chapter 9 resource masters - ktl math classes · chapter 9 test, form 3 . . . . . . . . . . ....

Chapter 9Resource Masters

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-828029-XSkills Practice Workbook 0-07-828023-0Practice Workbook 0-07-828024-9

ANSWERS FOR WORKBOOKS The answers for Chapter 9 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teacher, and families without charge; and be used solely in conjunction with Glencoe’s Algebra 2. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

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

ISBN: 0-07-828012-5 Algebra 2Chapter 9 Resource Masters

1 2 3 4 5 6 7 8 9 10 066 11 10 09 08 07 06 05 04 03 02

Glencoe/McGraw-Hill

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Lesson 9-1Study Guide and Intervention . . . . . . . . 517–518Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 519Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 520Reading to Learn Mathematics . . . . . . . . . . 521Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 522






Chapter 9 AssessmentChapter 9 Test, Form 1 . . . . . . . . . . . . 553–554Chapter 9 Test, Form 2A . . . . . . . . . . . 555–556Chapter 9 Test, Form 2B . . . . . . . . . . . 557–558Chapter 9 Test, Form 2C . . . . . . . . . . . 559–560Chapter 9 Test, Form 2D . . . . . . . . . . . 561–562Chapter 9 Test, Form 3 . . . . . . . . . . . . 563–564Chapter 9 Open-Ended Assessment . . . . . . 565Chapter 9 Vocabulary Test/Review . . . . . . . 566Chapter 9 Quizzes 1 & 2 . . . . . . . . . . . . . . . 567Chapter 9 Quizzes 3 & 4 . . . . . . . . . . . . . . . 568Chapter 9 Mid-Chapter Test . . . . . . . . . . . . . 569Chapter 9 Cumulative Review . . . . . . . . . . . 570Chapter 9 Standardized Test Practice . . 571–572

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A29

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 9 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 9 Resource Masters includes the core materials neededfor Chapter 9. These materials include 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 in theAlgebra 2 TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 9-1.Encourage them to add these pages to theirAlgebra 2 Study Notebook. Remind them to add definitions and examples as theycomplete each lesson.

Study Guide and InterventionEach lesson in Algebra 2 addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

© Glencoe/McGraw-Hill v Glencoe Algebra 2

Assessment OptionsThe assessment masters in the Chapter 9Resource Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

All of the above tests include a free-response Bonus question.

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Algebra 2. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of algebra concepts invarious formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and quantitative-comparison questions. Bubble-in andgrid-in answer sections are provided onthe master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 518–519. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

99

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 9.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to your AlgebraStudy Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

asymptote

A·suhm(p)·TOHT

complex fraction

constant of variation

continuity

KAHN·tuhn·OO·uh·tee

direct variation

inverse variation

IHN·VUHRS

joint variation

(continued on the next page)

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

point discontinuity

rational equation

rational expression

rational function

rational inequality

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

99

Study Guide and InterventionMultiplying and Dividing Rational Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

© Glencoe/McGraw-Hill 517 Glencoe Algebra 2

Less

on

9-1

Simplify Rational Expressions A ratio of two polynomial expressions is a rationalexpression. To simplify a rational expression, divide both the numerator and thedenominator by their greatest common factor (GCF).

Multiplying Rational Expressions For all rational expressions and , � � , if b � 0 and d � 0.

Dividing Rational Expressions For all rational expressions and , � � , if b � 0, c � 0, and d � 0.

Simplify each expression.

a.

� �

b. �

� � � �

c. �

� � �

� �


1. �(�220aabb

2

4)3

� � 2. 3.

4. � 2m2(m � 1) 5. �

6. � m 7. �

8. � 9. �

4��

p(4p � 1)��

4m2 � 1��4m � 8

2m � 1��m2 � 3m � 10

4p2 � 7p � 2��

7p516p2 � 8p � 1��

14p4

y5�18xz2

�5y6xy4�25z3

m3 � 9m��

m2 � 9(m � 3)2

��m2 � 6m � 9

c�c2 � 4c � 5

��c2 � 4c � 3

c2 � 3c�c2 � 25

4m5�m � 1

3m3 � 3m��

6m4

x � 2�x2 � x � 6

��x2 � 6x � 27

3 � 2x�4x2 � 12x � 9

��9 � 6x2a2b2�

x � 4�2(x � 2)

(x � 4)(x � 4)(x � 1)��2(x � 1)(x � 2)(x � 4)

x � 1��x2 � 2x � 8

x2 � 8x � 16��2x � 2

x2 � 2x � 8��x � 1

x2 � 8x � 16��2x � 2

x2 � 2x � 8��x � 1

x2 � 8x � 16��2x � 2

4s2�3rt2

2 � 2 � s � s��3 � r � t � t

3 � r � r � s � s � s � 2 � 2 � 5 � t � t��5 � t � t � t � t � 3 � 3 � r � r � r � s

20t2�9r3s

3r2s3�

5t4

20t2�9r3s

3r2s3�5t4

3a�2b2

2 � 2 � 2 � 3 � a � a � a � a � a � b � b��2 � 2 � 2 � 2 � a � a � a � a � b � b � b � b

24a5b2�(2ab)4

24a5b2�(2ab)4

ad�bc

c�d

a�b

c�d

a�b

ac�bd

c�d

a�b

c�d

a�b

ExampleExample

ExercisesExercises

1 1 1 1

1 1

1 1

1 1 1

1 1 1 1 1 1 1

11 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1


Simplify Complex Fractions A complex fraction is a rational expression whosenumerator and/or denominator contains a rational expression. To simplify a complexfraction, first rewrite it as a division problem.

Simplify .

� � Express as a division problem.

� � Multiply by the reciprocal of the divisor.

� Factor.

� Simplify.

Simplify.

1. 2. 3. (b � 1)2

4. 5.

6. a � 4 7. x � 3

8. 9.1

�

x2 � x � 2��x3 � 6x2 � x � 30��x � 1

�x � 3

b � 4��

�b2 �

b �6b

2� 8

�

��b2

b2�

�b

1�6

2�

�2x2

x�

�9x

1� 9

�

��105xx

2

2��

179xx�

�26

�

�aa

2 ��

126

�

��aa

2

2��

3aa

��

24

�

1��

�x2 �

x �6x

4� 9

�

��x2 �

3 �2x

x� 8

�

2(b � 10)��b(3b � 1)

�b2 �

b2100�

��3b2 � 3

21bb � 10�

�3bb

2 ��

12

�

��3b2

b�

�b1� 2

�

ac7�

�ax

2

2byc2

3�

��ca4xb

2

2

y�

xyz�

�xa

3

2yb

2

2z

�

��a3

bx2

2y�

s3�s � 3

(3s � 1)s4��s(3s � 1)(s � 3)

s4��3s2 � 8s � 3

3s � 1�s

3s2 � 8s � 3��

s43s � 1�s

�3s

s� 1�

��3s2 �

s84s � 3�

�3s

s� 1�

��3s2 �

s84s � 3�

Study Guide and Intervention (continued)

Multiplying and Dividing Rational Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

ExampleExample

1

1 1

s3

ExercisesExercises

Skills PracticeMultiplying and Dividing Rational Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1


Less

on

9-1


1. 2.

3. x6 4.

5. 6.

7. 8. �

9. � 6e 10. �

11. � 21g3 12. �

13. � x (x � 2) 14. �

15. � 16. �

(w � 8)(w � 7)

17. � (3x2 � 3x) 18. �

19. � 20.a � b�

�a2

4�a

b2�

��a

2�a

b�

5�

�2cd

2

2�

�

��5cd

6�

(4a � 5)(a � 4)��4a � 5

��a2 � 8a � 16

16a2 � 40a � 25��

3a2 � 10a � 81

�x2 � 5x � 4��2x � 8

t � 12�2t � 2

��t2 � 9t � 14

t2 � 19t � 84��4t � 4

w2 � 6w � 7��w � 3

w2 � 5w � 24��w � 1

q2�

q2 � 4�

3q2q2 � 2q�6q

3x�x2 � 4

3x2�x � 2

32z7�

25y5�14z12v5

80y4�49z5v7

1�

7g�y2

1��s � 2

�10s5

5s2�s2 � 4

10(ef)3�

8e5f24e3�5f 2

mn2�n3

�63m�2n

a � 8�3a2 � 24a

��3a2 � 12a

x � 2�x2 � 4

��(x � 2)(x � 1)9

�18�2x � 6

2�

8y2(y6)3�

4y24(x6)3�(x3)4

b�5ab3

�25a2b2

3x�

21x3y�14x2y2



1. 2. � 3.

4. 5. �

6. 7. � �

8. � 9. �

10. � n � w 11. � �

12. � 13. �

14. � �3� 15. �

16. � 17. �

18. � � 19.

20. �2(x � 3) 21.

22. GEOMETRY A right triangle with an area of x2 � 4 square units has a leg thatmeasures 2x � 4 units. Determine the length of the other leg of the triangle.x � 2 units

23. GEOMETRY A rectangular pyramid has a base area of square centimeters

and a height of centimeters. Write a rational expression to describe the

volume of the rectangular pyramid. cm3x � 5�

x2 � 3x��x2 � 5x � 6

x2 � 3x � 10��2x

x2 � 2x � 4��

�xx

2

3

��

22x

3�

��

�x2

(�x �

4x2

�)3

4�

�x2

4� 9�

��3 �

8x

�

2x � 1�

�2x

x� 1�

��4 �

xx

�

5�

2a � 6�5a � 10

9 � a2��a2 � 5a � 6

2s � 3��

s2 � 10s � 25��s � 4

2s2 � 7s � 15��

(s � 4)22

��6x2 � 12x��4x � 12

3x � 6�x2 � 9

1��

x2 � y2�3

x � y�6

xy3�

24x2�w5

2xy�w2

a2w2�

a3w2�w5y2

a5y3�wy7

5x � 1��

25x2 � 1��x2 � 10x � 25

x � 5�10x � 2

5x�

5x2�8 � x

x2 � 5x � 24��6x � 2x2

w2 � n2�y � a

a � y�w � n

1�

n2 � 6n�

n8n5

�n � 62�

4�y � a

a � y�6

5ux2�

25x3�14u2y2

�2u3y�15xz5

x � 2�

x4 � x3 � 2x2��

x4 � x3

v � 5�

25 � v2��3v2 � 13v � 10

2k � 5�

2k2 � k � 15��

k2 � 9

2y � 3�

10y2 � 15y��35y2 � 5y

4m4n2�

(2m3n2)3��18m5n4

1�

9a2b3�27a4b4c

Practice (Average)

Multiplying and Dividing Rational Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

Reading to Learn MathematicsMultiplying and Dividing Rational Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1


Less

on

9-1

Pre-Activity How are rational expressions used in mixtures?

Read the introduction to Lesson 9-1 at the top of page 472 in your textbook.

• Suppose that the Goodie Shoppe also sells a candy mixture of chocolatemints and caramels. If this mixture is made with 4 pounds of chocolate

mints and 3 pounds of caramels, then of the mixture is

mints and of the mixture is caramels.

• If the store manager adds another y pounds of mints to the mixture, whatfraction of the mixture will be mints?

Reading the Lesson

1. a. In order to simplify a rational number or rational expression, the

numerator and and divide both of them by their

.

b. A rational expression is undefined when its is equal to .

To find the values that make the expression undefined, completely

the original and set each factor equal to .

2. a. To multiply two rational expressions, the andmultiply the denominators.

b. To divide two rational expressions, by the of

the .

3. a. Which of the following expressions are complex fractions? ii, iv, v

i. ii. iii. iv. v.

b. Does a complex fraction express a multiplication or division problem? divisionHow is multiplication used in simplifying a complex fraction? Sample answer: To divide the numerator of the complex fraction by the denominator,multiply the numerator by the reciprocal of the denominator.

Helping You Remember

4. One way to remember something new is to see how it is similar to something youalready know. How can your knowledge of division of fractions in arithmetic help you tounderstand how to divide rational expressions? Sample answer: To dividerational expressions, multiply the first expression by the reciprocal ofthe second. This is the same “invert and multiply” process that is usedwhen dividing arithmetic fractions.

�r2 �

925

�

��r �

35

�

�z �

z1

�

�zr � 5�r � 5

�38

�

��156�

7�12

divisorreciprocalmultiply

numeratorsmultiply

0denominatorfactor

0denominatorgreatest common factor

denominatorfactor

4 � y�

�37

�

�47

�


Reading AlgebraIn mathematics, the term group has a special meaning. The followingnumbered sentences discuss the idea of group and one interesting example of a group.

01 To be a group, a set of elements and a binary operation must satisfy fourconditions: the set must be closed under the operation, the operationmust be associative, there must be an identity element, and everyelement must have an inverse.

02 The following six functions form a group under the operation of

composition of functions: f1(x) � x, f2(x) � �1x�, f3(x) � 1 � x,

f4(x) � �(x �

x1)

�, f5(x) � �(x �x

1)�, and f6(x) � �(1 �1

x)�.

03 This group is an example of a noncommutative group. For example,f3 � f2 � f4, but f2 � f3 � f6.

04 Some experimentation with this group will show that the identityelement is f1.

05 Every element is its own inverse except for f4 and f6, each of which is theinverse of the other.

Use the paragraph to answer these questions.

1. Explain what it means to say that a set is closed under an operation. Is the set of positive integers closed under subtraction?

2. Subtraction is a noncommutative operation for the set of integers. Writean informal definition of noncommutative.

3. For the set of integers, what is the identity element for the operation ofmultiplication? Justify your answer.

4. Explain how the following statement relates to sentence 05:

(f6 � f4)(x) � f6[ f4(x)] � f6��(1 �1

x)�� x � f1(x).1��1 � (x

x� 1)�

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-19-1

Study Guide and InterventionAdding and Subtracting Rational Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2


Less

on

9-2

LCM of Polynomials To find the least common multiple of two or more polynomials,factor each expression. The LCM contains each factor the greatest number of times itappears as a factor.

Find the LCM of 16p2q3r,40pq4r2, and 15p3r4.16p2q3r � 24 � p2 � q3 � r40pq4r2 � 23 � 5 � p � q4 � r2

15p3r4 � 3 � 5 � p3 � r4

LCM � 24 � 3 � 5 � p3 � q4 � r4

� 240p3q4r4

Find the LCM of 3m2 � 3m � 6 and 4m2 � 12m � 40.3m2 � 3m � 6 � 3(m � 1)(m � 2)4m2 � 12m � 40 � 4(m � 2)(m � 5)LCM � 12(m � 1)(m � 2)(m � 5)

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the LCM of each set of polynomials.

1. 14ab2, 42bc3, 18a2c 2. 8cdf3, 28c2f, 35d4f 2

126a2b2c3 280c2d4f 3

3. 65x4y, 10x2y2, 26y4 4. 11mn5, 18m2n3, 20mn4

130x4y4 1980m2n5

5. 15a4b, 50a2b2, 40b8 6. 24p7q, 30p2q2, 45pq3

600a4b8 360p7q3

7. 39b2c2, 52b4c, 12c3 8. 12xy4, 42x2y, 30x2y3

156b4c3 420x2y4

9. 56stv2, 24s2v2, 70t3v3 10. x2 � 3x, 10x2 � 25x � 15840s2t3v3 5x(x � 3)(2x � 1)

11. 9x2 � 12x � 4, 3x2 � 10x � 8 12. 22x2 � 66x � 220, 4x2 � 16(3x � 2)2(x � 4) 44(x � 2)(x � 2)(x � 5)

13. 8x2 � 36x � 20, 2x2 � 2x � 60 14. 5x2 � 125, 5x2 � 24x � 54(x � 5)(x � 6)(2x � 1) 5(x � 5)(x � 5)(5x � 1)

15. 3x2 � 18x � 27, 2x3 � 4x2 � 6x 16. 45x2 � 6x � 3, 45x2 � 56x(x � 3)2(x � 1) 15(5x � 1)(3x � 1)(3x � 1)

17. x3 � 4x2 � x � 4, x2 � 2x � 3 18. 54x3 � 24x, 12x2 � 26x � 12(x � 1)(x � 1)(x � 3)(x � 4) 6x(3x � 2)(3x � 2)(2x � 3)


Add and Subtract Rational Expressions To add or subtract rational expressions,follow these steps.

Step 1 If necessary, find equivalent fractions that have the same denominator.Step 2 Add or subtract the numerators.Step 3 Combine any like terms in the numerator.Step 4 Factor if possible.Step 5 Simplify if possible.

Simplify � .

�

� � Factor the denominators.

� � The LCD is 2(x � 3)(x � 2)(x � 2).

� Subtract the numerators.

� Distributive Property

� Combine like terms.

� Simplify.


