CChapter 11 hapter 11 RResource Mastersesource Mastersmrsaltsman.com/advdocs/ch11.pdfie s n c. Chapter 11 50 3.Explain const variable coefficient 2. Explain differen Commutative Properand

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Chapter 11 Chapter 11 Resource MastersResource Masters

ContentsChapter Resources• Family Letter • Are You Ready Worksheets• Diagnostic Test • Pretest

Language Arts Resources• Student Glossary

Practice and Reinforcement• Facts Practice

Leveled Lesson Resources• Explore• Reteach• Skills Practice• Homework Practice• Problem-Solving Practice• Enrich

Technology Resources• Graphing Calculator Activity• Scientific Calculator Activity• Spreadsheet Activity

Assessment Resources• Reflecting on the Chapter• Chapter Quizzes• Vocabulary Test• Chapter Tests• Standardized Test Practice• Extended-Response Test• Student Recording Sheet• Chapter Project Rubric

Answer Pages

ChapterResource Masters

are provided forevery chapter in both

print and digitalformats.

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Copyright © by the McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such materials be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Math Connects, Course 3. Any other reproduction, for use or sale, is expressly prohibited without prior written permission of the publisher.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240

ISBN: 978-0-07-892310-4MHID: 0-07-892310-7 Math Connects, Course 3

Printed in the United States of America.

2 3 4 5 6 7 8 9 10 032 18 17 16 15 14 13 12 11 10 09

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.CONTENTSCONTENTS

Teacher’s Guide . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Chapter 11 ResourcesFamily Letter . . . . . . . . . . . . . . . . . . . . . . 1Are You Ready?

Practice Worksheet . . . . . . . . . . . . . . . . . . . . . 5

AL Review Worksheet . . . . . . . . . . . . . . . . . . 6

BL Apply Worksheet . . . . . . . . . . . . . . . . . . . 7

Diagnostic Test . . . . . . . . . . . . . . . . . . . . . . 8

Pretest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Language Arts ResourcesStudent Glossary . . . . . . . . . . . . . . . . . . . . . . 10

Practice and Reinforcement Facts Practice . . . . . . . . . . . . . . . . . . . . . . . . . 11

Lesson Resources

A PropertiesAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 12

Skills Practice . . . . . . . . . . . . . . . . . . . . . . 13Homework Practice . . . . . . . . . . . . . . . . . . 14Problem-Solving Practice . . . . . . . . . . . . . 15BL Enrich. . . . . . . . . . . . . . . . . . . . . . . . . 16

B The Distributive PropertyAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 17


C Simplify Algebraic ExpressionsAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 22

Skills Practice . . . . . . . . . . . . . . . . . . . . . . 23Homework Practice . . . . . . . . . . . . . . . . . . 24

Problem-Solving Practice . . . . . . . . . . . . . 25BL Enrich. . . . . . . . . . . . . . . . . . . . . . . . . 26

D PSI: Solve a Simpler ProblemAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 27

Skills Practice . . . . . . . . . . . . . . . . . . . . . . 28Homework Practice . . . . . . . . . . . . . . . . . . 29Problem-Solving Practice . . . . . . . . . . . . . 30

Lesson Resources

A : Equations with Variables on Each Side

Explore . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

B Solve Equations with Variables on Each SideAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 32


TI-84 Plus Activity . . . . . . . . . . . . . . . . . . 37

C Solve Multi-Step EquationsAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 38


TI-84 Activity . . . . . . . . . . . . . . . . . . . . . . . 43

D Solve Multi-Step InequalitiesAL Reteach . . . . . . . . . . . . . . . . . . . . . . . 44


TI-84 Plus Activity . . . . . . . . . . . . . . . . . . 49

AL = Approaching Level BL = Beyond Level

Lesson

11-1

Lesson

11-2

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Assessment ResourcesReflecting on Chapter 11 . . . . . . . . . . . . . . . 50

Chapter Quizzes . . . . . . . . . . . . . . . . . . . . . . 51

Vocabulary Test . . . . . . . . . . . . . . . . . . . . . . . 52

Chapter Tests

AL 1A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

AL 1B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

BL 3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

BL 3B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Standardized Test Practice . . . . . . . . . . . . . . 65

Extended-Response Test . . . . . . . . . . . . . . . . 67

Extended-Response Rubric . . . . . . . . . . . . . . 68

Student Recording Sheet . . . . . . . . . . . . . . . 69

Chapter Project Rubric . . . . . . . . . . . . . . . . . 70

Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A1

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v

Teacher’s Guide to Using theChapter 11 Resource Masters

The Chapter 11 Resource Masters includes the core materials needed forChapter 11. These materials include information for families, student worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing from the online Teacher Edition.

Family ResourcesFamily Introduction to Course 3 (Available in Chapter 0)

• Talks about the focus of the grade level. • Gives Web site information.

Family Letter • English and Spanish • Overview of the chapter • Key vocabulary • Provides at home activities

Chapter ResourcesAre You Ready Worksheets • Use after the Are You Ready section in the Student Edition. • AL Review: Approaching-level students • Practice: On-level students • BL Apply: Beyond-level students

Chapter Diagnostic Test • Use to test skills needed for success in the upcoming chapter. • Retest approaching-level students after the Are You Ready worksheets.

Chapter Pretest • Quick check of the upcoming chapter’s concepts to determine pacing. • Use before the chapter to gauge students’ skill level. • Use to determine class grouping.

NAME ________________________________________ DATE _____________ PERIOD _____

Chapter 11 1

Course 3

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Family Letter

Key Vocabulary

Dear Parent or Guardian:

Today we began Chapter 11: Properties and Multi-Step Equations and Inequalities. In this chapter, your student will learn about and how to apply the Associative, Commutative, and Distributive Properties. We will also learn how to solve multi-step equations and inequalities with variables on both sides of the equal sign or inequality sign. Included in this letter are key vocabulary words and activities you can do with your student. You may also wish to log on to glencoe.com for other study help. If you have any questions or comments, feel free to contact me at school.

Sincerely,

Properties

Distributive Property The process of multiplying a sum or difference by a number.

equivalent expressions Two or more expressions that have the same value when the variable in each of them is replaced by the same number.

property A statement that is true for any numbers.

simplify To write an expression in simpler form.

Algebraic Expressions

coefficient The numerical factor of a term that contains a variable.

constant A term without a variable.

like terms Terms that contain the same variables to the same powers.

term Parts of an algebraic expression separated by plus or minus signs.

11Chapter

Distributive PropertyFor any numbers a, b, and c,

a(b + c) = ab + aca(b – c) = ab – ac.

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Chapter 11 14

Course 3

Homework PracticeProperties

Name the property shown by each statement. 1. 1 (a + 3) = a + 3 2. 2p + (3q + 2) = (2p + 3q) + 2

3. (ab) c = c (ab) 4. 2t 0 = 0

5. m (nr) = (mn) r 6. 0 + 2s = 2s

State whether the following conjectures are true or false. If false, provide a counterexample. 7. The product of an odd number and an even number is always odd. 8. The sum of two whole numbers is always larger than either whole number. Simplify each expression. Justify each step.

9. 2d (3)

10. 2y + (4 + 5y)

11. FAXES Marcellus sent four faxes to Gem. The first fax took 14 seconds to send, the second fax 19 seconds, the third 16 seconds, and the fourth 11 seconds. Use mental math to find out how many seconds it took to fax all four documents to Gem. Explain your reasoning.

12. SNOW The first four snowfalls of the year in Shawnee’s hometown measured 1.6 inches, 2.2 inches, 1.8 inches, and 1.4 inches. Use mental math to find the total amount of snow that fell. Explain your reasoning.

Get ConnectedGet Connected For more examples, go to glencoe.com.


1/13/09 7:22:05 PM

NAME ________________________________________ DATE _____________ PERIOD _____

Chapter 11 31

Course 3

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A11-2

Equations with Variables on Each SideUse algebra tiles to solve 4x + 1 = 2x + 3.Step 1 Model the equation.

Step 2 Remove the same number of x-tiles from each side of the mat until there are x-tiles on only one side.

Step 3 Remove the same number of 1-tiles from each side of the mat until the x-tiles are by themselves on one side.

Step 4 Separate the tiles into two equal groups.

Therefore x = 1. Since 4(1) + 1 = 2(1) + 3, the solution is correct.

Use algebra tiles to solve each equation. 1. 3x + 1 = 2x + 4 2. x – 1 = 2x + 3 3. 2x + 6 = 5x – 3 4. Explain why you can remove a 1-tile from each side of the mat.

5. MAKE A CONJECTURE In the set of algebra tiles, –x is represented by - Explain how you could use the –x-tiles and other algebra tiles to solve –2x – 1 = –x – 3.

1 11

1

4x + 1 2x + 3

=

=

1

=4x - 2x + 1 2x - 2x + 3

=11

1

1

1

1 1

=2x + 1 - 1 3 - 1

=

1

1

2x 2

=

=


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Language Arts ResourcesStudent Glossary • Includes key vocabulary terms from the chapter. • Students record definitions and/or examples for each term. • Students can use the page as a bookmark as they study the chapter.

Practice and ReinforcementFacts Practice • Quick recall of concepts needed in the upcoming chapter. • Use as a timed test to gauge student mastery of prior concepts.

Lesson Resources

Explore • Provides additional practice for the activities and exercises

found in the Student Edition. • Use as homework for same-day teaching.

Reteach • Provides vocabulary, key concepts, additional worked-out

examples, and exercises. • Use for students who have been absent.

