denominator is y.Subtract the numerators.

� or ��3y

�

�1137a� � �

21a7�

�1137a

� � �21a7� �

The common denominator is 17.Add the numerators.

�


a. �n3

� � �1n2� b. �

151x� � �

121x�

Find each sum or difference. Write in simplest form.

�15

8n� � �

154n�

�15

8n

� � �154n

� �The common denominator is 15n.Add the numerators.

�

� �1152n

� or Divide by the GCF, .

Your Turn

©G

lencoe/McG

raw-H

ill


15–4

• Add and subtractrational expressionswith like denominators.

WHAT YOU’LL LEARN

7 �

y

y

13a �

17

17

8 �

15n

15n

5n

Standard 13.0 Studentsadd, subtract, multiply,and divide rationalexpressions and functions.Students solve bothcomputationally andconceptually challengingproblems by using thesetechniques. (Key)

�25p� � �

21p3�

�25p� � �

21p3� � The common denominator is 2p.

Subtract the numerators.

�

� Divide by the GCF, .

� or �

�3x

1�0

1� � �

3x3� 1�

�3x

1�0

1� � �3x

3� 1� �

The common denominator is 3x � 1. Subtract the numerators.

�

�aa

��

33

� � �a �4

3�

�aa

��

33

� � �a �4

3� �

The common denominator is a � 3. Add the numerators.

�

�2xx

��

35

� � �xx

��

43

�

�2xx

��

35

� � �xx

��

43

�

�The common denominator is x � 3. Add the numerators.

�

©G

lencoe/McG

raw-H

ill

15–4


5 �

2p

2p

2p

p p

3x � 1

1

4

� 3

3x � 1

a � 3 �

a � 3

a �

a � 3

REMEMBER ITWhen adding orsubtracting rationalexpressions, always checkto see if your finalanswer can be simplified.

2x � 5 � x � 4

3x �

� Factor the numerator.

� Divide by the GCF, .

�

�2a

12�a

1� � �

22aa

��

51

�

�2a

12�a

1� � �

22aa

��

51

�

�The common denominator is 2a � 1.Subtract the numerators.

�

� Distributive Property

Find each sum or difference. Write insimplest form.

a. �1141y

� � �114y� b. �

230b� � �

32b� c. �

x �7

5� � �x �

65

�

d. �mm

��

120

� � �m �5

10� e. �

xx

��

41

� � �x3�x

1� f. �

4yy

��

13

� � �4yy� 3�

Your Turn

©G

lencoe/McG

raw-H

ill

15–4


Page(s):Exercises:

HOMEWORKASSIGNMENT

(x � 3)

(x � 3)

� 5

2a � 1

� (2a � 5)

2a � 1

12a � �

2a � 1

1

1

Find the LCM for each pair of expressions.

12m4n5, 14m2n

Factor each expression.

12m4n5� 2 � 2 � 3 � m � m � m � m � n � n � n � n � n

14m2n �

Use each factor the greatest number of times it appears ineither factorization.

LCM � � m � m � m � m � n � n � n � n � n

�

x2� 3x � 10, 3x2

� 7x � 2

Factor each expression.

x2� 3x � 10 � (x � 5)

3x2� 7x � 2 � (x � 2)

Use each factor the greatest number of times it appears ineither factorization.

LCM �

Find the LCM for each pair of expressions.a. 15x3y5, 18x2 b. x2

� 2x � 15, x2� 11x � 30

Your Turn

©G

lencoe/McG

raw-H

ill

Combining Rational Expressions with Unlike Denominators

15–5


• Add and subtractrational expressionswith unlikedenominators.

WHAT YOU’LL LEARN

Write two expressionsand explain the stepsyou would take to findthe LCM for theexpressions under thetab for Least CommonMultiple.

Rational Expression

Excluded Values

Rational Function






Rational Equation


ORGANIZE IT

Standard 13.0 Studentsadd, subtract, multiply,and divide rationalexpressions and functions.Students solve bothcomputationally andconceptually challengingproblems by using thesetechniques. (Key)

Write each pair of rational expressions with the same LCD.

�53m�, �

2m4

2�

First find the LCD.

5m � 5 � m 2m2�

LCD � 2 � 5 � or .

Then write each fraction with the same LCD.

�53m� � �2

2mm� � �

160mm2� �

2m4

2� � �

�x �

13

�, �2x

5�x

6�

First find the LCD.

x � 3 � x � 3 2x � 6 � 2

LCD �

Then write each fraction with same LCD.

�x �

13

� � � �2x

5�x

6� �

Write each pair of rational expressionswith the same LCD.

a. �32a3�, �

54a2� b. �

n �n

8�, �4n �

332

�

Your Turn

©G

lencoe/McG

raw-H

ill

15–5


The least number that is a of

two or more numbers is the least common multiple (LCM).

The least common multiple of the denominators is calledthe least common denominator (LCD).


10m2

A common denominatorcan always be found bymultiplying the twodenominators together.What are the reasonsthis method is notalways used?

WRITE IT

2(x � 3) 2(x � 3)

Write two rationalexpressions and explainthe steps you wouldtake to find the LCD for the expressions under the tab for LeastCommon Denominator.