1. � � 2. �

3. � 4. �

5. � 6. ��2x2 � 9x � 4��(2x � 1)(2x � 1)2

5x��20x2 � 5

4��4x2 � 4x � 1

4�x � 1

�x2 � 1

3x � 3��x2 � 2x � 1

4x � 14�4x � 5

�3x � 63

�x � 24a2 � 9b2��15b

�5ac4a�3bc

x � 1��1

�x � 12

�x � 3y�

4y2�2y

�7xy�3x

x��(x � 3)(x � 2)(x � 2)

2x��2(x � 3)(x � 2)(x � 2)

6x � 12 � 4x � 12��2(x � 3)(x � 2)(x � 2)

6(x � 2) � 4(x � 3)��2(x � 3)(x � 2)(x � 2)

2 � 2(x � 3)��2(x � 3)(x � 2)(x � 2)

6(x � 2)��2(x � 3)(x � 2)(x � 2)

2��(x � 2)(x � 2)

6��2(x � 3)(x � 2)

2�x2 � 4

6��2x2 � 2x � 12

2�x2 � 4

6��2x2 � 2x � 12


Adding and Subtracting Rational Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2

ExampleExample

ExercisesExercises

Skills PracticeAdding and Subtracting Rational Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2


Less

on

9-2


1. 12c, 6c2d 12c2d 2. 18a3bc2, 24b2c2 72a3b2c2

3. 2x � 6, x � 3 2(x � 3) 4. 5a, a � 1 5a(a � 1)

5. t2 � 25, t � 5 (t � 5)(t � 5) 6. x2 � 3x � 4, x � 1 (x � 4)(x � 1)


7. � 8. �

9. � 4 10. �

11. � 12. �

13. � 14. �

15. � 16. �

17. � 18. �

19. � 20. �

21. � 22. �

y � 12��

n � 2�

2��y2 � 6y � 8

3��y2 � y � 12

2n � 2��n2 � 2n � 3

n�n � 3

2x2 � 5x � 2��

4��x2 � 3x � 10

2x � 1�x � 5

x2 � x � 1��

x�x � 1

1��x2 � 2x � 1

5z2 � 4z � 16��

z � 4�z � 1

4z�z � 4

2m�

m�n � m

m�m � n

5 � 3t�

5�x � 2

3t�2 � x

3w � 7��

2�w2 � 9

3�w � 3

15bd � 6b � 2d��

2�3bd

5�3b � d

a � 6��

3�2a

2�a � 2

7h � 3g��

3�4h2

7�4gh

12z � 2y��

2�5yz

12�5y2

2 � 5m2��

5�n

2�m2n

2c � 5�

2c � 7�3

13�

5�4p2q

3�8p2q

5x � 3y�

5�y

3�x



1. x2y, xy3 2. a2b3c, abc4 3. x � 1, x � 3x2y3 a2b3c4 (x � 1)(x � 3)

4. g � 1, g2 � 3g � 4 5. 2r � 2, r2 � r, r � 1 6. 3, 4w � 2, 4w2 � 1(g � 1)(g � 4) 2r(r � 1) 6(2w � 1)(2w � 1)

7. x2 � 2x � 8, x � 4 8. x2 � x � 6, x2 � 6x � 8 9. d2 � 6d � 9, 2(d2 � 9)(x � 4)(x � 2) (x � 2)(x � 4)(x � 3) 2(d � 3)(d � 3)2


10. � 11. � 12. �

13. � 2 14. 2x � 5 � 15. �

16. � 17. � 18. �

19. � 20. � 21. � �

22. � � 23. 24.

25. GEOMETRY The expressions , , and represent the lengths of the sides of a

triangle. Write a simplified expression for the perimeter of the triangle.

26. KAYAKING Mai is kayaking on a river that has a current of 2 miles per hour. If rrepresents her rate in calm water, then r � 2 represents her rate with the current, and r � 2 represents her rate against the current. Mai kayaks 2 miles downstream and then

back to her starting point. Use the formula for time, t � , where d is the distance, to

write a simplified expression for the total time it takes Mai to complete the trip.

h4r��

d�r

5(x3 � 4x � 16)��

10�x � 4

20�x � 4

5x�2

r � 4�

3x � y�

12�

�r �

r6

� � �r �

12

�

��r2

r�2 �

4r2�r

3�

�x �

2y

� � �x �

1y

�

��x �

1y

�

36�a2 � 9

2a�a � 3

2a�a � 3

3(6 � 5n)��

2p2 � 2p � 1��

5��

7�10n

3�4

1�5n

5�p2 � 9

2p � 3��p2 � 5p � 6

20��x2 � 4x � 12

5�2x � 12

2y � 1��

7 � 9m�

2�

y��y2 � y � 2

y � 5��y2 � 3y � 10

4m � 5�9 � m

2 � 5m�m � 9

2�x � 4

16�x2 � 16

13a � 47��

2(x � 3)(x � 2)��

2(2 � 3n)��

3n

9�a � 5

4�a � 3

x � 8�x � 4

4m�3mn

2d 2 � 9c��

25y2 � 12x2��

20 � 21b��

24ab

3�4cd3

1�6c2d

1�5x2y3

5�12x4y

7�8a

5�6ab

Practice (Average)

Adding and Subtracting Rational Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2

Reading to Learn MathematicsAdding and Subtracting Rational Expressions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2


Less

on

9-2

Pre-Activity How is subtraction of rational expressions used in photography?


A person is standing 5 feet from a camera that has a lens with a focallength of 3 feet. Write an equation that you could solve to find how far thefilm should be from the lens to get a perfectly focused photograph.

� �

Reading the Lesson

1. a. In work with rational expressions, LCD stands for

and LCM stands for . The LCD is the of the denominators.

b. To find the LCM of two or more numbers or polynomials, each

number or . The LCM contains each the

number of times it appears as a .

2. To add and , you should first factor the of

each fraction. Then use the factorizations to find the of x2 � 5x � 6 and

x3 � 4x2 � 4x. This is the for the two fractions.

3. When you add or subtract fractions, you often need to rewrite the fractions as equivalentfractions. You do this so that the resulting equivalent fractions will each have a

denominator equal to the of the original fractions.

4. To add or subtract two fractions that have the same denominator, you add or subtract

their and keep the same .

5. The sum or difference of two rational expressions should be written as a polynomial or

as a fraction in .


6. Some students have trouble remembering whether a common denominator is needed toadd and subtract rational expressions or to multiply and divide them. How can yourknowledge of working with fractions in arithmetic help you remember this?

Sample answer: In arithmetic, a common denominator is needed to addand subtract fractions, but not to multiply and divide them. The situationis the same for rational expressions.

simplest form

denominatornumerators

LCD

LCD

LCM

denominatorx � 4��x3 � 4x2 � 4x

x2 � 3��x2 � 5x � 6

factorgreatestfactorpolynomial

factor

LCMleast common multipleleast common denominator

1�

1�

1�


SuperellipsesThe circle and the ellipse are members of an interesting family of curves that were first studied by the French physicist and mathematician Gabriel Lamé (1795–1870). The general equation for the family is

� �ax

� �n � � �by

� �n � 1, with a � 0, b � 0, and n � 0.

For even values of n greater than 2, the curves are called superellipses.

1. Consider two curves that are not superellipses.Graph each equation on the grid at the right.State the type of curve produced each time.

a. � �2x

� �2 � � �2y

� �2 � 1

b. � �3x

� �2 � � �2y

� �2 � 1

2. In each of the following cases you are given values of a, b, and n to use in the general equation. Write the resulting equation. Then graph. Sketch each graph on the grid at the right.

a. a � 2, b � 3, n � 4 b. a � 2, b � 3, n � 6 c. a � 2, b � 3, n � 8

3. What shape will the graph of � �2x

� �n � � �2y

� �napproximate for greater and greater even,whole-number values of n?

1–1–2–3 2 3

3

2

1

–1

–2

–3

1–1–2–3 2 3

3

2

1

–1

–2

–3

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-29-2

Study Guide and InterventionGraphing Rational Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-39-3


Less

on

9-3

Vertical Asymptotes and Point Discontinuity

Rational Function an equation of the form f(x) � , where p(x) and q(x) are polynomial expressions and q(x) � 0

Vertical Asymptote An asymptote is a line that the graph of a function approaches, but never crosses. of the Graph of a If the simplified form of the related rational expression is undefined for x � a, Rational Function then x � a is a vertical asymptote.

Point Discontinuity Point discontinuity is like a hole in a graph. If the original related expression is undefined of the Graph of a for x � a but the simplified expression is defined for x � a, then there is a hole in the Rational Function graph at x � a.

Determine the equations of any vertical asymptotes and the values

of x for any holes in the graph of f(x) � .

First factor the numerator and the denominator of the rational expression.

f(x) � �

The function is undefined for x � 1 and x � �1.

Since � , x � 1 is a vertical asymptote. The simplified expression is

defined for x � �1, so this value represents a hole in the graph.

Determine the equations of any vertical asymptotes and the values of x for anyholes in the graph of each rational function.

1. f(x) � 2. f(x) � 3. f(x) �

asymptotes: x � 2, hole: x � asymptote: x � 0; x � �5 hole x � 4

4. f(x) � 5. f(x) � 6. f(x) �

asymptote: x � �2; asymptotes: x � 1, asymptote: x � �3

hole: x � x � �7

7. f(x) � 8. f(x) � 9. f(x) �

asymptotes: x � 1, asymptote: x � �3; holes: x � 1, x � 3 x � 5 hole: x �

3�

x3 � 2x2 � 5x � 6��

x2 � 4x � 32x2 � x � 3��2x2 � 3x � 9

x � 1��x2 � 6x � 5

1�

3x2 � 5x � 2��x � 3

x2 � 6x � 7��x2 � 6x � 7

3x � 1��3x2 � 5x � 2

5�

x2 � x � 12��

x2 � 4x2x2 � x � 10��2x � 5

4��x2 � 3x � 10

4x � 3�x � 1

(4x � 3)(x � 1)��(x � 1)(x � 1)

(4x � 3)(x � 1)��(x � 1)(x � 1)

4x2 � x � 3��

x2 � 1

4x2 � x � 3��

x2 � 1

p(x)�q(x)

ExampleExample

ExercisesExercises


Graph Rational Functions Use the following steps to graph a rational function.

Step 1 First see if the function has any vertical asymptotes or point discontinuities.Step 2 Draw any vertical asymptotes.Step 3 Make a table of values.Step 4 Plot the points and draw the graph.

Graph f(x) � .

� or

Therefore the graph of f(x) has an asymptote at x � �3 and a point discontinuity at x � 1.Make a table of values. Plot the points and draw the graph.

Graph each rational function.

1. f(x) � 2. f(x) � 3. f(x) �

4. f(x) � 5. f(x) � 6. f(x) �

xO

f (x)

xO

f (x)

xO

f (x)

x2 � 6x � 8��x2 � x � 2

x2 � x � 6��x � 3

2�(x � 3)2

xO

f (x)

4 8

8

4

–4

–8

–4–8xO

f (x)

xO

f (x)

2x � 1�x � 3

2�x

3�x � 1

x �2.5 �2 �1 �3.5 �4 �5

f(x) 2 1 0.5 �2 �1 �0.5

1�x � 3

x � 1��(x � 1)(x � 3)

x � 1��x2 � 2x � 3

x

f (x)

O

x � 1��x2 � 2x � 3


Graphing Rational Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-39-3

ExampleExample

ExercisesExercises

Skills PracticeGraphing Rational Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-39-3


Less

on

9-3


1. f(x) � 2. f(x) �

asymptotes: x � 4, x � �2 asymptotes: x � 4, x � 9

3. f(x) � 4. f(x) �

asymptote: x � 2; hole: x � �12 asymptote: x � 3; hole: x � 1

5. f(x) � 6. f(x) �

hole: x � �2 hole: x � 3


7. f(x) � 8. f(x) � 9. f(x) �

10. f(x) � 11. f(x) � 12. f(x) �

xO

f (x)

xO

f (x)

xO

f (x)

x2 � 4�x � 2

x�x � 2

2�x � 1

xO

f (x)

xO

f (x)

2

2

xO

f (x)

�4�x

10�x

�3�x

x2 � x � 12��x � 3

x2 � 8x � 12��x � 2

x � 1��x2 � 4x � 3

x � 12��x2 � 10x � 24

10��x2 � 13x � 36

3��x2 � 2x � 8



1. f(x) � 2. f(x) � 3. f(x) �

asymptotes: x � 2, asymptote: x � 3; asymptote: x � �2x � �5 hole: x � 7

4. f(x) � 5. f(x) � 6. f(x) �

hole: x � �10 hole: x � 6 hole: x � �5


7. f(x) � 8. f(x) � 9. f(x) �

10. PAINTING Working alone, Tawa can give the shed a coat of paint in 6 hours. It takes her father x hours working alone to give the

shed a coat of paint. The equation f(x) � describes the

portion of the job Tawa and her father working together can

complete in 1 hour. Graph f(x) � for x 0, y 0. If Tawa’s

father can complete the job in 4 hours alone, what portion of the job can they complete together in 1 hour?

11. LIGHT The relationship between the illumination an object receives from a light source of I foot-candles and the square of the distance d in feet of the object from the source can be

modeled by I(d) � . Graph the function I(d) � for

0 I 80 and 0 d 80. What is the illumination in foot-candles that the object receives at a distance of 20 feet from the light source? 11.25 foot-candles

4500�

d24500�

d2

5�

6 � x�6x

6 � x�6x

xO

f (x)

xO

f (x)

xO

f (x)

3x�(x � 3)2

x � 3�x � 2

�4�x � 2

x2 � 9x � 20��x � 5

x2 � 2x � 24��x � 6

x2 � 100��x � 10

x � 2��x2 � 4x � 4

x � 7��x2 � 10x � 21

6��x2 � 3x � 10

Practice (Average)

Graphing Rational Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-39-3

Reading to Learn MathematicsGraphing Rational Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-39-3


Less

on

9-3

Pre-Activity How can rational functions be used when buying a group gift?


• If 15 students contribute to the gift, how much would each of them pay?$10

• If each student pays $5, how many students contributed?30 students

Reading the Lesson

1. Which of the following are rational functions? A and C

A. f(x) � B. g(x) � �x� C. h(x) �

2. a. Graphs of rational functions may have breaks in . These may occur

as vertical or as point .

b. The graphs of two rational functions are shown below.

I. II.

Graph I has a at x � .

Graph II has a at x � .

Match each function with its graph above.

f(x) � II g(x) � I


3. One way to remember something new is to see how it is related to something you alreadyknow. How can knowing that division by zero is undefined help you to remember how tofind the places where a rational function has a point discontinuity or an asymptote?

Sample answer: A point discontinuity or vertical asymptote occurs wherethe function is undefined, that is, where the denominator of the relatedrational expression is equal to 0. Therefore, set the denominator equal tozero and solve for the variable.

x2 � 4�x � 2

x�x � 2

�2vertical asymptote

�2point discontinuity

x

y

Ox

y

O

discontinuitiesasymptotescontinuity

x2 � 25��x2 � 6x � 9

1�x � 5


Graphing with Addition of y-CoordinatesEquations of parabolas, ellipses, and hyperbolas that are “tipped” with respect to the x- and y-axes are more difficult to graph than the equations you have been studying.

Often, however, you can use the graphs of two simplerequations to graph a more complicated equation. For example, the graph of the ellipse in the diagram at the right is obtained by adding the y-coordinate of each point on the circle and the y-coordinate of the corresponding point of the line.

Graph each equation. State the type of curve for each graph.

1. y � 6 � x � �4 � x2� 2. y � x � �x�

Use a separate sheet of graph paper to graph these equations. State the type ofcurve for each graph.

3. y � 2x � �7 � 6�x � x2� 4. y � �2x � ��2x�

1 4 5 6 72 3

8

7

6

5

4

3

2

1

–1

–2

y � x

y � ��x

x

y

O

1 4 5–1–2 2 3

9

8

7

6

5

4

3

2

1

–1

–2y � ��4x � x2

y � 6 � x

x

y

O

x

y

O

y � ��4x � x2

y � x � 6 � ��4x � x2

y � x � 6

A

B�

A�

B

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-39-3

Study Guide and InterventionDirect, Joint, and Inverse Variation

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4


Less

on

9-4

Direct Variation and Joint Variation

Direct Variationy varies directly as x if there is some nonzero constant k such that y � kx. k is called theconstant of variation.

Joint Variation y varies jointly as x and z if there is some number k such that y � kxz, where x � 0 and z � 0.

Find each value.ExampleExample

a. If y varies directly as x and y � 16when x � 4, find x when y � 20.

� Direct proportion

� y1 � 16, x1 � 4, and y2 � 20

16x2 � (20)(4) Cross multiply.

x2 � 5 Simplify.

The value of x is 5 when y is 20.

20�x2

16�4

y2�x2

y1�x1

b. If y varies jointly as x and z and y � 10when x � 2 and z � 4, find y when x � 4 and z � 3.

� Joint variation

� y1 � 10, x1 � 2, z1 � 4, x2 � 4, and z2 � 3

120 � 8y2 Simplify.

y2 � 15 Divide each side by 8.

The value of y is 15 when x � 4 and z � 3.

y2�4 � 310

�2 � 4

y2�x2z2

y1�x1z1

ExercisesExercises

Find each value.