Skills Practice • Focuses on the computational nature of the lesson. • Use as an additional practice. • Use as homework for second-day teaching.

Homework Practice • Mimics the types of problems found in the Practice

and Problem Solving of the Student Edition. • Use as an additional practice. • Use as homework for second-day teaching.

Problem-Solving Practice • Includes word problems that apply the concepts of the lesson. • Use as an additional practice. • Use as homework for second-day teaching.

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NAME ________________________________________ DATE _____________ PERIOD _____

SCORE _____

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11Chapter

Chapter 11 65

Course 3

Standardized Test Practice(Chapters 1–11)

Read each question. Then fill in the correct answer. 1. READING Kip read b books over the summer. Judson read 4 more books than Kip read. Ami read 3 fewer books than did Kip. Write an expression in simplest form to represent the total number of books read by all three people. A. b + 4 B. 3b + 1 C. b + 3 D. 3b + 7 2. Find the volume of the sphere in cubic inches. Round to the nearest whole number.

4 in.

3. Solve the equation. Check your solution.–2(2p – 4) = 3(p + 12)

F. –28 G. –4 H. 4 I. 28 4. Simplify and express your answer using a positive exponent. What is the exponent?

5–3 · 57

5. Complete the conversion. Round to the nearest tenth, if necessary. 14 in. ! cm

A. 5.5 B. 7 C. 30.2 D. 35.6 6. What is the slope of a line through points A(-2, 3) and B(1, 5) F. - 3 " 2 G. - 2 " 3 H. 2 " 3 I. 3 " 2 7. TRAVEL A car traveled 28 miles east and then 22 miles north. How far is the car from the starting point? A. 7.1 mi B. 17.3 mi C. 35.6 mi D. 50 m

1. A B C D

2.

3. F G H I

4.

5. A B C D

6. F G H I

7. A B C D


1/13/09 7:24:23 PM

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Chapter 11 50

Course 3

1. Explain the difference between a constant term and the coefficient of a variable.

2. Explain the difference between the Commutative Property and the Associative Property.

3. Explain how you would use the Distributive Property to multiply the numbers 6 and 99. Illustrate the process.


1/13/09 7:23:18 PM

NAME ________________________________________ DATE _____________ PERIOD _____

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Chapter 11 16

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Enrich

Other Algebraic PropertiesTwo properties of all real numbers are the Additive Inverse Property and the Multiplicative Inverse Property, as shown below.

Algebraic PropertiesProperty Symbols ExampleAdditive Inverse Property a + (–a) = 0 5 + (–5) = 0Multiplicative Inverse Property a · 1 " a = 1; a # 0 3 · 1 " 3 = 1

If an operation can be used for all numbers in a set of numbers, then the set is said to be closed under that operation. For example, the set of integers is closed under the operation of addition because the sum of any two integers is an integer. Questions are: Can you use these two properties with other sets of numbers? Would you be using a number that is not in the set? Is the set closed?The set of real numbers: The set of real numbers is closed when applying these properties. For example, choose any nonzero real number like 7. Both –7, the additive inverse of 7, and 1 " 7 , the multiplicative inverse of 7, are real numbers.

Additive Inverse Property: 7 + (–7) = 0Multiplicative Inverse Property: 7 · 1 " 7 = 1The set of real numbers is a closed set, and you can use these two properties because you are using numbers within the set of real numbers.Determine whether you use the Additive Inverse Property and/or the Multiplicative Inverse Property when working with the sets below. If you cannot use either or both properties, provide an explanation. 1. The set of whole numbers 2. {–1, –2, –3, …}

3. The set of integers 4. The set of rational numbers


1/13/09 7:22:13 PM

Enrich • Provides an extension of the concepts, offers a historical or

multicultural look at the concepts, or widens students’ perspectives on the mathematics.

• For use with all levels of students.

Technology Activities • Presents ways in which technology can be used with the

concepts in some of the lessons. • Use as an alternative approach to teaching the concept. • Use as part of the lesson presentation.

Assessment Resources

Reflecting on Chapter 11 • Three open-ended questions • Allows students to write about mathematics.

Chapter Quizzes • Free-response questions • One quiz for each multi-part lesson

Vocabulary Test • Includes a list of vocabulary words and questions to assess students’ knowledge of

those words. • Use in conjunction with one of the Chapter Tests.

Chapter Tests • AL 1A-1B Approaching-level students • Contains multiple choice questions. • 2A-2B On-level students • Contains both multiple-choice and free-response questions. • BL 3A-3B Beyond-level students • Contains free response questions. • Tests A and B are the same format with different numbers. • Use when students are absent or for different rows.

Standardized Test Practice • Test is cumulative. • Includes multiple-choice and short-response questions.

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NAME ________________________________________ DATE _____________ PERIOD _____

SCORE _____

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11Chapter

Chapter 11 69

Course 3

Use this recording sheet with pages 670–671 of the Student Edition.Fill in the correct answer. For gridded-response questions, write your answers in the boxes on the answer grid and fill in the bubbles to match your answers.

1. A B C D

2. F G H I

3.

4. A B C D

5.

6. F G H I

7. A B C D

8.

9.

10. F G H I

11. A B C D

12. F G H I

Extended ResponseRecord your answers for Exercise 13 on the back of this paper.

Student Recording Sheet


1/13/09 7:24:42 PM

Extended-Response Test • Contains performance-assessment tasks • Sample answers are included.

Extended-Response Rubric • The scoring rubric for the Extended-Response Test.

Student Recording Sheet • Corresponds with the Test Practice at the end of the

Student Edition chapter.

Chapter Project Rubric • The scoring rubric for the Chapter Project found in the

Teacher Edition.

Answers

Chapter and Lesson Resources • Chapter Resources, Facts Practice, and Lesson Resources are provided as reduced

pages with answers appearing in black.

Assessments • Full-size answer keys are provided for the assessment masters.

viii

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NAME ________________________________________ DATE _____________ PERIOD _____

PDF pass

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.Family Letter

Key Vocabulary

Dear Parent or Guardian:

Today we began Chapter 11: Properties and Multi-Step Equations and

Inequalities. In this chapter, your student will learn about and how to apply

the Associative, Commutative, and Distributive Properties. We will also learn

how to solve multi-step equations and inequalities with variables on both sides

of the equal sign or inequality sign. Included in this letter are key vocabulary

words and activities you can do with your student. You may also wish to log

on to glencoe.com for other study help. If you have any questions or

comments, feel free to contact me at school.

Sincerely,

Properties

Distributive Property The process of multiplying a sum or difference by a number.

equivalent expressions Two or more expressions that have the same value when the variable in each of them is replaced by the same number.

property A statement that is true for any numbers.

simplify To write an expression in simpler form.

Algebraic Expressions

coefficient The numerical factor of a term that contains a variable.

constant A term without a variable.

like terms Terms that contain the same variables to the same powers.

term Parts of an algebraic expression separated by plus or minus signs.

11Chapter

Distributive PropertyFor any numbers a, b, and c,

a(b + c) = ab + aca(b – c) = ab – ac.

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NAME ________________________________________ DATE _____________ PERIOD _____

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panies, Inc.At-Home Activities11

Chapter

Materials: construction paper, pencil, ruler, scissors

• On a piece of construction paper, draw a rectangle which is 4 inches by 3 inches and a rectangle which is x inches by 3 inches.

• Cut out both retangles. • Write the sum of the two areas. • Find the area of both rectangles. • Write the sum of the two areas. • Place the two rectangles side by side so that they form one larger

rectangle. • Find the length of the longer side. The shorter side is 3 inches. • Write an expression representing the area of the big rectangle.

Compare the areas of both rectangles.

• Ask your parents if they have ever used a shortcut or mental math to find the product of two numbers.

• If their answer is yes, ask them to explain what they did. If their answer is no, brainstorm some possible shortcuts.

• Have your parents explain how they would find the product of 9 and 29.

Hands-On Activity

Real-World Activity

3 in.

4 in.x in.

7 · 68 = 7(60 + 8) = 7 · 60 + 7 · 8 = 420 + 56 = 476


NOMBRE ______________________________________ FECHA ____________ PERÍODO ____

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.11Capítulo

Carta a la familia

Estimado padre o apoderado:

Hoy comenzamos el Capítulo 11: Ecuaciones y desigualdades de varios

pasos y sus propiedades. En este capítulo, su estudiante aprenderá qué son

y cómo se aplican las propiedades asociativa, conmutativa y distributiva.

También aprenderemos a resolver ecuaciones y desigualdades de varios pasos

con variables en ambos lados del signo de igualdad o desigualdad. En esta

carta se incluyen palabras del vocabulario clave y actividades que pueden

realizar con su estudiante. Si desean obtener más ayuda para el estudio,

visiten glencoe.com. Si tienen alguna pregunta o desean hacer algún

comentario, pueden contactarme en la escuela.

Sinceramente,

_____________________

Vocabulario clave

Propiedades

propiedad distributiva Proceso de multiplicar una suma

o diferencia por un número.

expresión equivalente Dos o más expresiones con el mismo valor cuando la variable

en cada una de ellas se reemplaza

con el mismo número.

propiedad Enunciado que se cumple para cualquier número.

simplificar Escribir una expresión de forma más simple.

Expresiones algebraicas

coeficiente Factor numérico de un término que contiene una variable.

constante Término sin variable.

términos semejantes Términos que contienen las mismas variables elevadas a las misma

potencias.

término Partes de una expresión algebraica separadas por signos de adición o sustracción.