Rational Expression

Excluded Values

Rational Function






Rational Equation


ORGANIZE IT


�4m

33� � �

8m5

2�

Find the LCD.

4m3� 2 � 2 � m � m � m 8m2

� 2 � 2 � 2 � m � m

LCD � � m � m � m or

Rename each expression with the LCD as the denominator.

�4m

33� � � �

8m5

2� � �mm

� �

Add.

�4m

33� � �

8m5

3� � � or

�x �

36

� � �x2

4�

x36

�

x � 6 � x � 6 x2� 36 �

LCD �

�x �

36

� � �x2

4�

x36

� � �x �

36

� � �((xx

�

�

66))

� � �(x � 6

4)(xx � 6)�

� � �(x � 6

4)(xx � 6)�

� � or �

Find each sum or difference. Write insimplest form.

a. �10

3b2� � �

55b3� b. �

x �

73

� � �x2

� 32xx

� 18�

Your Turn

©G

lencoe/McG

raw-H

ill

15–5


Page(s):Exercises:

HOMEWORKASSIGNMENT

8m3

8m3

5m �

8m3

8m3

8m3

3x �

(x � 6)(x � 6)

� 18

(x � 6)(x � 6) (x � 6)(x � 6)3x � 18 � 4x

Solving Rational Equations


�54x� � �

23

� � �71x2�

�54x� � �

23

� � �172x� The LCD is 12.

��54x� � �

23

�� 172x�� Multiply each side

by 12.

��54x��

23

�� 172x�� Distributive Property

12��54x�� 12��

23

�� 12��172x��

� � 7x

15x � 8 � � 7x � Subtract

from each side.

8 �

�

� x

So, the solution is .

©G

lencoe/McG

raw-H

ill


15–6

• Solve rational equations.

WHAT YOU’LL LEARN An equation that contains at least one rational

is a rational equation.


Tell the differencebetween a rationalexpression and arational equation underthe tab for RationalEquations. Thencompare the method of solving rationalequations with themethod of addingrational expressionswith unlikedenominators.

Rational Expression

Excluded Values

Rational Function






Rational Equation


ORGANIZE IT

1

111

8 �8x

�35x� � �

54x� � �

2165�

�35x� � �

54x� � �

2165� The LCD is 15x.

��35x� � �

54x��

2165�� Multiply each side

by the LCD.

��35x��

54x��

2165�� Distributive

Property

15x��35x�� 15x��

54x�� 15x��

2165��

� � 26x Simplify.

� 26x

� x Divide.

�x �

71

� � �x

2�x

1� � 5

�x �

71

� � �x2�x

1� � 5

The LCD is x � 1.

��x �

71

� � �x2�x

1�� 5

��x �

71

�� x

2�x

1�� 5

(x � 1)��x �

71

�� (x � 1)��x

2�x

1�� 5

7 � � 5x �

7 � 2x � � 5x � 5 �

7 � � �5

7 � � 7 � �5 �

�3x � �12

x �

©G

lencoe/McG

raw-H

ill

15–6


1 1

15 3

1

1 1

11


a. �67n� � �

n2

� � 5

b. �4y

� � �23y� � �

161�

c. �xx

��

11

� � �x �2

1� � �

8x

�

d. �a �

61

� � �1a2� � �a

��5

1�

Your Turn©

Glencoe/M

cGraw

-Hill

15–6


Page(s):Exercises:

HOMEWORKASSIGNMENT


©G

lencoe/McG

raw-H

ill


1. �278aa

2

bb

3� 2. �y4(yy

�

�

264)

� 3. �x2

�

x �

5x4� 4

� 4. �m

92m�

2

6�

m1�

8m16

�

Find each product or quotient.

5. �10

zx3y� � �

8xzy

2

2� 6. �3n

n�

215

� � �6n �

n30

�

7. �9a

c2b3� � 27ab 8. �

x2

2� 9� � �

x2 � 108x � 21�





You can use your completedVocabulary Builder (page 318) tohelp you solve the puzzle.


C H A P T E R

15STUDY GUIDE


15-1

Simplifying Rational Expressions

15-2

Multiplying and Dividing Rational Expressions


Glencoe/M

cGraw

-Hill

Find each quotient.

9. (12d � 30) � (2d � 5) 10. (x2� 6x � 7) � (x � 7)

11. (6y2� 7y � 5) � (3y � 2) 12. (a3

� 4a � 4) � (a � 1)


13. �71n0� � �

1n0� 14. �

193x� � �

95x�

15. �y

6�

y2

� � �yy

�

�

12

� 16. �x3�

x2

2� � �

2xx

�

�

28

�

15


15-4

Combining Rational Expressions with Like Denominators

15-3

Dividing Polynomials

©G

lencoe/McG

raw-H

ill


17. �32x� � �

x9�

x23

� 18. �58a� � �

a �

a2

�

19. �n �

24

� � �n7

� 20. �x

2�

x3

� � �x2

9� 9�


21. �25n� � �

n2

� � �130� 22. �

65x� � 3 � �

27x�

23. �52y� � �

35

� � �y �

y1

� 24. �a �

a2

� � �a �

81

� � 1



15-5

Combining Rational Expressions with Unlike Denominators

15-6

Solving Rational Equations

algebra 1 interactive notebook ebook

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