1. If y varies directly as x and y � 9 when 2. If y varies directly as x and y � 16 when x � 6, find y when x � 8. 12 x � 36, find y when x � 54. 24

3. If y varies directly as x and x � 15 4. If y varies directly as x and x � 33 when when y � 5, find x when y � 9. 27 y � 22, find x when y � 32. 48

5. Suppose y varies jointly as x and z. 6. Suppose y varies jointly as x and z. Find yFind y when x � 5 and z � 3, if y � 18 when x � 6 and z � 8, if y � 6 when x � 4when x � 3 and z � 2. 45 and z � 2. 36

7. Suppose y varies jointly as x and z. 8. Suppose y varies jointly as x and z. Find yFind y when x � 4 and z � 11, if y � 60 when x � 5 and z � 2, if y � 84 when when x � 3 and z � 5. 176 x � 4 and z � 7. 30

9. If y varies directly as x and y � 14 10. If y varies directly as x and x � 200 whenwhen x � 35, find y when x � 12. 4.8 y � 50, find x when y � 1000. 4000

11. If y varies directly as x and y � 39 12. If y varies directly as x and x � 60 whenwhen x � 52, find y when x � 22. 16.5 y � 75, find x when y � 42. 33.6

13. Suppose y varies jointly as x and z. 14. Suppose y varies jointly as x and z. Find yFind y when x � 6 and z � 11, if when x � 5 and z � 10, if y � 12 when y � 120 when x � 5 and z � 12. 132 x � 8 and z � 6. 12.5

15. Suppose y varies jointly as x and z. 16. Suppose y varies jointly as x and z. Find yFind y when x � 7 and z � 18, if when x � 5 and z � 27, if y � 480 when y � 351 when x � 6 and z � 13. 567 x � 9 and z � 20. 360


Inverse Variation

Inverse Variation y varies inversely as x if there is some nonzero constant k such that xy � k or y � .

If a varies inversely as b and a � 8 when b � 12, find a when b � 4.

� Inverse variation

� a1 � 8, b1 � 12, b2 � 4

8(12) � 4a2 Cross multiply.

96 � 4a2 Simplify.

24 � a2 Divide each side by 4.

When b � 4, the value of a is 24.

Find each value.

1. If y varies inversely as x and y � 12 when x � 10, find y when x � 15. 8











12. If y varies inversely as x and y � 3 when x � 8, find y when x � 40. 0.6




a2�12

8�4

a2�b1

a1�b2

k�x


Direct, Joint, and Inverse Variation

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4

ExampleExample

ExercisesExercises

Skills PracticeDirect, Joint, and Inverse Variation

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4


Less

on

9-4

State whether each equation represents a direct, joint, or inverse variation. Thenname the constant of variation.

1. c � 12m direct; 12 2. p � inverse; 4 3. A � bh joint;

4. rw � 15 inverse; 15 5. y � 2rst joint; 2 6. f � 5280m direct; 5280

7. y � 0.2s direct; 0.2 8. vz � �25 inverse; �25 9. t � 16rh joint; 16

10. R � inverse; 8 11. � direct; 12. C � 2�r direct; 2�

Find each value.

13. If y varies directly as x and y � 35 when x � 7, find y when x � 11. 55


15. If y varies directly as x and y � 540 when x � 10, find x when y � 1080. 20

16. If y varies directly as x and y � 12 when x � 72, find x when y � 9. 54

17. If y varies jointly as x and z and y � 18 when x � 2 and z � 3, find y when x � 5 and z � 6. 90

18. If y varies jointly as x and z and y � �16 when x � 4 and z � 2, find y when x � �1 and z � 7. 14




22. If y varies inversely as x and y � 3 when x � 14, find x when y � 6. 7

23. If y varies inversely as x and y � 27 when x � 2, find x when y � 9. 6

24. If y varies directly as x and y � �15 when x � 5, find x when y � �36. 12

1�

1�3

a�b

8�w

1�

1�2

4�q


State whether each equation represents a direct, joint, or inverse variation. Thenname the constant of variation.

1. u � 8wz joint; 8 2. p � 4s direct; 4 3. L � inverse; 5 4. xy � 4.5 inverse; 4.5

5. � � 6. 2d � mn 7. � h 8. y �

direct; � joint; inverse; 1.25 inverse;

Find each value.


10. If y varies directly as x and y � �16 when x � 6, find x when y � �4. 1.5


12. If y varies directly as x and y � 7 when x � 1.5, find y when x � 4.



15. If y varies jointly as x and z and y � 12 when x � �2 and z � 3, find y when x � 4 and z � �1. 8

16. If y varies inversely as x and y � 16 when x � 4, find y when x � 3.

17. If y varies inversely as x and y � 3 when x � 5, find x when y � 2.5. 6

18. If y varies inversely as x and y � �18 when x � 6, find y when x � 5. �21.6

19. If y varies directly as x and y � 5 when x � 0.4, find x when y � 37.5. 3

20. GASES The volume V of a gas varies inversely as its pressure P. If V � 80 cubiccentimeters when P � 2000 millimeters of mercury, find V when P � 320 millimeters ofmercury. 500 cm3

21. SPRINGS The length S that a spring will stretch varies directly with the weight F thatis attached to the spring. If a spring stretches 20 inches with 25 pounds attached, howfar will it stretch with 15 pounds attached? 12 in.

22. GEOMETRY The area A of a trapezoid varies jointly as its height and the sum of itsbases. If the area is 480 square meters when the height is 20 meters and the bases are28 meters and 20 meters, what is the area of a trapezoid when its height is 8 meters andits bases are 10 meters and 15 meters? 100 m2

64�

56�

3�

1�

3�4x

1.25�g

C�d

5�k

Practice (Average)

Direct, Joint, and Inverse Variation

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4

Reading to Learn MathematicsDirect, Joint, and Inverse Variation

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4


Less

on

9-4

Pre-Activity How is variation used to find the total cost given the unit cost?


• For each additional student who enrolls in a public college, the total

high-tech spending will (increase/decrease) by

.

• For each decrease in enrollment of 100 students in a public college, the

total high-tech spending will (increase/decrease) by

.

Reading the Lesson

1. Write an equation to represent each of the following variation statements. Use k as theconstant of variation.

a. m varies inversely as n. m �

b. s varies directly as r. s � kr

c. t varies jointly as p and q. t � kpq

2. Which type of variation, direct or inverse, is represented by each graph?

a. inverse b. direct


3. How can your knowledge of the equation of the slope-intercept form of the equation of aline help you remember the equation for direct variation?

Sample answer: The graph of an equation expressing direct variation isa line. The slope-intercept form of the equation of a line is y � mx � b. Indirect variation, if one of the quantities is 0, the other quantity is also 0,so b � 0 and the line goes through the origin. The equation of a linethrough the origin is y � mx, where m is the slope. This is the same asthe equation for direct variation with k � m.

x

y

Ox

y

O

k�

$14,900decrease

$149increase


Expansions of Rational ExpressionsMany rational expressions can be transformed into power series. A powerseries is an infinite series of the form A � Bx � Cx2 � Dx3 � …. Therational expression and the power series normally can be said to have thesame values only for certain values of x. For example, the following equationholds only for values of x such that �1 x 1.

�1 �1

x� � 1 � x � x2 � x3 � … for �1 x 1

Expand in ascending powers of x.

Assume that the expression equals a series of the form A � Bx � Cx2 � Dx3 � ….Then multiply both sides of the equation by the denominator 1 � x � x2.

�1 �

2 �

x �

3xx2� � A � Bx � Cx2 � Dx3 � …

2 � 3x � (1 � x � x2)(A � Bx � Cx2 � Dx3 � …)2 � 3x � A � Bx � Cx2 � Dx3 � …

� Ax � Bx2 � Cx3 � …� Ax2 � Bx3 � …

2 � 3x � A � (B � A)x � (C � B � A)x2 � (D � C � B)x3 � …

Now, match the coefficients of the polynomials.2 � A3 � B � A0 � C � B � A0 � D � C � B � A

Finally, solve for A, B, C, and D and write the expansion.A � 2, B � 1, C � �3, and D � 0

Therefore, �1 �

2 �

x �

3xx2� � 2 � x � 3x2 � …

Expand each rational expression to four terms.

1. �1 �

1x�

�

xx2�

2. �1 �2

x�

3. �1 �1

x�

2 � 3x��1 � x � x2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-49-4

ExampleExample

Study Guide and InterventionClasses of Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-59-5


Less

on

9-5

Identify Graphs You should be familiar with the graphs of the following functions.

Function Description of Graph

Constant a horizontal line that crosses the y-axis at a

Direct Variation a line that passes through the origin and is neither horizontal nor vertical

Identity a line that passes through the point (a, a), where a is any real number

Greatest Integer a step function

Absolute Value V-shaped graph

Quadratic a parabola

Square Root a curve that starts at a point and curves in only one direction

Rational a graph with one or more asymptotes and/or holes

Inverse Variationa graph with 2 curved branches and 2 asymptotes, x � 0 and y � 0 (special case of rational function)

Identify the function represented by each graph.

1. 2. 3.

quadratic rational direct variation4. 5. 6.

constant absolute value greatest integer7. 8. 9.

identity square root inverse variation

x

y

O

x

y

O

x

y

O

x

y

Ox

y

Ox

y

O

x

y

Ox

y

O

x

y

O

ExercisesExercises


Identify Equations You should be able to graph the equations of the following functions.

Function General Equation

Constant y � a

Direct Variation y � ax

Identity y � x

Greatest Integer equation includes a variable within the greatest integer symbol, � �

Absolute Value equation includes a variable within the absolute value symbol, | |

Quadratic y � ax2 � bx � c, where a � 0

Square Root equation includes a variable beneath the radical sign, ��

Rational y �

Inverse Variation y �

Identify the function represented by each equation. Then graph the equation.

1. y � inverse variation 2. y � x direct variation 3. y � � quadratic

4. y � |3x| � 1 absolutevalue 5. y � � inverse variation6. y � greatest

integer

7. y � �x � 2� square root 8. y � 3.2 constant 9. y � rational

x

y

Ox

y

Ox

y

O

x2 � 5x � 6��x � 2

x

y

Ox

y

Ox

y

O

x�2

2�x

x

y

Ox

y

Ox

y

O

x2�2

4�3

6�x

a�x

p(x)�q(x)


Classes of Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-59-5

ExercisesExercises

Skills PracticeClasses of Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-59-5


Less

on

9-5

Identify the type of function represented by each graph.

1. 2. 3.

constant direct variation quadratic

Match each graph with an equation below.

A. y � |x � 1| B. y � C. y � �1 � x� D. y � �x� � 1

4. B 5. C 6. A

Identify the type of function represented by each equation. Then graph theequation.

7. y � 8. y � 2�x� 9. y � �3x

inverse variation greatest integer direct variationor rational

x

y

Ox

y

OxO

y

2�x

x

y

O

x

y

Ox

y

O

1�x � 1

x

y

O

x

y

Ox

y

O



1. 2. 3.

rational square root absolute value

Match each graph with an equation below.

A. y � |2x � 1 | B. y � �2x � 1� C. y � D. y � ��x�

4. D 5. C 6. A

Identify the type of function represented by each equation. Then graph theequation.

7. y � �3 8. y � 2x2 � 1 9. y �

constant quadratic rational

10. BUSINESS A startup company uses the function P � 1.3x2 � 3x � 7 to predict its profit orloss during its first 7 years of operation. Describe the shape of the graph of the function.The graph is U-shaped; it is a parabola.

11. PARKING A parking lot charges $10 to park for the first day or part of a day. After that,it charges an additional $8 per day or part of a day. Describe the graph and find the cost

of parking for 6 days. The graph looks like a series of steps, similar to a greatest integer function, but with open circles on the left and closedcircles on the right; $58.

1�2

x

y

O

x

y

O

x2 � 5x � 6��x � 2

x

y

O

x

y

Ox

y

O

x � 3�2

x

y

O

x

y

O

x

y

O

Practice (Average)

Classes of Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-59-5

Reading to Learn MathematicsClasses of Functions

NAME ______________________________________________ DATE ____________ PERIOD _____

9-59-5


Less

on

9-5

Pre-Activity How can graphs of functions be used to determine a person’sweight on a different planet?


• Based on the graph, estimate the weight on Mars of a child who weighs40 pounds on Earth.about 15 pounds

• Although the graph does not extend far enough to the right to read itdirectly from the graph, use the weight you found above and yourknowledge that this graph represents direct variation to estimate theweight on Mars of a woman who weighs 120 pounds on Earth.about 45 pounds

Reading the Lesson

1. Match each graph below with the type of function it represents. Some types may be usedmore than once and others not at all.I. square root II. quadratic III. absolute value IV. rationalV. greatest integer VI. constant VII. identity

a. III b. I c. VI

d. II e. IV f. V


2. How can the symbolic definition of absolute value that you learned in Lesson 1-4 helpyou to remember the graph of the function f(x) � |x |? Sample answer: Using thedefinition of absolute value, f(x) � x if x 0 and f(x) � �x if x 0.Therefore, the graph is made up of pieces of two lines, one with slope 1and one with slope �1, meeting at the origin. This forms a V-shapedgraph with “vertex” at the origin.

x

y

Ox

y

Ox

y

O

x

y

Ox

y

Ox

y

O


Partial FractionsIt is sometimes an advantage to rewrite a rational expression as the sum oftwo or more fractions. For example, you might do this in a calculus coursewhile carrying out a procedure called integration.

You can resolve a rational expression into partial fractions if two conditionsare met:(1) The degree of the numerator must be less than the degree of the

denominator; and(2) The factors of the denominator must be known.

Resolve �x3

3� 1� into partial fractions.

The denominator has two factors, a linear factor, x � 1, and a quadraticfactor, x2 � x � 1. Start by writing the following equation. Notice that thedegree of the numerators of each partial fraction is less than itsdenominator.

�x3

3� 1� � �x �

A1� � �

x2B�

x �

x �

C1

�

Now, multiply both sides of the equation by x3 � 1 to clear the fractions andfinish the problem by solving for the coefficients A, B, and C.

�x3

3� 1� � �x �

A1� � �

x2B�

x �

x �

C1

�

3 � A(x2 � x � 1) � (x � 1)(Bx � C)3 � Ax2 � Ax � A � Bx2 � Cx � Bx � C3 � (A � B)x2 � (B � C � A)x � (A � C)

Equating each term, 0x2 � (A � B)x2

0x � (B � C � A)x3 � (A � C)

Therefore, A � 1, B � �1, C � 2, and �x3

3� 1� � �x �

11� � �

x2�

�

xx�

�

21

�.

Resolve each rational expression into partial fractions.

1. �x2

5�

x2�

x3� 3

� � �x �A

1� � �x �B

3�

2. �(6xx�

�

27)2� � �x �

A2� � �

(x �

B2)2�

3. � �Ax� � �

xB2� � �x �

C1� � �

(x �

D1)2�

4x3 � x2 � 3x � 2��

x2(x � 1)2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-59-5

ExampleExample

Study Guide and InterventionSolving Rational Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

9-69-6


Less

on

9-6

Solve Rational Equations A rational equation contains one or more rationalexpressions. To solve a rational equation, first multiply each side by the least commondenominator of all of the denominators. Be sure to exclude any solution that would producea denominator of zero.

Solve � � .

� � Original equation

10(x � 1)� � � � 10(x � 1)� � Multiply each side by 10(x � 1).

9(x � 1) � 2(10) � 4(x � 1) Multiply.

9x � 9 � 20 � 4x � 4 Distributive Property

5x � �25 Subtract 4x and 29 from each side.

x � �5 Divide each side by 5.

Check

� � Original equation

� � x � �5

� � Simplify.

� � Simplify.

� Simplify.

�

Solve each equation.

1. � � 2 5 2. � � 1 2 3. � � �

4. � � 4 � 5. � �7 6. � � 10

7. NAVIGATION The current in a river is 6 miles per hour. In her motorboat Marissa cantravel 12 miles upstream or 16 miles downstream in the same amount of time. What isthe speed of her motorboat in still water? 42 mph

8. WORK Adam, Bethany, and Carlos own a painting company. To paint a particular house

alone, Adam estimates that it would take him 4 days, Bethany estimates 5 days, and

Carlos 6 days. If these estimates are accurate, how long should it take the three of them

to paint the house if they work together? about 1 days2�

1�2

8�4

�x � 2x

�x � 2x � 1�12

4�x � 1

1�2m � 1

�2m3m � 2�5m

13�1

�2x � 5�4

2x � 1�3

4 � 2t�3

4t � 3�5

y � 3�6

2y�3

2�5

2�5

2�5

8�20

2�5

10�20

18�20

2�5

2��4

9�10

2�5

2��5 � 1

9�10

2�5

2�x � 1

9�10

2�5

2�x � 1

9�10

2�5

2�x � 1

9�10

2�5

2�x � 1

9�10

ExampleExample

ExercisesExercises


Solve Rational Inequalities To solve a rational inequality, complete the following steps.