Capítulo 11 3 Course 3

Propiedad distributivaPara cualquier número a,

b y c,

a(b + c) = ab + aca(b – c) = ab – ac.


NOMBRE ______________________________________ FECHA ____________ PERÍODO ____

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Actividades para el hogar

Actividad manual

Actividad concreta

Materiales: cartulina, lápiz, regla, tijeras

• En un trozo de cartulina, dibujen un rectángulo de 4 pulgadas por 3 pulgadase un rectángulo de x pulgadas por 3 pulgadas.

• Recorten ambos rectángulos.

• Hallenel área del ambos rectángulos.

• Calculen el área de ambos rectángulos.

• Escriban la suma de las dos áreas.

• Coloquen los dos rectángulos lado a lado de manera que formen un rectángulo más grande.

• Calculen la longitud del lado más largo. El lado más corto mide 3 pulgadas.

• Escriban una expresión que represente el área del rectángulo más grande. Compare las zonas de ambos rectángulos.

• Pregúntenles a sus padres si alguna vez han usado un atajo o el cálculo mental para obtener el producto de dos números.

• Su la respuesta es sí, pídanles que les expliquen qué hicieron. Si la respuesta es no, planteen ideas sobre posibles atajos.

• Pregúntenles a sus padres cómo calcularían el producto de 9 por 29.

Capítulo 11 4 Course 3

3 pulg

4 pulgx pulg

7 · 68 = 7(60 + 8) = 7 · 60 + 7 · 8 = 420 + 56 = 476


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Practice

Are You ReadyAre You Readyfor Chapter 11?for Chapter 11?

1.

2.

3.

4.

5.

6.

7.

8.

9.


Solve each equation. Check your solution.

1. -7 = m + 2

2. w ! 3 = 11

3. 0.5y = -12

4. p - 4 = -6

5. MERIT BADGES Kurt has 9 more merit badges than Johnny. If Kurt has 14 merit badges, write and solve an equation to determine the number of merit badges Johnny has.

Solve each inequality. Graph the solution set on a number line.

6. n - 4 " -3

7. 6 " m ! 2

8. w + 5 > -2

9. q !

-3 < 12

!1 !2 0 1 2 3 4

9 10 11 12 13 14 15

-10 -4-5-6-7-8-9

-39 -33-34-35-36-37-38

001_011_FLCRMC3C11_892310.indd 5001_011_FLCRMC3C11_892310.indd 5 7/25/09 9:41:49 AM7/25/09 9:41:49 AM

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Review


Exercises


1. m – 4 = – 9

2. 4 = p !

–2

3. 8 = x + 5

4. –16 = –2q

5. 18 = h – 12

6. y !

4 = – 6

7. b + 8 = –10

1.

2.

3.

4.

5.

6.

7.

Addition Property of Equality Subtraction Property of EqualityIf you add the same number to each side of an equation, the two sides remain equal.

If you subtract the same number from each side of an equation, the two sides remain equal.

Multiplication Property of Equality Division Property of EqualityIf you multiply each side of an equation by the same number, the two sides remain equal.

If you divide each side of an equation by the same nonzero number, the two sides remain equal.

Solve a + 3 = 7. Check your solution.

a + 3 = 7 Write the equation.

a + 3 – 3 = 7 – 3 Subtract 3 from each side.

a = 4 7 – 3 = 4

Solve m ! 2 = -3. Check

your solution.

m ! 2 = -3 Write the equation.

m ! 2 (2) = -3(2) Multiply each side by 2.

m = -6 -3 " 2 = - 6

Example 1 Example 2


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1. COLLECTING Paloma’s rock collection now has 112 rocks in it. At the beginning of the year she only had 41 rocks in her collection. Write and solve an equation to find out how many rocks she has added to her collection since the beginning of the year.

2. CANDY Kent bought 11 candy bars for $16.50. Write and solve an equation to find the cost of each candy bar.

3. SWIMMING Mrs. Nielsen wants to spend no more than $56 dollars for day passes to the public swimming pool this summer. One day pass costs $2.50. Write and solve an inequality to determine the number of day passes she can purchase if she wishes to stay within her budget.

4. SALES Sabastian sells magazine subscriptions. To date he has sold 13 subscriptions. To be successful, Sabastian believes he must sell at least 51 subscriptions. Write and solve an inequality to determine the minimum number of magazines he still needs to sell.

5. SHARING Carmita is sharing candy with her friends. She has already given away 21 pieces but still has 63 pieces left. Write and solve an equation to determine how many pieces of candy Carmita originally had.

6. TENNIS Jayson wants to spend less than $42 on tennis balls. Each canister of tennis balls costs $4. Write and solve an inequality to determine the maximum number of canisters of tennis balls he can buy.


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Diagnostic Test

1.

2.

3.

4.

5.

6.

7.

8.

9.


1. 0.2g = 8

2. f !

4 = –12

3. –6 = u + 9

4. n – 6 = –5

5. HOURS Lenore works at Happy Babies during the school year. Each week she would like to work exactly 18 hours. This week she has already worked 7 hours. Write and solve an equation to determine how many hours she still needs to work this week to meet her goal.


6. w – 6 > –2

7. c ! –2

" 3

8. –4 < h + 9

9. HEAT Reilly loves hot weather. Each summer he hopes to see at least 45 days with temperatures above 90°. This summer there have already been 12 days with temperatures above 90°. Write and solve an inequality to determine the minimum number of days needed to make Reilly happy.

1 2 3 4 5 6 7

-3-4-5-6-7-8-9

-16 -10-11-12-13-14-15


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.11Chapter

Pretest

Name the property shown by each statement.

1. m + n = n + m

2. (pq)r = p(qr)

3. –8(a – 2) = -8a + 16

Write each expression in simplest form.

4. n + 3n

5. 2p + 4 – 6p – 5

6. CHESS In a chess tournament, Zachary won m matches. His teammate won 2 more matches than he did. Write an expression in simplest form to represent the total number of matches they won.


7. 4n – 3 = –2n + 15

8. 2(2 – x) = 4(–2 + x)

9. VOLUNTEER At the county hospital, Goodwin was told that 4 students could each volunteer for 2 hours in 2 days. At this rate, how many total hours can 8 students volunteer in 10 days?

Solve each inequality. Graph the solution on a number line.

10. 5n ! 3n – 4

11. –3(y + 2) > 2(y + 7)

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

-5 10-1-2-3-4

-1-2-3-4-5-6-7


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Chapter

This is an alphabetical list of new vocabulary terms you will learn in Chapter 11. Fold the page vertically and use it as a bookmark. As you study the chapter, write each term's definition or description in as few words as possible.

Vocabulary Word Definition/Description/Example

coefficient

constant

counterexample

Distributive Property

equivalent expressions

like terms

property

simplest form

simplify

term

Fold over


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.Facts Practice11

Chapter

Solve each equation or inequality.

1. x + 2 = -4 2. 22 < -2m 3. 8 = h ! -3

4. 4r < 20

5. 4 < u + 7 6. 5p = 30 7. n - 11 " -2 8. b + 3 = -7

9. 2 = e ! 4 10. -3q # 12 11. n !

7 = -2 12. -12 # v + 2

13. y !

7 $ -14 14. d - 9 = -11 15. 14 # -6 + y 16. -8 = s + 3

17. c - 10 = -10 18. z - 7 " -9 19. -15 = -5k 20. g + 3 < -9


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A


ReteachProperties

Name the property shown by the statement u + v = v + u.

The order in which the variables are being added changed. This is the Commutative Property of Addition.

State whether the following conjecture is true or false. If false, provide a counterexample.

Subtraction of integers is commutative.

Write two subtraction expressions using the Commutative Property.

17 - 9 ! 9 – 17 State the conjecture.

8 " – 8 Subtract.

We found a counterexample. That is, 17 – 9 " 9 – 17. So, subtraction is not commutative. The conjecture is false.

Simplify the expression. Justify each step.

9 + (3x + 4)

9 + (3x + 4) = 9 + (4 + 3x) Commutative Property of Addition

= (9 + 4) + 3x Associative Property of Addition

= 13 + 3x Simplify.

Exercises


1. 7 · 1 = 7 2. 4 + (3y + 2) = (4 + 3y) + 2

State whether the following conjectures are true or false. If false, provide a counterexample.

3. The product of two even numbers is odd.

4. The difference of two odd numbers is even.

5. Simplify 4 + (5x + 2) . Justify each step.

Example 1

Example 2

Example 3


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Skills PracticeProperties


1. 9 · 6 = 6 · 9 2. m + 0 = m

3. 14 · 1 = 14 4. 2 + (8 + 3) = (2 + 8) + 3

5. x + y = y + x 6. (m + 2) + n = n + (m + 2)


7. The sum of an even whole number and an odd whole number is always odd.

8. Division of whole numbers is always commutative.

Simplify each expression. Justify each step.

9. 5 + (b + 2)

10. 8 (2q)

11. RAIN Piper recorded the amount of rain that fell for four nights in the table below. Use mental math to find the total amount of rain. Explain your reasoning.

Day Monday Tuesday Wednesday ThursdayRain (in.) 2.6 1.5 1.4 2.5


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Homework PracticeProperties


1. 1 · (a + 3) = a + 3 2. 2p + (3q + 2) = (2p + 3q) + 2

3. (ab) c = c (ab) 4. 2t · 0 = 0

5. m (nr) = (mn) r 6. 0 + 2s = 2s


7. The product of an odd number and an even number is always odd.

8. The sum of two whole numbers is always larger than either whole number.

Simplify each expression. Justify each step.

9. 2d (3)

10. 2y + (4 + 5y)

11. FAXES Marcellus sent four faxes to Gem. The first fax took 14 seconds to send, the second fax 19 seconds, the third 16 seconds, and the fourth 11 seconds. Use mental math to find out how many seconds it took to fax all four documents to Gem. Explain your reasoning.

12. SNOW The first four snowfalls of the year in Shawnee’s hometown measured 1.6 inches, 2.2 inches, 1.8 inches, and 1.4 inches. Use mental math to find the total amount of snow that fell. Explain your reasoning.



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Problem-Solving PracticeProperties

1. PROPERTY Alana’s house sits on a rectangular lot with dimensions 62.4 feet by 108.6 feet. Use mental math to find the perimeter.

2. SHOPPING Sera went to the mall and made four purchases. She spent $2.85, $5.11, $7.89, and $4.15. Use mental math to determine how much money Sera spent at the mall.

3. VIDEO GAME Porsche bought a new video game. The first time she played, it took her 24 minutes to reach level 2, the second time it took 18 minutes, the third time it took 16 minutes, and the fourth time it took 12 minutes. Use mental math to determine how many minutes she spent at level 1 while playing these four games.

4. FLOWERS Bethany placed a bouquet of roses in a vase full of water. Each day she recorded how much water had evaporated from the vase before refilling it. The results are shown in the table below. Over the course of five days how much water had evaporated? Use mental math to find your answer.

Day 1 2 3 4 5Evaporation (in.) 0.8 0.2 1.1 0.9 1

5. RECORDS Olympia listened to some old records. The first song lasted 2 minutes and 12 seconds, the second lasted 2 minutes and 16 seconds, the third 2 minutes and 18 seconds, and the fourth 3 minutes and 4 seconds. Use mental math to determine the total playing time for all four records.

6. DISTANCE Anza gave Angela directions to her house from school. Angela was to head south for 2.2 miles, then west for 3.5 miles, then south again for 5.8 miles. Use mental math to determine how far school is from Anza’s house. Explain your reasoning.

7. BUTCHER SHOP Tayshawn saw the following sign in a butcher shop. If he buys one of each item, how much will he spend? Use mental math to help find your answer. Explain your reasoning.

SALERoast - $7.19Bread - $1.56Milk - $2.81Yogurt - $0.44


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Enrich

Other Algebraic PropertiesTwo properties of all real numbers are the Additive Inverse Property and the Multiplicative Inverse Property, as shown below.

Algebraic PropertiesProperty Symbols Example

Additive Inverse Property a + (–a) = 0 5 + (–5) = 0

Multiplicative Inverse Property a · 1 ! a = 1; a " 0 3 · 1 ! 3 = 1

If an operation can be used for all numbers in a set of numbers, then the set is said to be closed under that operation. For example, the set of integers is closed under the operation of addition because the sum of any two integers is an integer.

Questions are: Can you use these two properties with other sets of numbers? Would you be using a number that is not in the set? Is the set closed?

The set of real numbers: The set of real numbers is closed when applying these properties. For example, choose any nonzero real number like 7. Both –7, the additive inverse of 7, and 1 !

7 , the multiplicative inverse of 7, are

real numbers.

Additive Inverse Property: 7 + (–7) = 0

Multiplicative Inverse Property: 7 · 1 ! 7 = 1

The set of real numbers is a closed set, and you can use these two properties because you are using numbers within the set of real numbers.

Determine whether you use the Additive Inverse Property and/or the Multiplicative Inverse Property when working with the sets below. If you cannot use either or both properties, provide an explanation.

1. The set of whole numbers 2. {–1, –2, –3, …}

3. The set of integers 4. The set of rational numbers


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BReteachThe Distributive Property

Distributive PropertyWords To multiply a sum or difference by a number, multiply each

term inside the parentheses by the number outside the parentheses.

Symbols a (b + c) = ab + ac a (b – c) = ab – ac

Examples 3 (2 + 5) = 3 · 2 + 3 · 5 6 (8 – 3) = 6 · 8 – 6 · 3

Use the Distributive Property to evaluate each expression.

5 (7 + 3) 5 (7 + 3) = 5 · 7 + 5 · 3 Distributive Property

= 35 + 15 Multiply.

= 50 Simplify.

(4 – 2)9 (4 – 2) 9 = 4 · 9 - 2 · 9 Distributive Property

= 36 - 18 Multiply.

= 18 Simplify.

MOVIES Alwyn is taking three of his friends to the movies. Tickets cost $8.90 per person. Find Alwyn’s total cost.

You can use the Distributive Property to find the total cost mentally.

4 ($9 – $0.10) = 4 ($9) – 4 ($0.10) Distributive Property

= $36 – $0.40 Multiply.

= $35.60 Subtract.

Alwyn will pay $35.60 for himself and three friends to go to the movies.

Exercises


1. 5 (7 + 4) 2. (3 – 5) (–2) 3. 7 (12 – 5)

4. –6 (4 + 8) 5. 8 (2 + 7) 6. (3 + 6) (–9)

7. BOOKS Mariah bought 7 books costing $11.20 each. Find the total cost of the 7 books. Justify your answer by using the Distributive Property.

Examples

Example 3


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Skills PracticeThe Distributive Property


1. 3 (2 + 8) 2. (–3 + 4) 2

3. –5 (4 – 2) 4. (12 + 13) (–2)

5. 8 (10 – 4) 6. (–4 + –7) (–3)

7. (–7 + 3) 4 8. -1 (18 - 11)

Use the Distributive Property to rewrite each expression.

9. 6 (t + 2) 10. –5 (4 + x)

11. (5 + v) (–3) 12. (w – 2) 4

13. –7 (8 – m) 14. (6 + d) (–6)

15. (4c + 2) (–2) 16. –2f (3f – 5)

17. TRAIN RIDE Mr. and Mrs. Caputo are taking their family into the city on the train. The cost per person is $5.80. If there are 4 members in their family, how much does the train trip cost? Justify your answer by using the Distributive Property.

18. CAMPING Chantee went camping over the weekend. The cost for the site was $16.95 a night for three nights. How much did it cost her to camp? Justify your answer by using the Distributive Property.

B


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B


1. (16 – 6) 2 2. 4 (12 + 3)

3. –3 (–7 + 2) 4. (8 + 3) (–1)

5. 5 (7 + 3) 6. –2 (8 – 5)

Use the Distributive Property to rewrite each expression.

7. (2 + g) 8 8. 4 (h – 5)

9. –7 (5 – n) 10. m (2m + 1)

11. 6x (y – z) 12. –3b (2b – 2a)

13. DINING OUT The table shows the different prices at a diner.

a. Write two equivalent expressions for the total cost if two customers order each of the items.

b. What is the total cost for both customers?

14. SUNDAES Carmine bought 5 ice cream sundaes for his friends. If each sundae costs $4.95, how much did he spend? Justify your answer by using the Distributive Property.

Homework PracticeThe Distributive Property

Item Cost ($)Sandwich $5Drink $2Dessert $3



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Problem-Solving PracticeThe Distributive Property

1. SCHOOL PLAY Marika and her three friends attended the school play. Tickets cost $5.75 each, and Marika paid for everyone. Find the total cost of the tickets. Justify your answer by using the Distributive Property.

2. LUNCH Althea buys a carton of milk each day at school. The milk costs $0.90. How much does she spend on milk during a typical 5-day week? Justify your answer by using the Distributive Property.

3. BOOKSTORE The sign below indicates the cost for several items at Ting’s middle school bookstore. If Ting wants to buy two of each item, how much will it cost? Justify your answer by using the Distributive Property.

Item Price ($)Pencil 1.00Pen 2.50Notebook 3.00

4. HOCKEY The table shows the price of a ticket and food items at a hockey game.

a. Suppose Coleman and two of his friends go to the game. Write an expression that could be used to find the total cost for them to go to the game and buy one of each item.

b. What is the total cost for all three people?

Item Cost ($)

Ticket 7.00Hot dog 3.00Fries 2.25Candy bar

1.50

5. PICTURES Belinda wants to buy 5 pictures to hang in her family room. If each picture costs $30.90, how much will it cost her to buy all five? Justify your answer by using the Distributive Property.

6. FLASH DRIVES Mr. Kaplan is ordering 30 flash drives for the students in his class. If each one costs $11.95, how much will he pay? Justify your answer by using the Distributive Property.

7. FORMULA Mr. and Mrs. Newby are buying baby formula. Each case of formula costs $59.89. If they want to purchase four cases, how much will they pay? Justify your answer by using the Distributive Property.

8. TIRES Mao needs four new tires for his car. Each tire costs $88.70. How much will it cost him to buy the tires? Justify your answer by using the Distributive Property.

B


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Enrich

Another Look at the Distributive PropertySuppose you want to multiply the sum or difference of two numbers or variables by the sum or difference of two numbers or variables. For example, to find x + 2 times x - 3, you would find the following product.

(x + 2) (x – 3)

We can use the Distributive Property to find this product.

(x + 2) (x – 3) = x (x – 3) + 2 (x – 3)

= (x · x – x · 3) + (2 · x – 2 · 3)

= (x2 – 3x) + (2x – 6)

= x2 + (–3x) + 2x – 6

= x2 + (–3x + 2x) – 6

= x2 + (–1x) – 6

= x2 – x – 6

Notice that in this example, the Distributive Property is applied twice. First the x in the expression x + 2 is multiplied by the expression x – 3, and then the 2 in the expression x + 2 is multiplied by the expression x – 3. Then the Distributive Property is applied again to simplify each product in the next step.