Step 1 State the excluded values.Step 2 Solve the related equation.Step 3 Use the values from steps 1 and 2 to divide the number line into regions. Test a value in each region to

see which regions satisfy the original inequality.

Solve � � .

Step 1 The value of 0 is excluded since this value would result in a denominator of 0.

Step 2 Solve the related equation.

� � Related equation

15n� � � � 15n� � Multiply each side by 15n.

10 � 12 � 10n Simplify.

22 � 10n Simplify.

2.2 � n Simplify.

Step 3 Draw a number with vertical lines at the excluded value and the solution to the equation.

Test n � �1. Test n � 1. Test n � 3.

� � �� is true. � is not true. � is true.

The solution is n 0 or n 2.2.

Solve each inequality.

1. 3 2. 4x 3. � �

�1 a � 0 x � � or 0 x � 0 p

4. � � 5. � 2 6. � 1 �

�2 x 0 x 0 or x 1 x �1 or 0 x 1

or x � 5 or x � 2

1�

2�x � 1

3�x2 � 1

5�x

4�x � 1

1�4

2�x

3�2x

39�

1�

1�

2�3

4�5p

1�2p

1�x

3�a � 1

2�3

4�15

2�9

2�3

4�5

2�3

2�3

4�5

2�3

�3 �2 �1 0 1 22.2

3

2�3

4�5n

2�3n

2�3

4�5n

2�3n

2�3

4�5n

2�3n


Solving Rational Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

9-69-6

ExampleExample

ExercisesExercises

Skills PracticeSolving Rational Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

9-69-6


Less

on

9-6

Solve each equation or inequality. Check your solutions.

1. � �1 2. 2 � �

3. � �1 4. 3 � z � 1, 2

5. � 5 6. � �5, 8

7. � �3 8. � � y � 7 3, 4

9. � 8 10. � � 0 k � 0

11. 2 � 0 v 4 12. n � n �3 or 0 n 3

13. � � 0 m 1 14. � 1 0 x

15. � � 9 3 16. � 4 � 4

17. 2 � � �5 18. 8 � �

19. � � �4 20. � �

21. � � 22. � � 2

23. � � �6 24. � � 52�t � 3

4�t � 3

8�t2 � 9

2�e � 2

1�e � 2

2e�e2 � 4

5�s � 4

3�s � 3

12s � 19��s2 � 7s � 12

2x � 3�x � 1

x�2x � 2

x � 8�2x � 2

4�w2 � 4

1�w � 2

1�w � 2

2�n � 3

5�n2 � 9

1�n � 3

2�8z � 8

�z � 24�z

2q�q � 1

5�2q

b � 2�b � 1

3b � 2�b � 1

9x � 7�x � 2

15�x

3�2

�x1

�2x5�2

3�m

1�2m

12�n

3�n

5�v

3�v

4�3k

3�k

x � 1�x � 10

x � 2�x � 4

12�y

3�2

2x � 3�x � 1

8�s

s � 3�5

1�d � 2

2�d � 1

2�z

�6�2

9�3x

12�1

�34�n

1�2

x�x � 1


Solve each equation or inequality. Check your solutions.

1. � � 16 2. � 1 � �1, 2

3. � � , 4 4. � s � 4

5. � � 1 all reals except 5 6. � � 0

7. t �5 or � t 0 8. � �

9. � �2 10. 5 � 0 a 2

11. � 0 x 7 12. 8 � � y 0 or y � 2

13. � p 0 or p � 14. � �

15. g � � �1 16. b � � 1 � �2

17. 2 � � 18. 5 � � 6

19. � � 20. � 4 � ��53

�, 5

21. � � 7 22. � � �1, �2

23. � � 0 24. � �

25. � � 26. 3 � � �2

all reals except �4 and 4

27. BASKETBALL Kiana has made 9 of 19 free throws so far this season. Her goal is to make60% of her free throws. If Kiana makes her next x free throws in a row, the function

f(x) � represents Kiana’s new ratio of free throws made. How many successful free

throws in a row will raise Kiana’s percent made to 60%? 6

28. OPTICS The lens equation � � relates the distance p of an object from a lens, the

distance q of the image of the object from the lens, and the focal length f of the lens.What is the distance of an object from a lens if the image of the object is 5 centimetersfrom the lens and the focal length of the lens is 4 centimeters? 20 cm

1�f

1�q

1�p

9 � x�19 � x

22�a � 5

6a � 1�2a � 7

r2 � 16�r2 � 16

4�r � 4

r�r � 4

2�x � 2

x�2 � x

x2 � 4�x2 � 4

14��y2 � 3y � 10

7�y � 5

y�y � 2

2��v2 � 3v � 2

5v�v � 2

4v�v � 1

25��k2 � 7k � 12

4�k � 4

3�k � 3

12��c2 � 2c � 3

c � 1�c � 3

3�3

�n2 � 41

�n � 21

�n � 2

2d � 4�d � 2

3d � 2�d � 1

14�x � 2

�x � 6x � 2�x � 3

b � 3�b � 1

2b�b � 1

2�g � 2

g�g � 2

2�x � 1

4�x � 2

6�x � 1

65�1

�51

�3p4�p

19�y

3�y

3�2x

1�10

4�5x

7�a

3�a

�1�w � 3

4�w � 2

11�3

�h � 15�h

1�2h

1�9

�2t � 15�t

5�5

�x1

�3x � 2y

�y � 55

�y � 5

5s � 8�s � 2

s�s � 2

2�4

�pp � 10�p2 � 2

x�2

x�x � 1

3�2

3�4

12�x

Practice (Average)

Solving Rational Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

9-69-6

Reading to Learn MathematicsSolving Rational Equations and Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

9-69-6


Less

on

9-6

Pre-Activity How are rational equations used to solve problems involving unitprice?Read the introduction to Lesson 9-6 at the top of page 505 in your textbook.

• If you increase total number of minutes of long-distance calls from Marchto April, will your long-distance phone bill increase or decrease? increase

• Will your actual cost per minute increase or decrease? decrease

Reading the Lesson1. When solving a rational equation, any possible solution that results in 0 in the

denominator must be excluded from the list of solutions.

2. Suppose that on a quiz you are asked to solve the rational inequality � � 0.Complete the steps of the solution.

Step 1 The excluded values are and .

Step 2 The related equation is � � 0

.

To solve this equation, multiply both sides by the LCD, which is .Solving this equation will show that the only solution is �4.

Step 3 Divide a number line into regions using the excluded values and thesolution of the related equation. Draw dashed vertical lines on the number linebelow to show these regions.

Consider the following values of � for various test values of z.

If z � �5, � � 0.2. If z � �3, � � �1.

If z � �1, � � 9. If z � 1, � � �5.

Using this information and your number line, write the solution of the inequality.

z �4 or �2 z 0

Helping You Remember3. How are the processes of adding rational expressions with different denominators and of

solving rational expressions alike, and how are they different? Sample answer:They are alike because both use the LCD of all the rational expressionsin the problem. They are different because in an addition problem, theLCD remains after the fractions are added, while in solving a rationalequation, the LCD is eliminated.

6�z

3�z � 2

6�z

3�z � 2

6�z

3�z � 2

6�z

3�z � 2

6�z

3�z � 2

�3�4�5�6 �2 �1 0 1 2 3 4 5 6

4

z (z � 2)

6�

3�

0�2

6�z

3�z � 2


LimitsSequences of numbers with a rational expression for the general term oftenapproach some number as a finite limit. For example, the reciprocals of thepositive integers approach 0 as n gets larger and larger. This is written usingthe notation shown below. The symbol ∞ stands for infinity and n → ∞ meansthat n is getting larger and larger, or “n goes to infinity.”

1, �12�, �

13�, �

14�, …, �n

1�, … lim

n→∞�n1

� � 0

Find limn→∞

�(n �

n2

1)2�

It is not immediately apparent whether the sequence approaches a limit ornot. But notice what happens if we divide the numerator and denominator ofthe general term by n2.

�(n �

n2

1)2� � �n2 �

n2

2

n � 1�

�

�

The two fractions in the denominator will approach a limit of 0 as n getsvery large, so the entire expression approaches a limit of 1.

Find the following limits.

1. limn→∞

�nn

3

4�

�56n

� 2. limn→∞

�1

n�

2n

�

3. limn→∞

�2(n

2�

n �1)

1� 1

� 4. limn→∞

�21n�

�

3n1

�

1��1 � �n

2� � �

n12�

�nn

2

2�

��nn

2

2� � �2

nn2� � �

n12�

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

9-69-6

ExampleExample

Chapter 9 Test, Form 1

NAME DATE PERIOD

SCORE


Ass

essm

ent

Write the letter for the correct answer in the blank at the right of each question.


1. �2148mm

n2�

A. �34mn�

B. �4m

3n

� C. �34mn� D. �

43� 1.

2. �6a �

512

� � �a1�0

2�

A. 12 B. 24 C. 12a � 12 D. 24a 2.

3. �x2 �y

y2� � �xy�

2

y�

A. �y(x1� y)� B. C. �

x �y

y� D. �y(x

1� y)� 3.

4.

A. 5mn B. �5mn�

C. �15�mn D. �

mn

2� 4.

5. �p10

q��

4q�

A. �10

p�q2

4p� B. �q(p

1�4

1)� C. �10p

pq� 4� D. �

10p�q

4p� 5.

6. �k �4

1� � �2(k9� 1)�

A. �2(k1�3

1)� B. �2(k1�7

1)� C. �k1�1

1� D. �89� 6.

For Questions 7 and 8, find the LCM of each set of polynomials.

7. 10x2, 30xy2

A. 30x2y2 B. 300x3y2 C. 10x D. 40x2y2 7.

8. 3z � 12, 6z � 24A. 18(z � 4) B. 3(z � 4) C. 6(z � 4) D. z � 4 8.

9. Which is an equation of the vertical asymptote of the graph of f(x) � �xx

��

12�?

A. y � 1 B. y � 2 C. x � 2 D. x � 1 9.

10. Which rational function is graphed?

A. f(x) � �x �2

1� B. f(x) � �x �2

1�

C. f(x) � �x �x

1� D. f(x) � �x �x

1� 10.

�5mn

2

3��

�nm

2�

y3��x3 � x2y � xy2 � y3

99

xO

f (x)


Chapter 9 Test, Form 1 (continued)

11. The equation z � 30x represents a(n) __?___ variation.A. direct B. joint C. inverse D. combined 11.

12. Suppose y varies jointly as x and z. If y � 24 when x � 2 and z � 3, find ywhen x � 1 and z � 5.A. 5 B. 20 C. 10 D. 4 12.

13. The equation m � �n4

� represents a(n) __?___ variation.

A. direct B. joint C. inverse D. reverse 13.

14. If y varies inversely as x and y � 2 when x � 10, find y when x � 5.A. 1 B. 4 C. 25 D. 100 14.

For Questions 15 and 16, identify the function represented by each graph.

15. A. absolute valueB. greatest integerC. direct variationD. quadratic 15.

16. A. identityB. constantC. inverse variationD. rational 16.

17. Identify the type of function represented by y � �16x�.A. direction variation B. quadraticC. inverse variation D. square root 17.

18. Solve �x �x

2� � �75�.

A. �7 B. 5 C. 7 D. ��57� 18.

19. Solve y � 4 � �5y�.

A. �5, 1 B. �1, 5 C. �1 D. � 19.

20. Solve �m9� 5� 3.

A. m 5 or m 8 B. m �2 or m 5C. �2 m 5 D. 5 m 8 20.

Bonus Determine the equations of any vertical asymptotes and B:

the values of x for any holes in the graph of f(x) � �xx2

2

��

39x�.

y

xO

y

xO

NAME DATE PERIOD

99

Chapter 9 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. For what value(s) of m is the expression �m2m2

2�

�2mm

��

13� undefined?

A. ��32�, 0, 1 B. �1, �

32� C. � �

32�, 1 D. �

32� 1.


2. �xx2

2��

52xx

��

41� � �

2xx��

42

�

A. �12� B. 2 C. �2

((xx��

41))2

2� D. �2(xx��

41)� 2.

3. �a �

3b

� � �a2

1�2

b2�

A. �4(aa

2��

bb2)� B. �a �

4b� C. �a �

4b� D. �

4a(2a

��

bb2)

� 3.

4.

A. �s

1�2

3� B. 12s � 36 C. �

ss

��

33� D. 3 4.

5. �n26�n

9� � �n �3

3�

A. �n �3

3� B. �n �3

3� C. �n26�n

n�

�3

12� D. �6nn2 �

�93

� 5.

6. �mm� 5� � �5 �

2m�

A. �m2�m

5� B. �mm

��

25� C. �

mm

��

25� D. �(m

2�m

5)2� 6.


7. 5p � 20, 15p � 60A. 75(p � 4) B. 15(p � 4) C. p � 4 D. 5(p � 4) 7.

8. t2 � 8t � 15, t2 � t � 20A. (t � 3)(t � 5)(t � 4) B. (t � 3)(t � 5)(t � 4)C. (t � 3)(t � 5)(t � 4) D. (t � 3)(t � 5)(t � 4) 8.

9. Determine the equations of any vertical asymptotes of the graph of

f(x) � �x2 �

x �5x

1� 6

�.

A. x � 1 B. x � �2C. x � �2, x � �3 D. y � 1 9.

10. Determine the values of x for any holes in the graph of f(x) � �x2 �x �

6x5� 5�.

A. x � 5 B. x � �5C. x � 1 D. x � �1, x � �5 10.

�84ss2

2

��

2346s�

��122ss2 �

�63s6

�

99


Chapter 9 Test, Form 2A (continued)


A. f(x) � �x �3

2� B. f(x) � �x �3

2�

C. f(x) � �x �x

2� D. f(x) � �x �x

2� 11.

12. If y varies directly as x and y � 4 when x � �2, find y when x � 30.

A. ��145�

B. 60 C. �60 D. �145�

12.

13. The area A of a triangle varies jointly as the lengths of its base b and height h. If A � 75 when b � 15 and h � 10, find A when b � 8 and h � 6.A. 12 B. 48 C. 24 D. 96 13.


A. �16� B. 6 C. 3 D. �

13� 14.

15. The distance a car can travel on a certain amount of fuel varies inversely with its speed. If a car traveling 50 miles per hour can travel 300 miles on 10 gallons of fuel, how far could the car travel on 10 gallons of fuel at 60 miles per hour?A. 250 mi B. 360 mi C. 275 mi D. 300 mi 15.

16. Identify the type of function represented by y � (x � 1)2 � 4.A. square root B. rationalC. inverse variation D. quadratic 16.

17. Identify the type of function represented by y � �xx2

��

39

�.

A. quadratic B. rationalC. inverse variation D. direct variation 17.

18. Solve �n �n

4� � n � �12n

��

44n

�.

A. �4, 3 B. �3, 4 C. �4 D. 3 18.

19. Solve 4 � �1b� �

3b�.

A. b 0 B. b 0 or b 1 C. 0 b 1 D. b 1 19.

20. Tomas can do a job in 4 hours. Julia can do the same job in 6 hours. How many hours will it take the two of them to do the job if they work together?A. 3.5 B. 2.4 C. 5 D. 2 20.

Bonus Simplify . B:1 � �

3x�

��1 � �

4x� � �x

32�

NAME DATE PERIOD

99

xO

f (x)

Chapter 9 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. For what value(s) of x is the expression �2xx2

2��

43xx��

42� undefined?

A. ��12�, 0, 2 B. ��

12�, 2 C. �2, �

12� D. ��

12� 1.


2. �t2 �

t22�t

1� 3

� � �t2 �3t

4�t �

33�

A. �t2 �

3t6�t �

39

� B. �3t(2t

��

13)

� C. 3 D. �t �3

1� 2.

3. �m �

62n

� � �m2

1�04n2�

A. �3(m5� 2n)� B. �3(m

5� 2n)�

C. �m �4

2n� D. 3.

4.

A. �bb

��

22� B. b � 2 C. 2b � 4 D. b � 2 4.

5. �m23�0

25� � �m3� 5�

A. �3mm2 �

�2255

� B. �m23�3

25� C. �m3� 5� D. �(m

3�(m

5)�(m

15�)

5)� 5.

6. �m7� 6� � �6 �

mm�

A. �7m

��

m6� B. �

mm

��

76� C. �

mm

��

76� D. �6 �

7m�

6.


7. 7m � 21, 14m � 42A. m � 3 B. 98(m � 3) C. 7(m � 3) D. 14(m � 3) 7.

8. t2 � t � 12, t2 � 2t � 24A. (t � 3)(t � 4)(t � 6) B. (t � 3)(t � 4)(t � 6)C. (t � 3)(t � 4)(t � 6) D. (t � 3)(t � 4)(t � 6) 8.

9. Determine the equations of any vertical asymptotes of the graph of

f(x) � �x22�x

2�x

3� 3�.

A. x � �1 B. x � 3 C. x � �3, x � 1 D. y � 2 9.

10. Determine the values of x for any holes in the graph of f(x) � �x2 �x �

5x3� 6�.