The Distributive Property can also be used in similar expressions without variables. For example, to find 12 · 14, rewrite 12 as 10 + 2 and 14 as 10 + 4 and multiply as shown below.

(10 + 2) (10 + 4) = 10 (10 + 4) + 2 (10 + 4)

= (10 · 10 + 10 · 4) + (2 · 10 + 2 · 4)

= 140 + 28

= 168

Exercises

Use the Distributive Property to find each product.

1. 13 · 15 2. (a + 2) (a + 5)

3. (b – 3) (b + 7) 4. (2x + 4) (x + 1)

5. (3y - 1)(y + 2) 6. (x - 5)(x + 5)

B


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ReteachSimplify Algebraic Expressions

When a plus or minus sign separates an algebraic expression into parts, each part is called a term. The numerical factor of a term that contains a variable is called the coefficient of the variable. A term without a variable is called a constant. Like terms contain the same variables to the same powers, such as 3x2 and 2x2.

Identify the terms, like terms, coefficients, and constants in the expression 7x - 5 + x - 3x.

7x - 5 + x - 3x = 7x + (-5) + x + (-3x) Definition of subtraction = 7x + (-5) + 1x + (-3x) Identity Property; x = 1x

The terms are 7x, -5, x, and -3x. The like terms are 7x, x, and -3x. The coefficients are 7, 1, and -3. The constant is -5.

An algebraic expression is in simplest form if it has no like terms and no parentheses.


5x + 3x

5x + 3x = (5 + 3) x or 8x Distributive Property; simplify.

-2m + 5 + 6m - 3

-2m and 6m are like terms. 5 and -3 are also like terms.

-2m + 5 + 6m - 3 = -2m + 5 + 6m + (-3) Definition of subtraction = -2m + 6m + 5 + (-3) Commutative Property = (-2 + 6) m + 5 + (-3) Distributive Property = 4m + 2 Simplify.

Exercises

Identify the terms, like terms, coefficients, and constants in each expression.

1. – 4y – 3 + 2y 2. –5g + 3 + 2g – g 3. 5 + 3a – 4 – a


4. 3d + 6d 5. 2 + 5s – 4 6. 2z + 3 – 9z – 8

Example

C

Examples


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1. 4e + 7e + 5 2. 5a + 2 - 7

3. -3h - 2h + 6h + 9 4. 4 - 4y + y - 3

5. 7 - 5y + 2 + 1 6. 2m + 3m - m

7. 9k + 7 -k + 4 8. -8p + 6p - 2


9. 3t + 6t 10. 4r + r 11. 7f - 2f

12. 9a - 8a 13. 5c + 8c 14. 2g - 5g

15. 8k + 3 + 4k 16. 7m - 5m - 6 17. 9 - 6x + 5

18. 7p - 1 - 9p + 5 19. -b - 3b + 8b + 4 20. 5h - 6 - 8 + 7h

21. 8b + 6 - 8b + 1 22. t - 5 - 2t + 5 23. 4w + 5w + w

24. 6m - 7 + 2m + 7 25. 5f - 7f + f 26. 12y - 8 + 4y + y

Write an expression in simplest form that represents the total amount in each situation.

27. RUNNING You run m miles on Friday, the same amount on Saturday, and 4 miles on Sunday.

28. READING Hendrick read b books in January, twice that amount in February, and 1 book in March.

Skills PracticeSimplify Algebraic ExpressionsC


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1. 4b + 7b + 5 2. 8 + 6t – 3t + t 3. –5x + 4 – x –1

4. 2z – z + 6 5. 4 + h – 8 – h 6. y – y – 2 + 2


7. h + 6h 8. 10k - k 9. 3b + 8 + 2b

10. - 3 ! 4 x - 1 !

3 + 7 !

8 x - 1 !

2 11. 5c - 3d - 12c + d 12. -y + 9z - 16y - 25z

MEASUREMENT Write an expression in simplest form for the perimeter of each figure.

13. 14. 15. 3a - 1

2a + 3

a 4x - 3

4h + 6

5h

2x

2y + 2

3y - 2

2y - 2

2y -1

y

16. SHOPPING Maggie bought c CDs for $12 each, b books for $7 each, and a purse costing $24.

a. Write an expression to show the total amount of money Maggie spent.

b. If Maggie bought 4 CDs and 3 books, how much money did she spend?

Homework PracticeSimplify Algebraic Expressions


C


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CProblem-Solving PracticeSimplify Algebraic Expressions

1. GAMES At the Beltway Outlet store, you buy x computer games for $13 each and a magazine for $4. Write an expression in simplest form that represents the total amount of money you spend.

2. TENNIS Two weeks ago, Star bought 3 cans of tennis balls. Last week, she bought 4 cans of tennis balls. This week, she bought 2 cans of tennis balls. The tennis balls cost d dollars per can. Write an expression in simplest form that represents the total amount that Star spent.

3. AMUSEMENT PARKS Sari and her friends played miniature golf. There were p people in the group. Each person paid $5 for a round of golf and together they spent $9 on snacks. Write an expression in simplest form that represents the total amount that Sari and her friends spent.

4. BICYCLING The bicycle path at the park is a loop that covers a distance of m miles. Dot biked 2 loops each on Monday and Wednesday and 3 loops on Friday. On Sunday, Dot biked 10 miles. Write an expression in simplest form that represents the total distance that Dot biked this week.

5. GEOMETRY Write an expression in simplest form for the perimeter of the triangle below.

2x + 3

4x - 2

2x

6. SIBLINGS Mala is y years old. Her sister is 4 years older than Mala. Write an expression in simplest form that represents the sum of the ages of the sisters.


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Olga Taussky-ToddOlga Taussky-Todd (1906–1995) had a rich and varied career as a research mathematician, mathematics professor, and author and editor of mathematical texts. Born in eastern Europe, she lived and worked in Austria, Germany, England, and the United States. She served as the consultant in mathematics for the National Bureau of Standards in Washington, D.C., for ten years. In 1957, she became the first woman appointed to the mathematics department of the California Institute of Technology.

Dr. Taussky-Todd made contributions in many areas of mathematics and physics. The exercises below will help you learn some more about her life.

Find each product. Circle the correct solution. The phrase following the solution will complete the statement correctly.

1. (x + 3)(x + 7) Her paper on sums of squares won the Ford Prize of the Mathematical Association of America in ?

x2 + 10x + 21: 1971

x2 + 21x: 1981

2. (x + 4)(2x + 1) In 1978, she was elected ? of the Austrian Academy of Sciences.

3x + 5: Correspondent

2x2 + 9x + 4: Corresponding Member

3. (2x + 1)(x + 7) In 1978, the government of Austria awarded her the ? .

2x2 + 15x + 7: Cross of Honor in Science and Arts, First Class

3x2 + 8x + 7: Purple Cross

4. (x + 6)(x + 6) In 1988, the University of Southern California awarded her an ? .

2x + 12: honorary science degree

x2 + 12x + 36: honorary Doctor of Science degree

CEnrich


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Gift cards come in packages of 12 and envelopes come in packages of 15. Meagan needs to send 600 cards in envelopes. How many packages of each kind should she buy?

Understand Meagan needs the same number of cards and envelopes.

Plan Find out how many packages are needed for 300 cards in envelopes.

Solve 12c = 300 15e = 300

c = 25 e = 20

Multiply the answers by 2.

Check 2 ! 25 = 50 packages of cards 2 ! 20 = 40 packages of envelopes

Meagan should buy 50 packages of cards and 40 packages of envelopes.

How many triangles of any size are in the figure at the right?

Understand We need to find how many triangles are in the figure.

Plan Draw a simpler diagram.

Solve 9 Count the smallest triangles, which have 1 triangle per side.

3 Count the next largest triangles, which have 2 triangles per side.

+1 Count the largest triangle, which has 3 triangles per side.

13 Add together to find the total triangles of any size.

Check Now repeat the steps for the original problem.

16 Count the smallest triangles, which have 1 triangle per side.



+1 Count the largest triangle, which has 4 triangles per side.

27 Add together to find the total triangles of any size.

Exercise

Hot dogs come in packages of 10 and buns come in packages of 8. How many packages of each will Blair need to provide 640 hot dogs for a street fair? Use the solve a simpler problem strategy.

ReteachProblem-Solving Investigation: Solve a Simpler ProblemD

Example 1

Example 2


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Use the solve a simpler problem strategy to solve Exercises 1–3.

1. Earline has a square-shaped deep-dish pizza. What is the maximum number of pieces that can be made by using 6 cuts?

2. CDs come in packages of 25 and CD cases come in packages of 16. How many of each type of package will Amanda need to buy in order to make 400 CDs and put them in cases with none left of either?

3. A restaurant has 10 triangular tables that can be pushed together in an alternating up-and-down pattern as shown below to form one long table for large parties. Each triangular table can seat 3 people per side. How many people can be seated at the combined tables?

For Exercises 4–15, rewrite to solve a simpler problem and solve. Find a reasonable answer.

4. 13 ! 29 5. 48 + 32 + 87

6. 74 ! (18 - 9) 7. 33 ÷ 9

8. 57 " 113

9. 55 + 44 + 33

10. 63 ! 17 11. 532 - 389

12. 78 ! 41 - 276 13. 52 + 39 + 111

14. 452 - 377 15. 67 ! 34 ! 12

Skills PracticeProblem-Solving Investigation: Solve a Simpler ProblemD


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Homework PracticeProblem-Solving Investigation: Solve a Simpler Problem

Use the solve a simpler problem strategy to solve Exercises 1 and 2.