A. x � �3 B. x � 3 C. x � �2, x � �3 D. x � �2 10.

�63bb2

2

��

1122b�

��10

5bb2��

1200b�

m3 � 4mn2 � 2m2n � 8n3��60

99


Chapter 9 Test, Form 2B (continued)


A. f(x) � �xx

��

31� B. f(x) � �(x � 3)

3(x � 1)�

C. f(x) � �xx

��

31� D. f(x) � �(x � 3)

3(x � 1)� 11.

12. If y varies jointly as x and z and y � 60 when x � 10 and z � �3, find y when x � 8 and z � 15.A. �240 B. 15 C. 240 D. �15 12.

13. SALES An appliance store manager noted that weekly sales varied directly with the amount of money spent on advertising. If last week’s sales were $10,000 and $2000 was spent on advertising, what should sales be during a week that $1200 was spent on advertising?A. $4800 B. $6000 C. $16,667 D. $50,000 13.


A. �32� B. �

23� C. �

59� D. �

95� 14.

15. The distance a car can travel on a certain amount of fuel varies inversely with its speed. If a car traveling 50 miles per hour can travel 336 miles on 10 gallons of fuel, how far could the car travel on 10 gallons of fuel at 60 miles per hour?A. 315 mi B. 320 mi C. 403.2 mi D. 280 mi 15.

16. Identify the type of function represented by y � � x � 5 �.A. direct variation B. absolute valueC. inverse variation D. constant 16.

17. Identify the type of function represented by y � 4.A. greatest integer B. direct variationC. constant D. identity 17.

18. Solve �n �n

3� � n � �7nn

��

138

�.

A. 3 B. 6 C. 3, 6 D. �3, 6 18.

19. Solve 7 � �m3� �

1m8�.

A. m 0 or m 3 B. 0 m 3C. m 3 D. m 0 19.

20. The sum of a number and 16 times its reciprocal is 10. Find the number(s).A. �8 or �2 B. 2 or 8 C. 4 D. �4 20.

Bonus Simplify . B:1 � �

2x�

��1 � �

1x� � �x

22�

NAME DATE PERIOD

99

xO

f (x)

Chapter 9 Test, Form 2C


1. For what value(s) of x is the expression �2x2x�

2 �3x

9� 9� 1.

undefined?


2. �x2 �x3

64� � �xx�

2

8� 2.

3. �3bb2

2

��

63bb

��

56

� � �6bb2 �

�2152� 3.

4. 4.

5. �x �2

2� � �x28� 4� 5.

6. �3m5� 1� � �1 �

23m� 6.


7. 4m3n, 9mn4, 18m4n2 7.

8. n2 � 2n � 8, n2 � 2n � 24 8.

For Questions 9 and 10, determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function.

9. f(x) � �xx

��

13� 9.

10. f(x) � �x2 �

x �2x

2� 8

� 10.

11. Graph the rational function f(x) � �xx

��

32�. 11.

12. If y varies jointly as x and z and y � 6 when x � 4 and 12.z � 12, find y when x � 24 and z � 5.

�63mm

2

2

��

3705m�

��9m

4m2 �

�4250m�

NAME DATE PERIOD

SCORE 99

Ass

essm

ent

xO

f (x)


Chapter 9 Test, Form 2C (continued)

13. PHOTOGRAPHS A film-developing company noted that, in 13.a particular town, the number of customers requesting online delivery of their vacation pictures varied directly with the number of households having high-speed Internet access. Currently, 5000 households in the town have high-speed Internet access and 80 customers request online delivery of their photographs. If this trend continues, how many customers should the film-developing company expect to request online delivery when 12,000 households have high-speed Internet access?

14. If y varies inversely as x and y � 25 when x � 6, find y 14.when x � 150.

15. WILDFIRES Firefighters battling wildfires in western states 15.noted that the percentage P of the fire remaining uncontained varied inversely with the amount of precipitation A that fell the previous day. If k is the constant of variation, write an equation that expresses P as a function of A.

16. Identify the type of function 16.represented by the graph.

17. Identify the type of function represented by y � ��23�x. 17.

For Questions 18 and 19, solve each equation or inequality.

18. x � �x2�x

2� � �3xx��

22

� 18.

19. 9 � �m2� �

4m7� 19.

20. PAINTING Alice can paint a room in 8 hours. Her assistant 20.can paint the same room in 12 hours. How long will it take if the two of them work together?

Bonus Solve � 1. B:�x �

12� � �x �

13�

��x �

12� � �x �

13�

y

xO

NAME DATE PERIOD

99

Chapter 9 Test, Form 2D


1. For what value(s) of x is the expression �2xx

2

2��

xx

��

610� 1.

undefined?


2. �x2 �x4

25� � �xx�

2

5� 2.

3. �3mm

2

2

��

125mm

��

812�� 8m

4m2 �

2 �16

4m� 3.

4. 4.

5. �x �3

3� � �x21�8

9� 5.

6. �2n3� 1� � �1 �

22n� 6.


7. 7s2t, 6st4, 14s3t2 7.

8. n2 � 6n � 5, n2 � 3n � 10 8.

For Questions 9 and 10, determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function.

9. f(x) � �x2 �x

2�x

6� 24� 9.

10. f(x) � �x2 � 73x � 10� 10.

11. Graph the rational function f(x) � �x �x

2�. 11.

12. If y varies jointly as x and z and y � 12 when x � 18 and 12.z � 6, find y when x � 81 and z � 7.

xO

f (x)

�182y2y2

��1468y�

��49yy2��

188y�

NAME DATE PERIOD

SCORE 99

Ass

essm

ent


Chapter 9 Test, Form 2D (continued)

13. RESTAURANTS In a certain county, the planning 13.commission noted that the number of restaurant permits renewed each year varied directly with the number of tourists visiting the county during the previous year. Last year, 400,000 tourists visited the county and 1200 restaurants renewed their permits. This year, 350,000 tourists are projected to visit the county. How many restaurant permits will be renewed if the trend continues?

14. If y varies inversely as x and y � 12 when x � 6, find y 14.when x � 8.

15. GOVERNMENT Part of a model used by a state government 15.indicates that revenue R varies inversely with the percentage of eligible workers who are unemployed U. If the constant of variation is k, write an equation that expresses R as a function of U.


17. Identify the type of function represented by 17.

y � �1x1�.

For Questions 18 and 19, solve each equation or inequality.

18. �x2�x

3� � �12� � �2x

2� 6� 18.

19. �8r

r� 3� �

4r5� 19.

20. GARDENING Joyce can plant a garden in 120 minutes, 20.and Jim can do the same job in 80 minutes. How long will it take to plant the garden if both of them work together?

Bonus Solve � 1. B:�x �

15� � �x �

11�

��

�x �1

5� � �x �1

1�

y

xO

NAME DATE PERIOD

99

Chapter 9 Test, Form 3


1. For what value(s) of x is the expression �6x23x�

2 �13

xx�2 �

105x� 1.

undefined?

For Questions 2–6, simplify each expression.

2. �3x22

x�2 �

12xx��

612

��3x34�x2

x2�

�9

10x� 2.

3. �g2

5�g

5�g

5� 4

� � �gg2

2��

8gg

��

1126

� 3.

4. 4.

5. �99aa

2

2��

44bb

2

2� � �2b3�a

3a� � �3a2�b

2b� 5.

6. 6.

7. Find the LCM of c2 � 2cd � d2, c2 � d2, and c � d. 7.

For Questions 8 and 9, determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function. Then graph each function.

8. f(x) � �(x��

23)2�

8.

9. f(x) � �2xx2 �

�44� 9.

10. If y varies jointly as x and z and y � �15� when x � �

13� and 10.

z � 15, find y when x � 10 and z � �14�.

(2 � n)��12� � �n

1��

��

�34mm

��

43nn�

��34mm

��

43nn�

NAME DATE PERIOD

SCORE 99

Ass

essm

ent

xO

f (x)

xO

f (x)


Chapter 9 Test, Form 3 (continued)

TELECOMMUNICATIONS For Questions 11 and 12,use the information below and in the table.

The average number of daily phone calls C between two cities is directly proportional to the product of the populations P1 and P2 of the cities and inversely proportional to the square of the distance d

between the cities. That is, C � �kP

d12

P2�.

11. Atlanta and Charleston are located approximately 11.324 miles apart and the average number of daily phone calls between the cities is 7700. Find the constant of variation k to the nearest hundredth.

12. About 17,100 calls are made each day between Atlanta and 12.Tallahassee. Find the distance between the cities to the nearest mile.

13. The current I in an electrical circuit varies inversely with 13.the resistance R in the circuit. If the current is 1.2 when the resistance is 6, write an equation relating the current and the resistance. Then find the current when the resistance is 0.18.


15. Identify the type of function 15.represented by xy � 0.3.

For Questions 16–19, solve each equation or inequality.

16. �y �5

3� � �y2 �1y0

� 6� � �y �y

2� 16.

17. �n �2

5� � �n2 �3n

3�n �

110� � �n �

12� 17.

18. �61x�

� �32x�

�59� 19. �1 �

4z� z � 3 18.

20. NUMBER THEORY A fraction has a value of �35�. If the

numerator is decreased by 8 and the denominator is

increased by 3, its value is �14�. Find the original fraction.

Bonus Simplify � and state any value(s) of x B:

for which the expression is undefined.

�x32� � �

2x�

��x32� � �x

23�

2 � �3x�

��2x� � �x

32�

NAME DATE PERIOD

99

CityPopulation

in 2000

Atlanta 416,000

Charleston 97,000

Raleigh 276,000

Tallahassee 151,000

19.

20.

y

xO

Chapter 9 Open-Ended Assessment


Demonstrate your knowledge by giving a clear, concise solutionto each problem. Be sure to include all relevant drawings andjustify your answers. You may show your solutions in more thanone way or investigate beyond the requirements of the problem.

1. Write three different rational expressions that are equivalent to theexpression �a �

a5�.

2. The volume of the rectangular box shown is given by V � (2x3 � 26x2 � 60x) cubic inches.a. Explain how to find an expression in terms

of x for the height h of the box.b. In terms of x, h � _______?________ in simplest form.c. Explain how you could check the expression you found

in part b. Then check your expression.

3. Write two polynomials for which the LCM is 3y2 � 12.

4. Compare and contrast the graphs of the rational functions

f(x) � �(x �

x2�)(x

2� 3)

� and g(x) � �(x �

x(x2)

�(x

2�)

3)�.

5. You decide to invest 10% of your before-tax income in a retirement fund,so you have your employer deduct this money from your weekly paycheck.a. Write an equation to represent the amount deducted from your paycheck

d for investment in your retirement fund for a week during which youworked h hours at r dollars per hour.

b. Is your equation a direct, joint, or inverse variation? Explain your choice.c. If you earn $9.50 per hour and worked 36 hours last week, explain how to

determine the amount deducted last week for your retirement fund.

6. The Franklin Electronics Company has determined that, after its first 50 CDplayers are produced, the average cost of producing one CD player can be

approximated by the function C(x) � �60x

x��

1570,000

�, where x represents the

number of CD players produced. Consumer research has indicated that thecompany should charge the consumer $80 per CD player in order to maximizeits profit. Thus, the revenue from the sale of each CD player can be representedby the function R(x) � 80.a. Identify the function represented by C(x). Explain your choice.b. Identify the function represented by R(x). Explain your choice.c. The company wants to determine how many CD players must be produced

and sold in order to ensure that the revenue from each one is greater thanthe average cost of producing each one. Write an inequality whose solutionrepresents the information for which the company is looking.

d. Solve your inequality and interpret your solution in the context of the problem.

2x in.h

(x � 10) in.

NAME DATE PERIOD

SCORE 99

Ass

essm

ent


Chapter 9 Vocabulary Test/Review

Underline or circle the correct word or phrase to complete each sentence.

1. The equation y � �3x� is an example of (direct variation, inverse variation,

joint variation).

2. r(x) � �xx

2

2��

65xx

��

96� is an example of a (complex fraction, rational function,

rational expression).

3. The graph of y � �x �3

5� has a(n) (asymptote, point discontinuity,

constant of variation).

4. Adding or subtracting rational expressions requires you to find a(n) (least common denominator, asymptote, complex fraction).

5. The formula for simple interest, I � Prt, is an example of (direct variation, inverse variation, joint variation).

6. The graph of y � �xx

��

53� has a break in (asymptote, discontinuity, continuity)

at x � 3.

7. �2t�

� �t32� 1 is an example of a (rational inequality, rational equation,

rational function).

8. If you walk at a steady speed, your speed and the time it takes towalk 1 mile are (asymptotes, inversely proportional, direct variations)to each other.

9. The equation C � �d gives the circumference of a circle in terms of itsdiameter. Here, � is called the (constant of variation, point discontinuity,asymptote).

10. If the rational expression in a rational function is not written in lowest terms, the graph of the function may have a (continuity,constant of variation, point discontinuity).

In your own words—Define each term.

11. rational expression

12. complex fraction

asymptotecomplex fractionconstant of variation

continuitydirect variationinverse variation

joint variationpoint discontinuityrational equation

rational expressionrational functionrational inequality

NAME DATE PERIOD

SCORE 99

Chapter 9 Quiz (Lessons 9–1 and 9–2)

99


NAME DATE PERIOD

SCORE

Chapter 9 Quiz (Lesson 9–3)

For Questions 1–3, determine the equations of any vertical asymptotes and the values of x for any holes in the graph of each rational function.

1. f(x) � �x2 �3x � 2�

2. f(x) � �x2 �x �

2x3� 3�

3. f(x) � �x2 �

x �2x

4� 8

�

4. Graph f(x) � �x �4

3�.

NAME DATE PERIOD

SCORE 99

Ass

essm

ent

9


1. �1x22an3

4n

� � �96ax7

5nn

5

2� 2. �x2

3�x �

6x1�2

8� � �x2 �

x2

5�x

4� 6�

3. �2x2

x��

x4� 3

� � �x2 �

x2�x

1� 24� 4.

5. Standardized Test Practice For what value(s) of x is the

expression �xx2

2��

57xx

��

1140� undefined?

A. �5, 2 B. 0, 2, 5 C. �2 D. 0, 2 E. �5, �2


6. 12a2, 15b3, 20ab2 7. 5x2 � 20, 3x � 6

8. 2t2 � 3t � 1, 2t2 � 7t � 4


9. �m72n�

� �5m2

n� 10. �y25�y

3y� � �3 �7

y�

�p2p�

2 �6p

3�p

9��

�4p2�0

12�

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1.

2.

3.4.

xO

f (x)


1. State whether rt � 30 represents a direct, joint, or inverse 1.variation. Then name the constant of variation.

2. Suppose y varies jointly as x and z. Find y when x � 1 and 2.z � 4, if y � 96 when x � 4 and z � 8.


3. 4. 3.

5. Identify the type of function represented by y � 3� x � � 2. 5.Then graph the equation.

xO

yy

xO

Chapter 9 Quiz (Lesson 9–6)

For Questions 1–4, solve each equation or inequality.

1. �x �6

2� � �xx

��

72� � �

14� 1.

2. �tt

��

53� � �

tt

��

33� � �t �

13� 2.

3. 3 � �2t�

�8t�

3.

4. �m6� 5� 2 4.

5. NUMBER THEORY The ratio of two less than a number 5.to six more than that number is 2 to 3. Find the number.

NAME DATE PERIOD

SCORE

Chapter 9 Quiz (Lessons 9–4 and 9–5)

99

NAME DATE PERIOD

SCORE

99

4.

y

xO

Chapter 9 Mid-Chapter Test (Lessons 9–1 through 9–3)



1. For what value(s) of x is the expression �(x �2x

4(x)(

�x2

3�)

9)� undefined?

A. �4, 9 B. �4, �3, 0, 3 C. �4, 0, 3, 9 D. �4, �3, 3 1.


2. �92yy2

��

11

� � �13y

��

21y

�

A. �3y � 1 B. 3y � 1 C. �3y � 1 D. 3y � 1 2.

3. �cc2

2��

c6c

��205� � �

c32

c��

136

�

A. �c �3

4� B. �c �3

4� C. �c �

34

� D. �c �

34

� 3.

4.

A. �169m(m

2(m�

�2)

2)� B. �m(mm2

��2

4)� C. m � 2 D. �

4(m3� 2)� 4.

5. �15� � �4

3w�

� �103w�

A. �4w

20�w

21� B. �

4w20

�w

9� C. �20

1w�

D. ��41w�

5.

6. Simplify �x2 �xx � 6� � �x2 � 6

1x � 8�. 6.

For Questions 7 and 8, find the LCM for each set of polynomials.

7. 12s3, 18s2t, 24t4 7.

8. 9c � 15, 21c � 35 8.

9. Determine the equations of any vertical asymptotes and the 9.

values of x for any holes in the graph of f(x) � �x2 �x �

x �3

12�.

10. Graph f(x) � �(x �4

2)2�. 10.

�43mm

2

2

��

81m2

��8m

6m2 �

�1162m�

Part I

NAME DATE PERIOD

SCORE 99

Ass

essm

ent

xO

f (x)

Part II


Chapter 9 Cumulative Review (Chapters 1–9)

1. Determine whether C � � � and D � � � are 1.

inverses. (Lesson 4-7)

2. Simplify the expression �w�13��

�25�

. (Lesson 5-7) 2.

3. Solve x2 � 2x � 2 � 0 by completing the square. (Lesson 6-4) 3.

4. Graph y � x2 � 4x. (Lesson 6-7) 4.

5. Use synthetic substitution to find f(3) for 5.f(x) � 3x3 � 7x2 � 5x � 10. (Lesson 7-4)

6. List all of the possible rational zeros of 6.2x4 � 5x3 � 3x2 � 12x � 6. (Lesson 7-6)

7. Write an equation for a circle with center at (0, �3) that 7.passes through (5, 7). (Lessons 8-1 and 8-3)

8. Write an equation for the ellipse whose major axis is 8.10 units long and parallel to the x-axis, whose minor axis is 6 units long, and whose center is at (1, �2). (Lesson 8-4)

9. State whether the graph of 5x2 � 5y2 � 10x � 15y � 10 is an 9.ellipse, circle, parabola, or hyperbola. (Lesson 8-6)

10. Simplify . (Lesson 9-1) 10.

11. Suppose y varies jointly as x and z. Find y when x � 16 and 11.z � 5, if y � 9 when x � 3 and z � 12. (Lesson 9-4)

12. Evita adds a 75% acid solution to 8 milliliters of solution 12.that is 15% acid. The function that represents the percent

of acid in the resulting solution is f(x) ��8(0.15

8) �

�xx(0.75)

�,

where x is the amount of 75% acid solution added. How much 75% acid solution should be added to create a solution that is 50% acid? (Lesson 9-6)

�59yy2

2

��

1306y�

��10

6yy2��

1220y�

y

xO

�116�

��156�

1 5�3 1

NAME DATE PERIOD

99

Standardized Test Practice (Chapters 1–9)


1. If 6 more than the product of a number and �2 is greater than 10, which of the following could be that number?A. �3 B. �2 C. 0 D. 3 1.

2. If the diameter of a circle is doubled, then the area is multiplied by _______.E. 2 F. 4 G. 8 H. 16 2.

3. Which represents an irrational number?

A. ��13� B. 1 C. �2� D. �9� 3.

4. If a 0, which of the following must be true?E. a � 2 2 � a F. �2a a2

G. a � 2 2a H. a2 a � 2 4.

5. A cube is equal in volume to a rectangular solid with edges that measure 4, 6, and 9. What is the measure of an edge of the cube?A. 216 B. 36 C. 108 D. 6 5.

6. If abc � 30 and b � c, then a equals which of the following?

E. �3c02� F. �

1c5� G. 30c2 H. 15c 6.

7. What is the value of (a � b)3 if b � a � 2?A. �8 B. �6 C. 6 D. 8 7.

8. In the figure, WXZ and XYZ are isosceles right triangles. If XY � 8, find the perimeter of quadrilateral WXYZ.E. 16 � 16�2� F. 24 � 8�2� G. 32 � 8�2� H. 32 � 16�2� 8.

9. In a 30-day month, how many weekend days fall on dates that are prime numbers if the first day of the month is Thursday?A. 2 B. 3 C. 4 D. 5 9.

10. Sonia purchased 5 pencils and 2 pens for $5.10. Wai purchased 8 of the same type of pencil and 6 of the same type of pen, and spent $13.20. What is the cost of 2 pencils and one pen?E. $2.10 F. $3.90 G. $1.80 H. $2.40 10. HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

99

Ass

essm

ent

Part 1: Multiple Choice

Instructions: Fill in the appropriate oval for the best answer.

Z Y

W X


Standardized Test Practice (continued)

11. 3 is 12% of what number? 11. 12.

12. If w � 4x, y � 10z, x � 3, and z � �12�, what

is the value of �2y� � �w

3�?

13. How many rectangles can be found in the figure shown?

13. 14.

14. What is the value of a in the figure shown?

Column A Column B

15. R h � 1 15.

16.m � n � p � 3m

16.

17. �3t�

� 2; �2s� � 3 17.

18. y 0 18.

�3y3

y�2

y2�3y � 1

DCBA

st

DCBA

pn

DCBAm �

p �n �

hR

DCBA

Part 3: Quantitative Comparison

Instructions: Compare the quantities in columns A and B. Shade in if the quantity in column A is greater; if the quantity in column B is greater; if the quantities are equal; or if the relationship cannot be determined from the information given.

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

NAME DATE PERIOD

99

NAME DATE PERIOD

Part 2: Grid In

Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.

a�

61�

25�18�

A

D

C

B

Standardized Test PracticeStudent Record Sheet (Use with pages 518–519 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

NAME DATE PERIOD

99

An

swer

s

Select the best answer from the choices given and fill in the corresponding oval.

1 4 7

2 5 8

3 6 9

Solve the problem and write your answer in the blank.

For Questions 14–20, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.

10 15 17 19

11

12

13

14 16 18 20

Select the best answer from the choices given and fill in the corresponding oval.

21 23 25

22 24 DCBADCBA

DCBADCBADCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

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87654321

0 0 0

.. ./ /

.

99 9 987654321

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0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

DCBADCBADCBA

DCBADCBADCBA

DCBADCBADCBA

Part 2 Short Response/Grid InPart 2 Short Response/Grid In

Part 1 Multiple ChoicePart 1 Multiple Choice

Part 3 Quantitative ComparisonPart 3 Quantitative Comparison


Answers (Lesson 9-1)

Stu

dy G

uid

e a

nd I

nte

rven

tion

Mu

ltip

lyin

g a

nd

Div

idin

g R

atio

nal

Exp

ress

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-1

9-1

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Lesson 9-1

Sim

plif

y R

atio

nal

Exp

ress

ion

sA

rat

io o

f tw

o po

lyn

omia

l ex

pres

sion

s is

a r

atio

nal

exp

ress

ion

.To

sim

plif

y a

rati

onal

exp

ress

ion

,div

ide

both

th

e n

um

erat

or a

nd

the

den

omin

ator

by

thei

r gr

eate

st c

omm

on f

acto

r (G

CF

).

Mu

ltip

lyin

g R

atio

nal

Exp

ress

ion

sF

or a

ll ra

tiona

l exp

ress

ions

an

d ,

��

, if b

�0

and

d�

0.

Div

idin

g R

atio

nal

Exp

ress

ion

sF

or a

ll ra

tiona

l exp

ress

ions

an

d ,

��

, if b

�0,

c�

0, a

nd d

�0.

Sim

pli

fy e

ach

exp

ress

ion

.

a.

��

b.

� ��

��

c.� �

��

�

�

Sim

pli

fy e

ach

exp

ress

ion

.

1.�(�

22 0a ab b2 4)3�

�2.

3.

4.�

2m2 (

m�

1)5.

�

6.�

m7.

�

8.�

9.�

4�

�(2

m�

1)(m

�5)

p(4

p�

1)�

�2(

p�

2)

4m2

�1

��

4m�

82m

�1

��

m2

�3m

�10

4p2

�7p

�2

��

7p5

16p2

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�1

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y5

� 15z5

18xz

2�

5y6x

y4� 25

z3m

3�

9m�

�m

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9(m

�3)

2�

�m

2�

6m�

9

c� c

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c2�

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5�

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c2�

3c� c2

�25

4m5

� m�

13m

3�

3m�

�6m

4

x�

2� x

�9

x2�

x�

6�

�x2

�6x

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3 �

2x�

34x

2�

12x

�9

��

9 �

6x2a

2 b2

�5

x�

4� 2(

x�

2)(x

�4)

(x�

4)(x

�1)

��

�2(

x�

1)(x

�2)

(x�

4)x�

1�

�x2

�2x

�8

x2�

8x�

16�

�2x

�2

x2�

2x�

8�

�x

�1

x2�

8x�

16�

�2x

�2

x2�

2x�

8�

�x

�1

x2�

8x�

16�

�2x

�2

4s2

� 3rt2

2 �

2 �

s�

s�

�3

�r

�t

�t

3 �

r�

r�

s�

s�

s�

2 �

2 �

5 �

t�

t�

��

�5

�t

�t

�t

�t

�3

�3

�r

�r

�r

�s

20t2

� 9r3 s

3r2 s

3�

5t4

20t2

� 9r3 s

3r2 s

3�

5t4

3a � 2b2

2 �

2 �

2 �

3 �

a�

a�

a�

a�

a�

b�

b�

��

��

2 �

2 �

2 �

2 �

a�

a�

a�

a�

b�

b�

b�

b24

a5 b2

� (2ab

)4

24a

5 b2

� (2a

b)4

ad � bcc � d

a � bc � d

a � b

ac � bdc � d

a � bc � d

a � b

Exam

ple

Exam

ple

Exer

cises

Exer

cises

11

11

11

11

11

1

11

11

11

1

11

11

11

11

1

11

11

11

11

1

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Sim

plif

y C

om

ple

x Fr

acti

on

sA

com

ple

x fr

acti

onis

a r

atio

nal

exp

ress

ion

wh

ose

nu

mer

ator

an

d/or

den

omin

ator

con

tain

s a

rati

onal

exp

ress

ion

.To

sim

plif

y a

com

plex

frac

tion

,fir

st r

ewri

te i

t as

a d

ivis

ion

pro

blem

.

Sim

pli

fy

.

��

Exp

ress

as

a di

visi

on p

robl

em.

��

Mul

tiply

by

the

reci

proc

al o

f th

e di

viso

r.

�F

acto

r.

�S

impl

ify.

Sim

pli

fy.

1.2.

3.(b

�1)

2

4.5.

6.a

�4

7.x

�3

8.9.

1� x

�5

x2�

x�

2�

��

x3�

6x2

�x

�30

��

�x

�1

� x�

3

b�

4�

�(b

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(b�

2)

� b2�b

� 6b2 �

8�

��

�b2

b2��b

1� 62

�

�2x2

x��9x

1�9

�

��

�10 5x x2 2

� �1 79 xx �

�26

�

�a a2� �

1 26�

��

�a a2 2� �3 aa ��

24�

1�

�(x

�3)

(x�

2)

� x2�x

� 6x4 �

9�

��

�x2� 3

�2xx�

8�

2(b

�10

)�

�b

(3b

�1)

�b2� b210

0�

��

�3b2

�3 21 bb

�10

�

� 3b b2� �

1 2�

��

� 3b2b �

�b1 �

2�

ac7

� by

�a x2 2b yc 23�

� � ca 4 xb 22 y�

xyz

� a5

�x a3 2y b2 2z�

� �a3

bx 22 y �

s3� s

�3(3s

�1)

s4�

�s(

3s�

1)(s

�3)s4

��

3s2

�8s

�3

3s�

1�

s

3s2

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1�

s

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dy G

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rven

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(c

onti

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)

Mu

ltip

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nd

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idin

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atio

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s

NA

ME

____

____

____

____

____

____

____

____

____

____

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AT

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____

____

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IOD

____

_

9-1

9-1

Exam

ple

Exam

ple

1

11

s3

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Mu

ltip

lyin

g a

nd

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idin

g R

atio

nal

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ress

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

E__

____

____

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ER

IOD

____

_

9-1

9-1

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Lesson 9-1

Sim

pli

fy e

ach

exp

ress

ion

.

1.2.

3.x

64.

5.6.

7.8.

�

9.�

6e10

.�

11.

�21

g312

.�

13.

�x

(x�

2)14

.�

15.

�16

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(w�

8)(w

�7)

17.

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x2�

3x)

18.

�

19.

�20

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�b

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�a2

4� ab2

�

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2� ab

�

5� 2c

4 d

� 2c d2 2�

� �� 5c d6 �

(4a

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(a�

4)�

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4a�

5�

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16a2

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a�

25�

��

3a2

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a�

81 � 6x

x2�

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t�

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t�

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2 y25

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z12v5

80y4

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71

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10(e

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mn

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a

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318

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6

2 � y4

8y2 (

y6 )3

�4y

24(x

6 )3

� (x3 )

4

b � 5a5a

b3� 25

a2 b2

3x � 2y21

x3 y� 14

x2 y2

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pli

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.

1.2.

�3.

4.5.

�

6.7.

��

8.�

9.�

10.

�n

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11.

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12.

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9-1

9-1



Readin

g t

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earn

Math

em

ati

csM

ult

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ing

an

d D

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9-1

9-1

©G

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1G

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lgeb

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Lesson 9-1

Pre-

Act

ivit

yH

ow a

re r

atio

nal

exp

ress

ion

s u

sed

in

mix

ture

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

9-1

at

the

top

of p

age

472

in y

our

text

book

.

•S

upp

ose

that

th

e G

oodi

e S

hop

pe a

lso

sell

s a

can

dy m

ixtu

re o

f ch

ocol

ate

min

ts a

nd

cara

mel

s.If

th

is m

ixtu

re i

s m

ade

wit

h 4

pou

nds

of

choc

olat

e

min

ts a

nd

3 po

un

ds o

f ca

ram

els,

then

of

th

e m

ixtu

re i

s

min

ts a

nd

of t

he

mix

ture

is

cara

mel

s.

•If

th

e st

ore

man

ager

add

s an

oth

er y

pou

nds

of

min

ts t

o th

e m

ixtu

re,w

hat

frac

tion

of

the

mix

ture

wil

l be

min

ts?

Rea

din

g t

he

Less

on

1.a.

In o

rder

to

sim

plif

y a

rati

onal

nu

mbe

r or

rat

ion

al e

xpre

ssio

n,

the

nu

mer

ator

an

d an

d di

vide

bot

h o

f th

em b

y th

eir

.

b.

A r

atio

nal e

xpre

ssio

n is

und

efin

ed w

hen

its

is e

qual

to

.

To

fin

d th

e va

lues

th

at m

ake

the

expr

essi

on u

nde

fin

ed,c

ompl

etel

y

the

orig

inal

an

d se

t ea

ch f

acto

r eq

ual

to

.

2.a.

To

mu

ltip

ly t

wo

rati

onal

exp

ress

ion

s,th

e an

dm

ult

iply

th

e de

nom

inat

ors.

b.

To

divi

de t

wo

rati

onal

exp

ress

ion

s,by

th

e of

the

.

3.a.

Wh

ich

of

the

foll

owin

g ex

pres

sion

s ar

e co

mpl

ex f

ract

ion

s?ii,

iv,v

i.ii

.ii

i.iv

.v.

b.

Doe

s a

com

plex

fra

ctio

n e

xpre

ss a

mu

ltip

lica

tion

or

divi

sion

pro

blem

?d

ivis

ion

How

is

mu

ltip

lica

tion

use

d in

sim

plif

yin

g a

com

plex

fra

ctio

n?

Sam

ple

an

swer

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div

ide

the

nu

mer

ato

r o

f th

e co

mp

lex

frac

tio

n b

y th

e d

eno

min

ato

r,m

ult

iply

th

e n

um

erat

or

by t

he

reci

pro

cal o

f th

e d

eno

min

ato

r.

Hel

pin

g Y

ou

Rem

emb

er

4.O

ne

way

to

rem

embe

r so

met

hin

g n

ew i

s to

see

how

it

is s

imil

ar t

o so

met

hin

g yo

ual

read

y kn

ow.H

ow c

an y

our

know

ledg

e of

div

isio

n o

f fr

acti

ons

in a

rith

met

ic h

elp

you

to

un

ders

tan

d h

ow t

o di

vide

rat

ion

al e

xpre

ssio

ns?

Sam

ple

an

swer

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div

ide

rati

on

alex

pre

ssio

ns,

mu

ltip

ly t

he

firs

t ex

pre

ssio

n b

y th

e re

cip

roca

l of

the

seco

nd

.Th

is is

th

e sa

me

“inv

ert

and

mu

ltip

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pro

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th

at is

use

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hem

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s,th

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rm g

rou

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as a

spe

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scu

ss t

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idea

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grou

p an

d on

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tere

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ampl

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a g

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01T

o be

a g

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set

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ents

an

d a

bin

ary

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atio

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ust

sat

isfy

fou

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t m

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be

clos

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r th

e op

erat

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erat

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st b

e as

soci

ativ

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st b

e an

ide

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ent,

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ever

yel

emen

t m

ust

hav

e an

in

vers

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02T

he

foll

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x fu

nct

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s fo

rm a

gro

up

un

der

the

oper

atio

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f

com

posi

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of

fun

ctio

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) �

x,f 2

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��1 x� ,

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1 �

x,

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�(x� x

1)�

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d f 6

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his

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04S

ome

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f1.

05E

very

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ow

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4an

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th

e p

arag

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h t

o an

swer

th

ese

qu

esti

ons.