1. ASSEMBLY A computer company has two locations that assemble computers. One location assembles 13 computers in an hour and the other location assembles 12 computers in an hour. Working together, how long will it take both locations to assemble 80 computers?

2. AREA Determine the area of the shaded region if the radii of the six circles are 1, 2, 3, 4, 5, and 10 centimeters. Round to the nearest tenth if necessary.

Use any strategy to solve Exercises 3–6. Some strategies are shown below.

PROBLEM-SOLVING STRATEGIESSolve a simpler problem.Look for a pattern.Work backward.Choose an operation.

••••

3. NUMBER SENSE Find the sum of all the even numbers from 2 to 50, inclusive.

4. ANALYZE TABLES Mr. Brown has $1,050 to spend on computer equipment. Does Mr. Brown have enough money to buy the computer, scanner, and software if a 20% discount is given and the sales tax is 5%? Explain.

Item CostComputer $899Scanner $54Software $278

5. COPIER The counter on a business copier read 18,678 at the beginning of the week and read 20,438 at the end of the week. If the business was in operation 40 hours that week, what was the average number of copies made each hour?

6. HUMMINGBIRD In normal flight a hummingbird can flap its wings 75 times each second. At this rate, how many times does a hummingbird flap it wings in a 20-minute flight?

Mixed Problem Solving

D



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Problem-Solving PracticeProblem-Solving Investigation: Solve a Simpler Problem

1. GEOMETRY Alejandro has a large pizza. What is the maximum number of pieces that can be made by using 12 cuts?

2. TABLES A picnic area has 21 square tables that can be pushed together to form one long table for a large group. Each square table can seat 4 people per side. How many people can be seated at the combined tables?

3. PACKAGES Postcards come in packages of 12 and stamps come in packages of 20. How many of each type of package will Jessica need to buy in order to send 300 postcards with no stamps or postcards left over?

4. JOBS Larry can stuff 150 envelopes in one hour. Harold can stuff 225 envelopes in one hour. About how long will it take them to stuff 10,000 envelopes?

5. BUILDING Alexy can lay 40 bricks in one hour. Vashawn can lay 30 bricks in one hour. Jesse can lay 20 bricks in one hour. About how long will it them to build a wall that uses 900 bricks?

6. GEOMETRY How many squares of any size are in the figure?

For Exercises 1 –6, use the solve a simpler problem strategy.

D


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A11-2

Equations with Variables on Each Side

Use algebra tiles to solve 4x + 1 = 2x + 3.

Step 1 Model the equation.

Step 2 Remove the same number of x-tiles from each side of the mat until there are x-tiles on only one side.

Step 3 Remove the same number of 1-tiles from each side of the mat until the x-tiles are by themselves on one side.

Step 4 Separate the tiles into two equal groups.

Therefore x = 1. Since 4(1) + 1 = 2(1) + 3, the solution is correct.

Use algebra tiles to solve each equation.

1. 3x + 1 = 2x + 4 2. x – 1 = 2x + 3 3. 2x + 6 = 5x – 3

4. Explain why you can remove a 1-tile from each side of the mat.

5. MAKE A CONJECTURE In the set of algebra tiles, –x is represented by -

Explain how you could use the –x-tiles and other algebra tiles to solve –2x – 1 = –x – 3.

1 11

1

4x + 1 2x + 3

=

=

1

=4x - 2x + 1 2x - 2x + 3

=11

1

1

1

1 1

=2x + 1 - 1 3 - 1

=

1

1

2x 2

=

=


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BReteachSolve Equations with Variables on Each Side

Example

Solve 3x - 9 = 6x. Check your solution.

3x - 9 = 6x Write the equation.

3x - 3x - 9 = 6x - 3x Subtraction Property of Equality

-9 = 3x Simplify by combining like terms.

-3 = x Mentally divide each side by 3.

To check your solution, replace x with -3 in the original equation.

Check 3x - 9 = 6x Write the equation.

3(-3) - 9 ! 6(-3) Replace x with -3.

-18 = -18 " The sentence is true.

The solution is -3.

Example

Solve 4a - 7 = 5 - 2a.

4a - 7 = 5 - 2a Write the equation.

4a + 2a - 7 = 5 - 2a + 2a Addition Property of Equality

6a - 7 = 5 Simplify by combining like terms.

6a - 7 + 7 = 5 + 7 Addition Property of Equality

6a = 12 Simplify.

a = 2 Mentally divide each side by 6.

The solution is 2. Check this solution.

Exercises


1. 6s - 10 = s 2. 8r = 4r - 16 3. 25 - 3u = 2u

4. 14t - 8 = 6t 5. k + 20 = 9k - 4 6. 11m + 13 = m + 23

7. -4b - 5 = 3b + 9 8. 6y - 1 = 27 - y 9. 1.6h - 72 = 4h - 30

10. 8.5 - 3z = -8z 11. 10x + 8 = 5x - 3 12. 16 - 7d = -3d + 2

Some equations, like 3x - 9 = 6x, have variables on each side of the equals sign. Use the Addition or Subtraction Property of Equality to write an equivalent equation with the variables on one side of the equals sign. Then solve the equation.


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BSkills PracticeSolve Equations with Variables on Each Side


1. 3w + 6 = 4w 2. a + 18 = 7a

3. 8c = 5c + 21 4. 11d + 10 = 6d

5. 2e = 4e - 16 6. 7v = 2v - 20

7. 4n - 6 = 10n 8. 2y + 27 = 5y

9. 8h = 6h - 14 10. 18 - 2g = 4g

11. 4x - 9 = 6x - 13 12. 5c - 15 = 2c + 6

13. t + 10 = 7t - 14 14. 8z + 6 = 7z + 4

15. 2e - 12 = 7e + 8 16. 9k + 6 = 8k + 13

17. 2d + 10 = 6d - 10 18. -2a - 9 = 6a + 15

19. 8 - 3k = 3k + 2 20. 7t - 4 = 10t + 14

21. 3c - 15 = 17 - c 22. 14 + 3n = 5n - 6

23. 3y + 5.2 = 2 - 5y 24. 10b - 2 = 7b - 7.4

25. 2m - 2 = 6m - 4 26. 3g + 5 = 7g + 4

27. 4s - 1 = 8 - 2s 28. 9w + 3 = 4w - 9

29. 6z - 7 = 2z - 2 30. 3 - a = 4a + 12


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1. 9m + 14 = 2m 2. 13x = 32 + 5x 3. 8d - 25 = 3d

4. t - 27 = 4t 5. 7p - 5 = 6p + 8 6. 11z - 5 = 9z + 7

7. 12 - 5h = h + 6 8. 4 - 7f = f -12 9. -6y + 17 = 3y -10

10. 3x - 32 = -7x + 28 11. 3.2a - 16 = 4a 12. 16.8 - v = 6v

Define a variable, write an equation, and solve to find each number.

13. Fourteen less than five times a number is three times the number.

14. Twelve more than seven times a number equals the number less six.

Write an equation to find the value of x so that each pair of polygons has the same perimeter. Then solve.

15.

+

+

+

+

+

16.

+ +

+

Exercises

Write and solve an equation to solve each exercise.

17. GOLF For an annual membership fee of $500, Mr. Bailey can join a country club that would allow him to play a round of golf for $35. Without the membership, the country club charges $55 for each round of golf. How many rounds of golf would Mr. Bailey have to play for the cost to be the same with and without a membership?

18. MUSIC Marc has 45 CDs in his collection, and Corinna has 61. If Marc buys 4 new CDs each month and Corinna buys 2 new CDs each month, after how many months will Marc and Corinna have the same number of CDs?

Homework PracticeSolve Equations with Variables on Each SideB



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Write and solve an equation to solve each exercise.

Problem-Solving PracticeSolve Equations with Variables on Each Side

1. PLUMBING A1 Plumbing Service charges $35 per hour plus a $25 travel charge for a service call. Good Guys Plumbing Repair charges $40 per hour for a service call with no travel charge. How long must a service call be for the two companies to charge the same amount?

2. EXERCISE Mike’s Fitness Center charges $30 per month for a membership. All-Day Fitness Club charges $22 per month plus an $80 initiation fee for a membership. After how many months will the total amount paid to the two fitness clubs be the same?

3. SHIPPING The Lone Star Shipping Company charges $14 plus $2 a pound to ship an overnight package. Discount Shipping Company charges $20 plus $1.50 a pound to ship an overnight package. For what weight is the charge the same for the two companies?

4. MONEY Deanna and Lise are playing games at the arcade. Deanna started with $15, and the machine she is playing costs $0.75 per game. Lise started with $13, and her machine costs $0.50 per game. After how many games will the two girls have the same amount of money remaining?

5. MONEY The Wayside Hotel charges its guests $1 plus $0.80 per minute for long distance calls. Across the street, the Blue Sky Hotel charges its guests $2 plus $0.75 per minute for long distance calls. Find the length of a call for which the two hotels charge the same amount.

6. COLLEGE Duke is a part-time student at Horizon Community College. He currently has 22 credits, and he plans to take 6 credits per semester until he is finished. Duke’s friend Kila is also a student at the college. She has 4 credits and plans to take 12 credits per semester. After how many semesters will Duke and Kila have the same number of credits?

B


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BEnrich

Famous Scientific EquationsMany important laws or principles in physical science are described by equations. You may have already studied some of the equations on this page, or you may learn about them in future science classes.

Match each statement with its equation. Then write the variables for the quantities listed.

SCIENTIFIC PRINCIPLE

1. Law of the Lever A lever will balance if the mass of

object 1 times its distance from the fulcrum equals the mass of object 2 times its distance from the fulcrum.

2. Newton’s Second Law of Motion The acceleration of an object equals the

applied force divided by the object’s mass.

3. Ohm’s Law The amount of current in an electrical

circuit equals the voltage divided by the resistance.

4. Boyle’s Law For a gas at a constant temperature,

the product of the pressure and the volume remains constant.

5. Law of Universal Gravitation To compute the force of gravity between

two objects, multiply their masses by the gravitational constant and then divide by the square of the distance between the objects.

EQUATION AND VARIABLES

P1V1 = P2V2

pressure at first time =

volume at first time =

pressure at second time =

volume at second time =

I = V ! R

voltage =

resistance =

current =

A = F ! m applied force =

mass of object =

acceleration =

F = G m1m2 !

d2

mass of first object =

mass of second object =

distance between objects =

gravitational constant =

force of gravity =

m1d1= m2d2

mass of first object =

distance of first object from

fulcrum =

mass of second object =

distance of second object

from fulcrum =


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BTI-84 Plus ActivityEvaluating Equations

You can use your TI-84 Plus Graphing Calculator to see if a number is the solution to an equation. If the number is the solution to the equation the calculator will display the number 1. If the number is not the solution to the equation the calculator will display the number 0.

Is -5 the solution to the equation 5x + 2 = 3x -8?

Use the following keystrokes to help answer the question.

)( 5 X,T,!,n

Now enter the equation 5x + 2 = 3x - 8 on your TI-84 Plus view screen and press the key. (Note: the = sign is under the TEST menu.) The number

1 will show up on the right side of your view screen, indicating that the equation is true for x = -5. Thus -5 is the solution to the equation.

Is 2 the solution to the equation -4x - 1 = -3x + 5?

Use the following keystrokes to help answer the question.

2 X,T,!,n

Now enter the equation -4x - 1 = -3x + 5 on your TI-84 Plus view screen and press the key. The number 0 will show up on the right side of your

view screen, indicating that the equation is false for x = 2. Thus 2 is not the solution to the equation. Repeat the process with x = -6 to verify that !6 is the solution to the equation.

Exercises

Check to see if the given number is the solution to the equation.

1. 7x - 1 = 5x - 9; x = -4 2. -4x + 2 = 2x + 14; x = -3

3. -2x + 8 = 5x - 6; x = -2 4. 0.5x + 3 = 1.5x - 2; x = 5

Example 1

Example 2


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CReteachSolve Multi-Step Equations

Example 1

Solve 2(4a ! 5) = 30.

2(4a - 5) = 30 Write the equation.

8a - 10 = 30 Distributive Property

8a - 10 + 10 = 30 + 10 Addition Property of Equality

8a = 40 Simplify.

8a ! 8 = 40 !

8 Division Property of Equality

a = 5 Simplify.

Example 2

BOOKS Roland has

p + 11 p + 11 p + 11 p + 11 p + 11

hardcover hardcover hardcover hardcoverhardcover

p p p

paperback paperbackpaperback 3 paperback books and 5 hardcover books. Each hardcover book is worth $11 more than each paperback book. If the value of all of his books is $95, what is the cost of one paperback book?

Write an equation to represent the bar model.

3p + 5(p + 11) = 95 Write the equation.

3p + 5p + 55 = 95 Distributive Property

8p + 55 = 95 Simplify.

8p + 55 + (–55) = 95 + (–55) Addition Property of Equality

8p = 40 Simplify.

8p

! 8 = 40 !

8 Division Property of Equality

p = 5 Simplify.

Exercises


1. 2(3b – 1) = 40 2. 49 = –7(t + 1) 3. 5(1 – n) = 75

4. 4(x – 2) = 3(x – 3) 5. –5(p + 2) = 2(2p – 15) + p 6. 4z – 6 = 6(z + 2) + 8


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CSkills PracticeSolve Multi-Step Equations


1. 4(2 + 3c) = 56 2. 63 = –3(1 – 2n)

3. –29 = 5(2a – 1) + 2a 4. –g + 2(3 + g) = –4(g + 1)

5. 7p – (3p + 4) = –2(2p – 1) + 10 6. –3(t + 5) + (4t + 2) = 8

7. 1 ! 2 (–4 + 6x) = 1 !

3 x + 2 !

3 (x + 9) 8. –8 – n = –3(2n – 4)

9. 2 ( 1 ! 2 q + 1) = –3(2q – 1) + 4(2q + 1) 10. –4(2 – y) + 3y = 3(y – 4)

11. HEALTH CLUB Currently, 96 members participate in the morning workout, and this number has been increasing by 2 people per week. Currently, 80 members participate in the afternoon workout, and this number has been decreasing by 3 people per week. In how many weeks will the number of people working out in the morning be double the number of people working out in the afternoon?

12. DISTANCE Two cyclists leave town at the same time on the same road going in the same direction. Cyclist A is going 6 miles per hour faster than cyclist B. After 8 hours cyclist A has traveled three times the distance as cyclist B. How fast is cyclist B traveling?


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Solve Multi-Step EquationsC



1. 5(x - 3) + 2x = 41 2. 4a - 3(a - 2) = 2(3a - 2)

3. (7t - 2) - (-3t + 1) = –3(1 – 3t) 4. 14 - 2(3p + 1) = 6(4 + p)

5. 2 ! 7 (14q + 7 !

2 ) - 3q = 9 6. x - (4x - 7) = 5x - (x + 21)

7. BACKPACKING Guido and Raoul each went backpacking in Glacier National Park. The expressions 4(d + 2) – 2d and 3(2 + d) represent the respective distances Guido and Raoul hiked each day. On what day number d will their distance hiking be the same?

8. SAVINGS The table at the right shows the savings Sibling Account Balance

Cindy s

Petros 2(s + 3)

Nila 4s – 5

account balance of each of the Alvarez siblings.

a. Write an equation to find the amount of money in Petros’s account if the total of all of their accounts is $148.

b. Solve the equation from part a to find the amount of money in Petros’s account.

9. LAWNS Luisa mows lawns during the summer. She charges $15 if she cuts the grass but charges $5 more if she also trims the grass. Last week she trimmed 5 more yards than she cut. If she made $415 last week, how many yards did she trim?


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CProblem-Solving PracticeSolve Multi-Step Equations

1. AGES Mel is 3 years older than Rahfat and Aurelio is twice as old as Mel. The sum of their ages is 57. How old is Mel?

2. SALES Ye has his own business. He checks his sales receipts three times a day. One day, his afternoon sales were $50 more than his morning sales, and his evening sales were three times his afternoon sales. If his total sales for the day were $1,000, what were his evening sales?

3. POLYGONS The triangle and square shown below have the same perimeter. What is the length of one side of the square?

x + 2

x + 2

3x

4x

5x

4. PRESENTS Torrance is buying presents for members of his family. He wants to spend $10 less on his brother than he spends on his sister, and six dollars more than twice the amount he spends on his sister on his mother. If Torrance has $100 to spend, how much does he intend to spend on his brother?

5. NUMBERS Pasha is thinking of a number such that when twice the number is added to three times one more than the number she gets the same result as when she multiplies four times one less than the number. What number is Pasha thinking about?

6. SAVINGS Garland put 2b + 3 dollars in the bank in the first week. The following week he doubled the first week’s savings and put that amount in the bank. The next week he doubled what was in the bank and put that amount in the bank. If he now has $477 in the bank, how much did he put in the bank the first week?

7. FOOD Nendell saw the following sign at a diner. If he bought one of each item and spent $7.50, how much did the drink cost?

Item Cost ($)

Burger 3x + 0.05

Fries x

Drink x + 0.10

8. WORK Colby worked three more hours on Tuesday than he did on Monday. On Wednesday, he worked one hour more than twice the number of hours that he worked on Monday. If the total number of hours is two more than five times the number of hours worked on Monday, how many hours did he work on Monday?


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Solve Equations with FractionsYou want to solve the equation 1 !

2 x + 1 !

3 x = 5.

Method 1 Find equivalent fractions that have common denominators.

1 ! 2 x + 1 !

3 x = 5

3 ! 3 ( 1 !

2 ) x + 2 !

2 ( 1 !

3 ) x = 5 Use equlivalent forms of one.

3 ! 6 x + 2 !

6 x = 5 Simplify.

5 ! 6 x = 5 Use the Distributive Property.

6 ! 5 ( 5 !

6 ) x = 6 !

5 (5) Multiplication Property of Equality

x = 6 Simplify.

Method 2 Use the Multiplication Property of Equality to multiply each term on both sides of the equation by the common denominator of the fractions.

1 ! 2 x + 1 !

3 x = 5

6 ( 1 ! 2 ) x + 6 ( 1 !

3 ) x = 6(5) The LCD is 6.

3x + 2x = 30 Simplify.

5x = 30 Use the Distributive Property.

5x ! 5 = 30 !

5 Divide each side by 5.

x = 6 Simplify.

The second method eliminates the fractions prior to combining like terms.

Exercises

Solve each equation by using Method 2.

1. 1 ! 5 x + 1 !

4 x = 18 2. 2 !

3 m + 1 !

5 m = 26

3. 3 ! 4 y – 5 !

6 y = 2 4. 2 !

3 x + 3 !

4 x = 0.34

5. 2 ! 3 m + 3 !

4 m = 17 6. 1 !

3 p " 1 !

6 p = -3


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You can use your TI-84 Plus Graphing Calculator to solve multi-step equations.