1.E

xpla

in w

hat

it

mea

ns

to s

ay t

hat

a s

et i

s cl

osed

un

der

an o

pera

tion

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the

set

of p

osit

ive

inte

gers

clo

sed

un

der

subt

ract

ion

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erfo

rmin

g t

he

op

er-

atio

n o

n a

ny t

wo

ele

men

ts o

f th

e se

t re

sult

s in

an

ele

men

t o

f th

esa

me

set.

No

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nd

4 a

re p

osi

tive

inte

ger

s bu

t 3

� 4

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ot.

2.S

ubt

ract

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a n

onco

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tive

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on f

or t

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set

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in

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al d

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itio

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f n

onco

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tive

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er in

wh

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th

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re u

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h t

he

op

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can

aff

ect

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resu

lt.

3.F

or t

he

set

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nte

gers

,wh

at i

s th

e id

enti

ty e

lem

ent

for

the

oper

atio

n o

fm

ult

ipli

cati

on?

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ify

you

r an

swer

.1,

bec

ause

,fo

r ev

ery

inte

ger

a,a

�1

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and

1 �

a�

a.

4.E

xpla

in h

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atem

ent

rela

tes

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ente

nce

05:

(f6

�f 4

)(x)

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[f4(

x)]

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inve

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rich

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t

NA

ME

____

____

____

____

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AT

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____

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ER

IOD

____

_

9-1

9-1


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Ad

din

g a

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Su

btr

acti

ng

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ion

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xpre

ssio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

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____

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ER

IOD

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_

9-2

9-2

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�2)

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x�

3)�

��

2(x

�3)

(x�

2)(x

�2)

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2(x

�3)

��

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x�

3)(x

�2)

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2)6(

x�

2)�

��

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�3)

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2)(x

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(x�

2)6

��

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2)

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dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

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din

g a

nd

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btr

acti

ng

Rat

ion

al E

xpre

ssio

ns

NA

ME

____

____

____

____

____

____

____

____

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____

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AT

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____

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ER

IOD

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_

9-2

9-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Ad

din

g a

nd

Su

btr

acti

ng

Rat

ion

al E

xpre

ssio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

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AT

E__

____

____

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ER

IOD

____

_

9-2

9-2

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Lesson 9-2

Fin

d t

he

LC

M o

f ea

ch s

et o

f p

olyn

omia

ls.

1.12

c,6c

2 d12

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pli

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ach

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.

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Fin

d t

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LC

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f ea

ch s

et o

f p

olyn

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pli

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ach

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24.

25. G

EOM

ETRY

The

exp

ress

ions

,

,and

re

pres

ent

the

leng

ths

of t

he s

ides

of

a

tria

ngle

.Wri

te a

sim

plif

ied

expr

essi

on f

or t

he p

erim

eter

of

the

tria

ngle

.

26.K

AYA

KIN

GM

ai i

s ka

yaki

ng

on a

riv

er t

hat

has

a c

urr

ent

of 2

mil

es p

er h

our.

If r

repr

esen

ts h

er r

ate

in c

alm

wat

er,t

hen

r�

2 re

pres

ents

her

rat

e w

ith

th

e cu

rren

t,an

d r

�2

repr

esen

ts h

er r

ate

agai

nst

th

e cu

rren

t.M

ai k

ayak

s 2

mil

es d

own

stre

am a

nd

then

back

to

her

sta

rtin

g po

int.

Use

th

e fo

rmu

la f

or t

ime,

t�

,wh

ere

dis

th

e di

stan

ce,t

o

wri

te a

sim

plif

ied

expr

essi

on f

or t

he

tota

l ti

me

it t

akes

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to

com

plet

e th

e tr

ip.

h4r

��

(r�

2)(r

�2)

d � r

5(x3

�4x

�16

)�

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x�

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e (

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rag

e)

Ad

din

g a

nd

Su

btr

acti

ng

Rat

ion

al E

xpre

ssio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

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AT

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ER

IOD

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9-2

9-2


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csA

dd

ing

an

d S

ub

trac

tin

g R

atio

nal

Exp

ress

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

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ER

IOD

____

_

9-2

9-2

©G

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lenc

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lgeb

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Lesson 9-2

Pre-

Act

ivit

yH

ow i

s su

btr

acti

on o

f ra

tion

al e

xpre

ssio

ns

use

d i

n p

hot

ogra

ph

y?

Rea

d th

e in

trod

uct

ion

to

Les

son

9-2

at

the

top

of p

age

479

in y

our

text

book

.

A p

erso

n i

s st

andi

ng

5 fe

et f

rom

a c

amer

a th

at h

as a

len

s w

ith

a f

ocal

len

gth

of

3 fe

et.W

rite

an

equ

atio

n t

hat

you

cou

ld s

olve

to

fin

d h

ow f

ar t

he

film

sh

ould

be

from

th

e le

ns

to g

et a

per

fect

ly f

ocu

sed

phot

ogra

ph.

��

Rea

din

g t

he

Less

on

1.a.

In w

ork

wit

h r

atio

nal

exp

ress

ion

s,L

CD

sta

nds

for

and

LC

M s

tan

ds f

or

.Th

e L

CD

is

the

of t

he

den

omin

ator

s.

b.

To

fin

d th

e L

CM

of

two

or m

ore

nu

mbe

rs o

r po

lyn

omia

ls,

each

nu

mbe

r or

.T

he

LC

M c

onta

ins

each

th

e

nu

mbe

r of

tim

es i

t ap

pear

s as

a

.

2.T

o ad

d an

d ,y

ou s

hou

ld f

irst

fac

tor

the

of

each

fra

ctio

n.T

hen

use

th

e fa

ctor

izat

ion

s to

fin

d th

e of

x2

�5x

�6

and

x3�

4x2

�4x

.Th

is i

s th

e fo

r th

e tw

o fr

acti

ons.

3.W

hen

you

add

or

subt

ract

fra

ctio

ns,

you

oft

en n

eed

to r

ewri

te t

he

frac

tion

s as

equ

ival

ent

frac

tion

s.Yo

u d

o th

is s

o th

at t

he

resu

ltin

g eq

uiv

alen

t fr

acti

ons

wil

l ea

ch h

ave

a

den

omin

ator

equ

al t

o th

e of

th

e or

igin

al f

ract

ion

s.

4.T

o ad

d or

su

btra

ct t

wo

frac

tion

s th

at h

ave

the

sam

e de

nom

inat

or,y

ou a

dd o

r su

btra

ct

thei

r an

d ke

ep t

he

sam

e .

5.T

he

sum

or

diff

eren

ce o

f tw

o ra

tion

al e

xpre

ssio

ns

shou

ld b

e w

ritt

en a

s a

poly

nom

ial

or

as a

fra

ctio

n i

n

.

Hel

pin

g Y

ou

Rem

emb

er

6.S

ome

stu

den

ts h

ave

trou

ble

rem

embe

rin

g w

het

her

a c

omm

on d

enom

inat

or i

s n

eede

d to

add

and

subt

ract

rat

ion

al e

xpre

ssio

ns

or t

o m

ult

iply

an

d di

vide

th

em.H

ow c

an y

our

know

ledg

e of

wor

kin

g w

ith

fra

ctio

ns

in a

rith

met

ic h

elp

you

rem

embe

r th

is?

Sam

ple

an

swer

:In

ari

thm

etic

,a c

om

mo

n d

eno

min

ato

r is

nee

ded

to

ad

dan

d s

ub

trac

t fr

acti

on

s,bu

t n

ot

to m

ult

iply

an

d d

ivid

e th

em.T

he

situ

atio

nis

th

e sa

me

for

rati

on

al e

xpre

ssio

ns.

sim

ple

st f

orm

den

om

inat

or

nu

mer

ato

rs

LC

D

LC

D

LC

M

den

om

inat

or

x�

4�

�x3

�4x

2�

4xx2

�3

��

x2�

5x�

6

fact

or

gre

ates

tfa

cto

rp

oly

no

mia

lfa

cto

r

LC

Mle

ast

com

mo

n m

ult

iple

leas

t co

mm

on

den

om

inat

or

1 � 51 � 3

1 � q

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Su

per

ellip

ses

Th

e ci

rcle

an

d th

e el

lips

e ar

e m

embe

rs o

f an

in

tere

stin

g fa

mil

y of

cu

rves

th

at w

ere

firs

t st

udi

ed b

y th

e F

ren

ch p

hys

icis

t an

d m

ath

emat

icia

n G

abri

el

Lam

é (1

795–

1870

).T

he

gen

eral

equ

atio

n f

or t

he

fam

ily

is

�� ax � �n�

�� by � �n�

1,w

ith

a�

0,b

�0,

and

n�

0.

For

eve

n v

alu

es o

f n

grea

ter

than

2,t

he

curv

es a

re c

alle

d su

per

elli

pse

s.

1.C

onsi

der

two

curv

es t

hat

are

not

supe

rell

ipse

s.G

raph

eac

h e

quat

ion

on

th

e gr

id a

t th

e ri

ght.

Sta

te t

he

type

of

curv

e pr

odu

ced

each

tim

e.

a.�� 2x � �2

�� 2y � �2

�1

circ

le

b.

�� 3x � �2�

�� 2y � �2�

1el

lipse

2.In

eac

h o

f th

e fo

llow

ing

case

s yo

u a

re

give

n v

alu

es o

f a,

b,an

d n

to u

se i

n t

he

gen

eral

equ

atio

n.W

rite

th

e re

sult

ing

equ

atio

n.T

hen

gra

ph.S

ketc

h e

ach

gra

ph

on t

he

grid

at

the

righ

t.

a.a

�2,

b�

3,n

�4

b.

a�

2,b

�3,

n�

6 c.

a�

2,b

�3,

n�

8

See

stu

den

ts’g

rap

hs.

3.W

hat

sh

ape

wil

l th

e gr

aph

of

�� 2x � �n�

�� 2y � �n

appr

oxim

ate

for

grea

ter

and

grea

ter

even

,w

hol

e-n

um

ber

valu

es o

f n

?

a re

ctan

gle

th

at is

6 u

nit

s lo

ng

an

d4

un

its

wid

e,ce

nte

red

at

the

ori

gin

1–1

–2–3

23

3 2 1 –1 –2 –3

1–1

–2–3

23

3 2 1 –1 –2 –3

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-2

9-2



Stu

dy G

uid

e a

nd I

nte

rven

tion

Gra

ph

ing

Rat

ion

al F

un

ctio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-3

9-3

©G

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ill52

9G

lenc

oe A

lgeb

ra 2

Lesson 9-3

Ver

tica

l Asy

mp

tote

s an

d P

oin

t D

isco

nti

nu

ity

Rat

ion

al F

un

ctio

nan

equ

atio

n of

the

for

m f

(x)

�,

whe

re p

(x)

and

q(x

) ar

e po

lyno

mia

l exp

ress

ions

and

q

(x)

�0

Ver

tica

l Asy

mp

tote

A

n as

ympt

ote

is a

line

tha

t th

e gr

aph

of a

fun

ctio

n ap

proa

ches

, bu

t ne

ver

cros

ses.

o

f th

e G

rap

h o

f a

If th

e si

mpl

ified

for

m o

f th

e re

late

d ra

tiona

l exp

ress

ion

is u

ndef

ined

for

x�

a,

Rat

ion

al F

un

ctio

nth

en x

�a

is a

ver

tical

asy

mpt

ote.

Po

int

Dis

con

tin

uit

y P

oint

dis

cont

inui

ty is

like

a h

ole

in a

gra

ph.

If th

e or

igin

al r

elat

ed e

xpre

ssio

n is

und

efin

ed

of

the

Gra

ph

of

a fo

r x

�a

but

the

sim

plifi

ed e

xpre

ssio

n is

def

ined

for

x�

a, t

hen

ther

e is

a h

ole

in t

he

Rat

ion

al F

un

ctio

ngr

aph

at x

�a.

Det

erm

ine

the

equ

atio

ns

of a

ny

vert

ical

asy

mp

tote

s an

d t

he

valu

es

of x

for

any

hol

es i

n t

he

grap

h o

f f(

x) �

.

Fir

st f

acto

r th

e n

um

erat

or a

nd

the

den

omin

ator

of

the

rati

onal

exp

ress

ion

.

f(x)

��

Th

e fu

nct

ion

is

un

defi

ned

for

x�

1 an

d x

��

1.

Sin

ce

�,x

�1

is a

ver

tica

l as

ympt

ote.

Th

e si

mpl

ifie

d ex

pres

sion

is

defi

ned

for

x�

�1,

so t

his

val

ue

repr

esen

ts a

hol

e in

th

e gr

aph

.

Det

erm

ine

the

equ

atio

ns

of a

ny

vert

ical

asy

mp

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9-3

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9-3

9-3



Readin

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9-3

9-3

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Lesson 9-3

Pre-

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how

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e pl

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9-3

9-3


An

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Stu

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9-4

9-4

©G

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Lesson 9-4

Dir

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Dir

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e va

lue

of y

is 1

5 w

hen

x�

4 an

d z

�3.

y 2� 4

�3

10� 2

�4

y 2� x 2

z 2

y 1� x 1z

1

Exer

cises

Exer

cises

Fin

d e

ach

val

ue.

1.If

yva

ries

dir

ectl

y as

xan

d y

�9

wh

en

2.If

yva

ries

dir

ectl

y as

xan

d y

�16

wh

en

x�

6,fi

nd

yw

hen

x�

8.12

x�

36,f

ind

yw

hen

x�

54.

24

3.If

yva

ries

dir

ectl

y as

xan

d x

�15

4.

If y

vari

es d

irec

tly

as x

and

x�

33 w

hen

w

hen

y�

5,fi

nd

xw

hen

y�

9.27

y�

22,f

ind

xw

hen

y�

32.

48

5.S

upp

ose

yva

ries

join

tly

as x

and

z.6.

Su

ppos

e y

vari

es jo

intl

y as

xan

d z.

Fin

d y

Fin

d y

wh

en x

�5

and

z�

3,if

y�

18

wh

en x

�6

and

z�

8,if

y�

6 w

hen

x�

4w

hen

x�

3 an

d z

�2.

45an

d z

�2.

36

7.S

upp

ose

yva

ries

join

tly

as x

and

z.8.

Su

ppos

e y

vari

es jo

intl

y as

xan

d z.

Fin

d y

Fin

d y

wh

en x

�4

and

z�

11,i

f y

�60

w

hen

x�

5 an

d z

�2,

if y

�84

wh

en

wh

en x

�3

and

z�

5.17

6x

�4

and

z�

7.30

9.If

yva

ries

dir

ectl

y as

xan

d y

�14

10

.If

yva

ries

dir

ectl

y as

xan

d x

�20

0 w

hen

wh

en x

�35

,fin

d y

wh

en x

�12

.4.

8y

�50

,fin

d x

wh

en y

�10

00.

4000

11.I

f y

vari

es d

irec

tly

as x

and

y�

39

12.I

f y

vari

es d

irec

tly

as x

and

x�

60 w

hen

wh

en x

�52

,fin

d y

wh

en x

�22

.16

.5y

�75

,fin

d x

wh

en y

�42

.33

.6

13.S

upp

ose

yva

ries

join

tly

as x

and

z.14

.Su

ppos

e y

vari

es jo

intl

y as

xan

d z.

Fin

d y

Fin

d y

wh

en x

�6

and

z�

11,i

f

wh

en x

�5

and

z�

10,i

f y

�12

wh

en

y�

120

wh

en x

�5

and

z�

12.

132

x�

8 an

d z

�6.

12.5

15.S

upp

ose

yva

ries

join

tly

as x

and

z.16

.Su

ppos

e y

vari

es jo

intl

y as

xan

d z.

Fin

d y

Fin

d y

wh

en x

�7

and

z�

18,i

f w

hen

x�

5 an

d z

�27

,if

y�

480

wh

en

y�

351

wh

en x

�6

and

z�

13.

567

x�

9 an

d z

�20

.36

0

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lenc

oe/M

cGra

w-H

ill53

6G

lenc

oe A

lgeb

ra 2

Inve

rse

Var

iati

on

Inve

rse

Var

iati

on

yva

ries

inve

rsel

y as

xif

ther

e is

som

e no

nzer

o co

nsta

nt k

such

tha

t xy

�k

or y

�.

If a

vari

es i

nve

rsel

y as

ban

d a

�8

wh

en b

�12

,fin

d a

wh

en b

�4.

�In

vers

e va

riatio

n

�a 1

�8,

b1

�12

, b 2

�4

8(12

) �

4a2

Cro

ss m

ultip

ly.

96 �

4a2

Sim

plify

.

24 �

a 2D

ivid

e ea

ch s

ide

by 4

.

Wh

en b

�4,

the

valu

e of

ais

24.

Fin

d e

ach

val

ue.

1.If

yva

ries

in

vers

ely

as x

and

y�

12 w

hen

x�

10,f

ind

yw

hen

x�

15.

8

2.If

yva

ries

in

vers

ely

as x

and

y�

9 w

hen

x�

45,f

ind

yw

hen

x�

27.