Example

Solve 2x – 1 = (4 – 4x) + (x + 10).

You will need to use the following keystrokes.

= 2 X,T,!,n 1

( 4 4 X,T,!,n ) ( X,T,!,n 10 )

6

[CALC] 5

You will now see two intersecting lines and a blinking cursor. Press the calculator key several times and observe that the blinking cursor gets

closer and closer to the point of intersection. Continue pressing the key until the blinking cursor is close to the point of intersection of the two lines. The blinking cursor does not have to be exactly on the point of intersection. (Note: On some problems you may have to press the calculator key to get the

cursor to move towards the point of intersection.) Now press three times. At the bottom of the view screen you will see x = 3 and y = 5. The x-value 3 is the solution to the equation.

Check :

2x - 1 = (4 - 4x) + (x + 10) 2x - 1 = 14 - 3x 2x - 1 + 3x = 14 - 3x + 3x 5x - 1 = 14 5x - 1 + 1 = 14 + 1 5x = 15 x = 3 !

Exercises

Solve each multi-step equation.

1. 4x - (3x + 1) = -5 - x 2. x + 7 = (5 + 2x) - (4 + 3x) 3. 2x - (x + 5) = -1(x + 1)

TI-84 Plus ActivitySolve Multi-Step Equations


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DReteachSolve Multi-Step Inequalities

Example

BASKETBALL Millie has scored 13, 10, 15, 9, and 18 points in her first five of six games. She would like to finish with an average of at least 14 points per game. Write and solve an inequality to find the minimum number of points she must score in the last game. Graph the solution set on a number line.

Words The total points divided by the number of games is at least 14.

Variables Let g represent the number of points needed.

Inequality 13 + 10 + 15 + 9 + 18 + g

!! 6 " 14

13 + 10 + 15 + 9 + 18 + g

!! 6 " 14 Write the inequality.

65 + g

! 6 " 14 Add.

65 + g

! 6 # 6 " 14 # 6 Multiplication Property of Inequality

65 + g " 84 Simplify.

65 + g - 65 " 84 - 65 Subtraction Property of Inequality

g " 19 Simplify.

So, Millie must score 19 points in her last game to have a scoring average of at least 14 points.

212019181716 22

Exercises


1. 3(2x - 1) < 39 2. 1 - 5x " -2(x + 4)

101 2 3 4 5 6 7 8 9

3. 2(4x - 4) " 16 4. 2 - 4x $ 2(3x + 6)

43210-1 5

210-1-2-3 3

-5 43210-1-2-3-4 5


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1. 2(4q + 2) > 52 2. -3(3 + 2t) > 45

3. 1 ! 2 (6p - 8) " 17 4. -2(4 - 2m) < 32

5. 4(2n - 1) # -2(n - 13) 6. -1(1 + 3p) < -1(-4 - 2p)

7. 2(3g - 4) " 4(g - 4) 8. -3(2 + a) > -2(a + 1)

9. ELECTRICIAN Patty called an electrician to fix a wiring problem. The electrician charges $80 for the service call and $45 dollars an hour. If the electrician told Patty that her bill would be no more than $260, what is the most number of hours he will work?

10. CHARITY Cain collects money on the weekends for his favorite charity. The past four weekends he collected $105, $170, $132, and $165. How much does he have to collect on the fifth weekend to average at least $150 for the five weekends?

Skills PracticeSolve Multi-Step Inequalities


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DHomework PracticeSolve Multi-Step Inequalities


1. -3(2b - 1) ! 45 2. -1(4 - 2c) > 18

3. 4(3m + 2) < 56 4. 2(3p + 1) " 5(p - 2)

5. -2(n ! 3) > -4(-1 - n) 6. 5(1 - 2e) ! -11(e - 2)

7. DIVING Fredrico has earned a score of 7.2, 8.4, and 8.4 on his first three dives. He has one dive left. What score must he get on his last dive to have an average of at least 7.4 on all four dives?

8. PERIMETER A square has side lengths of x + 3 inches. If the perimeter of the square is at least 100 inches, what is the minimum length of each side of the square?

9. CARS Neva is renting a motor home to use while she is on vacation. The rental store charges a $200 deposit plus a $90 rental fee per day. If Neva has at most $1,100 to spend on a motor home rental, how many days can she go on vacation?



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1. BOWLING Hardy and his brother Ralph went bowling. Ralph’s average score for his three games is 110. Hardy scored 101 and 113 in his first two games. If Hardy wants his average score for three games to be greater than his brother’s average, what is the least score for the third game?

2. LOANS Carmen borrowed money from her sister. Each month she makes four payments, with an average payment of no more than $200. This month she has already paid her sister $225, $245, and $235. What is the maximum amount she can pay for the fourth payment?

3. BUDGET Kjel has budgeted no more than $55 a week for lunches. The table shows how much he spent for lunch on four of five days last week. If Kjel stayed within his budget, what is the maximum cost for lunch on Wednesday?

Day Lunch ($)Monday $12.00Tuesday $10.50Wednesday ?Thursday $11.25Friday $10.00

4. GROCERIES Lila wants to spend no more than $22 at the grocery store. The receipt below shows what Lila bought and what each item cost. The price of the last item is missing. What is the maximum cost of the pizza?

5. RENTALS Breana is renting skis. The rental store charges $30 plus $9 for each hour or partial hour. If she has $92 dollars to spend, how many hours can she rent the skis?

6. BASEBALL Jacob plays on his high school baseball team. Jacob got 42, 53, and 47 hits for the first three seasons. If Jacob wants to average at least 50 hits per season over his high school career, what is the minimum number of hits he needs to fulfill his goal?

Problem-Solving PracticeSolve Multi-Step InequalitiesD

Sales RecepitBread $2.79Roast $9.11

Coffee $6.50Pizza


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Multi-Step Inequalities and Absolute ValueConsider the equation |x| = 4. x = 4 and -4 are both solutions. Now consider the inequality |x| < 7. You may think that the solution is all numbers less than 7, but -8 is not a solution. The solution is all numbers smaller than 7 or larger than -7, or -7 < x < 7. Does this work for other inequalities?

Example 1

Find the solution to the inequality |x| ! 2.

You may think that the solution is all numbers less than or equal to 2, but -6 is less than or equal to 2 but it is not a solution. The solution is all numbers between and including -2 and 2, or -2 ! x ! 2. Verify the solutions for yourself.

Use this idea to solve the inequality |m - 4| < 9. Think of m - 4 as x, and write the following: |m - 4| < 9 means -9 < m - 4 < 9. Now use the Addition Property of Inequality to get the following.

-9 < m - 4 < 9 Write the inequality.

-9 + 4 < m - 4 + 4 < 9 + 4 Addition Property of Inequality

-5 < m < 13 Simplify.

The solution is -5 < m < 13, or all numbers between !5 and 13.

Example 2

Find the solution to the inequality !2(x + 5)" ! 8.

-8 ! 2(x + 5) ! 8 Write the inequality.

-4 ! x + 5 ! 4 Division Property of Inequality; simplify.

-9 ! x ! -1 Subtraction Property of Inequality; simplify.

The solution is -9 ! x ! -1, or all numbers between and including !9 and !1.

Exercises

Solve each inequality.

1. " t # < 3 2. "3m# ! 18 3. "y $ 2# ! 12

4. "q + 3# < 5 5. "3(2x - 4)# ! 12 6. "-2(x + 1)# < 14


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DTI-84 Plus ActivitySolve Multi-Step Inequalities

You can use your TI-84 Plus Graphing Calculator to solve multi-step inequalities.

Example

Solve 2(x - 1) ! 3(x + 1).

You will need to use the following keystrokes.

= 2 ( X,T,!,n 1 )

[TEST] 6 3 ( X,T,!,n 1 )

6

You will see the screen at the right on your TI-84 Plus view screen.

Notice that the horizontal line graphed starts at x = -5 and goes to the right one unit above the x-axis. This graph tells us that the solution to the inequality is x " -5.

Exercises

Use a calculator to solve each multi-step inequality.

1. -2(t - 3) < 4 2. 1 - 5x " -2(x + 4)

3. -1(1 + 3x) < -1(-4 - 2x) 4. 2(q + 3) > 3(q - 1) + 2q

5. 6(-2 + 3x) < -2(x - 4) 6. -3(1 - 2x) -3x > 3x + 2

7. (12 - 4x)5 > -3(-x - 4) + x 8. 6(-1- 3x) < 7(x +2) + 5x

1"5


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1. Explain the difference between a constant term and the coefficient of a variable.

2. Explain the difference between the Commutative Property and the Associative Property.

3. Explain how you would use the Distributive Property to multiply the numbers 6 and 99. Illustrate the process.


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ISBN: 978-0-07-892310-4MHID: 0-07-892310-7

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Contents Chapter 0 Start Smart Chapter 1 Rational Numbers and Percent Chapter 2 Expressions and Functions Chapter 3 Linear Functions and Systems of Equations Chapter 4 Equations and Inequalities Chapter 5 Operations on Real Numbers Chapter 6 Angles and Lines Chapter 7 Similar Triangles and the Pythagorean Theorem Chapter 8 Data Analysis Chapter 9 Units of Measure Chapter 10 Measurement: Area and Volume Chapter 11 Properties and Multi-Step Equations and InequalitiesChapter 12 Nonlinear Functions and Polynomials

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CChapter 11 hapter 11 RResource Mastersesource Mastersmrsaltsman.com/advdocs/ch11.pdfie s n c. Chapter 11 50 3.Explain const variable coefficient 2. Explain differen Commutative Properand