15

3.If

yva

ries

in

vers

ely

as x

and

y�

100

wh

en x

�38

,fin

d y

wh

en x

�76

.50

4.If

yva

ries

in

vers

ely

as x

and

y�

32 w

hen

x�

42,f

ind

yw

hen

x�

24.

56

5.If

yva

ries

in

vers

ely

as x

and

y�

36 w

hen

x�

10,f

ind

yw

hen

x�

30.

12

6.If

yva

ries

in

vers

ely

as x

and

y�

75 w

hen

x�

12,f

ind

yw

hen

x�

10.

90

7.If

yva

ries

in

vers

ely

as x

and

y�

18 w

hen

x�

124,

fin

d y

wh

en x

�93

.24

8.If

yva

ries

in

vers

ely

as x

and

y�

90 w

hen

x�

35,f

ind

yw

hen

x�

50.

63

9.If

yva

ries

in

vers

ely

as x

and

y�

42 w

hen

x�

48,f

ind

yw

hen

x�

36.

56

10.I

f y

vari

es i

nve

rsel

y as

xan

d y

�44

wh

en x

�20

,fin

d y

wh

en x

�55

.16

11.I

f y

vari

es i

nve

rsel

y as

xan

d y

�80

wh

en x

�14

,fin

d y

wh

en x

�35

.32

12.I

f y

vari

es i

nve

rsel

y as

xan

d y

�3

wh

en x

�8,

fin

d y

wh

en x

�40

.0.

6

13.I

f y

vari

es i

nve

rsel

y as

xan

d y

�16

wh

en x

�42

,fin

d y

wh

en x

�14

.48

14.I

f y

vari

es i

nve

rsel

y as

xan

d y

�9

wh

en x

�2,

fin

d y

wh

en x

�5.

3.6

15.I

f y

vari

es i

nve

rsel

y as

xan

d y

�23

wh

en x

�12

,fin

d y

wh

en x

�15

.18

.4

a 2� 12

8 � 4

a 2� b 1

a 1� b 2

k � x

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Dir

ect,

Join

t,an

d In

vers

e V

aria

tio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-4

9-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Dir

ect,

Join

t,an

d In

vers

e V

aria

tio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-4

9-4

©G

lenc

oe/M

cGra

w-H

ill53

7G

lenc

oe A

lgeb

ra 2

Lesson 9-4

Sta

te w

het

her

eac

h e

qu

atio

n r

epre

sen

ts a

dir

ect,

join

t,or

in

vers

eva

riat

ion

.Th

enn

ame

the

con

stan

t of

var

iati

on.

1.c

�12

md

irec

t;12

2.p

�in

vers

e;4

3.A

�bh

join

t;

4.rw

�15

inve

rse;

155.

y�

2rst

join

t;2

6.f

�52

80m

dir

ect;

5280

7.y

�0.

2sd

irec

t;0.

28.

vz�

�25

inve

rse;

�25

9.t

�16

rhjo

int;

16

10.R

�in

vers

e;8

11.

�d

irec

t;12

.C�

2�r

dir

ect;

2�

Fin

d e

ach

val

ue.

13.I

f y

vari

es d

irec

tly

as x

and

y�

35 w

hen

x�

7,fi

nd

yw

hen

x�

11.

55

14.I

f y

vari

es d

irec

tly

as x

and

y�

360

wh

en x

�18

0,fi

nd

yw

hen

x�

270.

540

15.I

f y

vari

es d

irec

tly

as x

and

y�

540

wh

en x

�10

,fin

d x

wh

en y

�10

80.

20

16.I

f y

vari

es d

irec

tly

as x

and

y�

12 w

hen

x�

72,f

ind

xw

hen

y�

9.54

17.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

18 w

hen

x�

2 an

d z

�3,

fin

d y

wh

en x

�5

and

z�

6.90

18.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

�16

wh

en x

�4

and

z�

2,fi

nd

yw

hen

x�

�1

and

z�

7.14

19.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

120

wh

en x

�4

and

z�

6,fi

nd

yw

hen

x�

3 an

d z

�2.

30

20.I

f y

vari

es i

nve

rsel

y as

xan

d y

�2

wh

en x

�2,

fin

d y

wh

en x

�1.

4

21.I

f y

vari

es i

nve

rsel

y as

xan

d y

�6

wh

en x

�5,

fin

d y

wh

en x

�10

.3

22.I

f y

vari

es i

nve

rsel

y as

xan

d y

�3

wh

en x

�14

,fin

d x

wh

en y

�6.

7

23.I

f y

vari

es i

nve

rsel

y as

xan

d y

�27

wh

en x

�2,

fin

d x

wh

en y

�9.

6

24.I

f y

vari

es d

irec

tly

as x

and

y�

�15

wh

en x

�5,

fin

d x

wh

en y

��

36.

12

1 � 31 � 3

a � b8 � w

1 � 21 � 2

4 � q

©G

lenc

oe/M

cGra

w-H

ill53

8G

lenc

oe A

lgeb

ra 2

Sta

te w

het

her

eac

h e

qu

atio

n r

epre

sen

ts a

dir

ect,

join

t,or

in

vers

eva

riat

ion

.Th

enn

ame

the

con

stan

t of

var

iati

on.

1.u

�8w

zjo

int;

82.

p�

4sd

irec

t;4

3.L

�

inve

rse;

54.

xy�

4.5

inve

rse;

4.5

5.�

�6.

2d�

mn

7.�

h8.

y�

dir

ect;

�jo

int;

inve

rse;

1.25

inve

rse;

Fin

d e

ach

val

ue.

9.If

yva

ries

dir

ectl

y as

xan

d y

�8

wh

en x

�2,

fin

d y

wh

en x

�6.

24

10.I

f y

vari

es d

irec

tly

as x

and

y�

�16

wh

en x

�6,

fin

d x

wh

en y

��

4.1.

5

11.I

f y

vari

es d

irec

tly

as x

and

y�

132

wh

en x

�11

,fin

d y

wh

en x

�33

.39

6

12.I

f y

vari

es d

irec

tly

as x

and

y�

7 w

hen

x�

1.5,

fin

d y

wh

en x

�4.

13.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

24 w

hen

x�

2 an

d z

�1,

fin

d y

wh

en x

�12

an

d z

�2.

288

14.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

60 w

hen

x�

3 an

d z

�4,

fin

d y

wh

en x

�6

and

z�

8.24

0

15.I

f y

vari

es jo

intl

y as

xan

d z

and

y�

12 w

hen

x�

�2

and

z�

3,fi

nd

yw

hen

x�

4 an

d z

��

1.8

16.I

f y

vari

es i

nve

rsel

y as

xan

d y

�16

wh

en x

�4,

fin

d y

wh

en x

�3.

17.I

f y

vari

es i

nve

rsel

y as

xan

d y

�3

wh

en x

�5,

fin

d x

wh

en y

�2.

5.6

18.I

f y

vari

es i

nve

rsel

y as

xan

d y

��

18 w

hen

x�

6,fi

nd

yw

hen

x�

5.�

21.6

19.I

f y

vari

es d

irec

tly

as x

and

y�

5 w

hen

x�

0.4,

fin

d x

wh

en y

�37

.5.

3

20.G

ASE

ST

he

volu

me

Vof

a g

as v

arie

s in

vers

ely

as i

ts p

ress

ure

P.I

f V

�80

cu

bic

cen

tim

eter

s w

hen

P�

2000

mil

lim

eter

s of

mer

cury

,fin

d V

wh

en P

�32

0 m

illi

met

ers

ofm

ercu

ry.

500

cm3

21.S

PRIN

GS

Th

e le

ngt

h S

that

a s

prin

g w

ill

stre

tch

var

ies

dire

ctly

wit

h t

he

wei

ght

Fth

atis

att

ach

ed t

o th

e sp

rin

g.If

a s

prin

g st

retc

hes

20

inch

es w

ith

25

pou

nds

att

ach

ed,h

owfa

r w

ill

it s

tret

ch w

ith

15

pou

nds

att

ach

ed?

12 in

.

22.G

EOM

ETRY

Th

e ar

ea A

of a

tra

pezo

id v

arie

s jo

intl

y as

its

hei

ght

and

the

sum

of

its

base

s.If

th

e ar

ea i

s 48

0 sq

uar

e m

eter

s w

hen

th

e h

eigh

t is

20

met

ers

and

the

base

s ar

e28

met

ers

and

20 m

eter

s,w

hat

is

the

area

of

a tr

apez

oid

wh

en i

ts h

eigh

t is

8 m

eter

s an

dit

s ba

ses

are

10 m

eter

s an

d 15

met

ers?

100

m2

64 � 3

56 � 3

3 � 41 � 2

3 � 4x1.

25�

gC � d

5 � k

Pra

ctic

e (

Ave

rag

e)

Dir

ect,

Join

t,an

d In

vers

e V

aria

tio

n

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-4

9-4


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csD

irec

t,Jo

int,

and

Inve

rse

Var

iati

on

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-4

9-4

©G

lenc

oe/M

cGra

w-H

ill53

9G

lenc

oe A

lgeb

ra 2

Lesson 9-4

Pre-

Act

ivit

yH

ow i

s va

riat

ion

use

d t

o fi

nd

th

e to

tal

cost

giv

en t

he

un

it c

ost?

Rea

d th

e in

trod

uct

ion

to

Les

son

9-4

at

the

top

of p

age

492

in y

our

text

book

.

•F

or e

ach

add

itio

nal

stu

den

t w

ho

enro

lls

in a

pu

blic

col

lege

,th

e to

tal

hig

h-t

ech

spe

ndi

ng

wil

l (i

ncr

ease

/dec

reas

e) b

y

.

•F

or e

ach

dec

reas

e in

en

roll

men

t of

100

stu

den

ts i

n a

pu

blic

col

lege

,th

e

tota

l h

igh

-tec

h s

pen

din

g w

ill

(in

crea

se/d

ecre

ase)

by

.

Rea

din

g t

he

Less

on

1.W

rite

an

equ

atio

n t

o re

pres

ent

each

of

the

foll

owin

g va

riat

ion

sta

tem

ents

.Use

kas

th

eco

nst

ant

of v

aria

tion

.

a.m

vari

es i

nve

rsel

y as

n.

m�

b.

sva

ries

dir

ectl

y as

r.

s�

kr

c.t

vari

es jo

intl

y as

pan

d q.

t�

kpq

2.W

hic

h t

ype

of v

aria

tion

,dir

ect

or i

nve

rse,

is r

epre

sen

ted

by e

ach

gra

ph?

a.in

vers

eb

.d

irec

t

Hel

pin

g Y

ou

Rem

emb

er

3.H

ow c

an y

our

know

ledg

e of

th

e eq

uat

ion

of

the

slop

e-in

terc

ept

form

of

the

equ

atio

n o

f a

lin

e h

elp

you

rem

embe

r th

e eq

uat

ion

for

dir

ect

vari

atio

n?

Sam

ple

an

swer

:Th

e g

rap

h o

f an

eq

uat

ion

exp

ress

ing

dir

ect

vari

atio

n is

alin

e.T

he

slo

pe-

inte

rcep

t fo

rm o

f th

e eq

uat

ion

of

a lin

e is

y�

mx

�b

.In

dir

ect

vari

atio

n,i

f o

ne

of

the

qu

anti

ties

is 0

,th

e o

ther

qu

anti

ty is

als

o 0

,so

b�

0 an

d t

he

line

go

es t

hro

ug

h t

he

ori

gin

.Th

e eq

uat

ion

of

a lin

eth

rou

gh

th

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9-5

9-5

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9-5

9-5



Readin

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the

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(B�

C�

A)x

�(A

�C

)

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atin

g ea

ch t

erm

,0x2

�(A

�B

)x2

0x�

(B�

C�

A)x

3 �

(A�

C)

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eref

ore,

A�

1,B

��

1,C

�2,

and

� x33 �

1�

�� x

�11

�� x2� �x

x�

�21

�.

Res

olve

eac

h r

atio

nal

exp

ress

ion

in

to p

arti

al f

ract

ion

s.

1.� x2

5 �

x2� x

3 �3

��

� x�A

1�

�� x

�B3

�A

�2,

B�

3

2.� (6 xx ��

27 )2�

�� x

�A2

��

� (x�B

2)2

�A

�6,

B�

�5

3.�

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� xB 2��

� x�C

1�

�� (x

�D1)

2�

A�

1,B

��

2,C

�3,

D�

�4

4x3

�x2

�3x

�2

��

�x2

(x�

1)2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-5

9-5

Exam

ple

Exam

ple


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g R

atio

nal

Eq

uat

ion

s an

d In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-6

9-6

©G

lenc

oe/M

cGra

w-H

ill54

7G

lenc

oe A

lgeb

ra 2

Lesson 9-6

Solv

e R

atio

nal

Eq

uat

ion

sA

rat

ion

al e

qu

atio

nco

nta

ins

one

or m

ore

rati

onal

expr

essi

ons.

To

solv

e a

rati

onal

equ

atio

n,f

irst

mu

ltip

ly e

ach

sid

e by

th

e le

ast

com

mon

den

omin

ator

of

all

of t

he

den

omin

ator

s.B

e su

re t

o ex

clu

de a

ny

solu

tion

th

at w

ould

pro

duce

a de

nom

inat

or o

f ze

ro.

Sol

ve

��

.

��

Orig

inal

equ

atio

n

10(x

�1)

��

��10

(x�

1)�

�M

ultip

ly e

ach

side

by

10(x

�1)

.

9(x

�1)

�2(

10)

�4(

x�

1)M

ultip

ly.

9x�

9 �

20 �

4x�

4D

istr

ibut

ive

Pro

pert

y

5x�

�25

Sub

trac

t 4x

and

29

from

eac

h si

de.

x�

�5

Div

ide

each

sid

e by

5.

Ch

eck �

�O

rigin

al e

quat

ion

��

x�

�5

��

Sim

plify

.

��

Sim

plify

.

�S

impl

ify.

�

Sol

ve e

ach

eq

uat

ion

.

1.�

�2

52.

��

12

3.�

��

4.�

�4

�5.

��

76.

��

10

7.N

AV

IGA

TIO

NT

he

curr

ent

in a

riv

er i

s 6

mil

es p

er h

our.

In h

er m

otor

boat

Mar

issa

can

trav

el 1

2 m

iles

ups

trea

m o

r 16

mil

es d

own

stre

am i

n t

he

sam

e am

oun

t of

tim

e.W

hat

is

the

spee

d of

her

mot

orbo

at i

n s

till

wat

er?

42 m

ph

8.W

OR

KA

dam

,Bet

han

y,an

d C

arlo

s ow

n a

pai

nti

ng

com

pan

y.T

o pa

int

a pa

rtic

ula

r h

ouse

alon

e,A

dam

est

imat

es t

hat

it

wou

ld t

ake

him

4 d

ays,

Bet

han

y es

tim

ates

5da

ys,a

nd

Car

los

6 da

ys.I

f th

ese

esti

mat

es a

re a

ccu

rate

,how

lon

g sh

ould

it

take

th

e th

ree

of t

hem

to p

ain

t th

e h

ouse

if

they

wor

k to

geth

er?

abo

ut

1d

ays

2 � 3

1 � 2

8 � 34

� x�

2x

� x�

2x

�1

�12

4� x

�1

1 � 242m

�1

�2m

3m�

2�

5m

13 � 51 � 2

x�

5�

42x

�1

�3

4 �

2t�

34t

�3

�5

y�

3�

62y � 3

2 � 52 � 5

2 � 58 � 20

2 � 510 � 20

18 � 20

2 � 52

� �4

9 � 10

2 � 52

� �5

�1

9 � 10

2 � 52

� x�

19 � 10

2 � 52

� x�

19 � 10

2 � 52

� x�

19 � 10

2 � 52

� x�

19 � 10

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill54

8G

lenc

oe A

lgeb

ra 2

Solv

e R

atio

nal

Ineq

ual

itie

sTo

sol

ve a

rat

iona

l ine

qual

ity,

com

plet

e th

e fo

llow

ing

step

s.

Ste

p 1

Sta

te t

he e

xclu

ded

valu

es.

Ste

p 2

Sol

ve t

he r

elat

ed e

quat

ion.

Ste

p 3

Use

the

val

ues

from

ste

ps 1

and

2 t

o di

vide

the

num

ber

line

into

reg

ions

. Tes

t a

valu

e in

eac

h re

gion

to

see

whi

ch r

egio

ns s

atis

fy t

he o

rigin

al in

equa

lity.

Sol

ve

��

.

Ste

p 1

Th

e va

lue

of 0

is

excl

ude

d si

nce

th

is v

alu

e w

ould

res

ult

in

a d

enom

inat

or o

f 0.

Ste

p 2

Sol

ve t

he

rela

ted

equ

atio

n.

��

Rel

ated

equ

atio

n

15n�

��

15n�

�M

ultip

ly e

ach

side

by

15n.

10 �

12 �

10n

Sim

plify

.

22 �

10n

Sim

plify

.

2.2

�n

Sim

plify

.

Ste

p 3

Dra

w a

nu

mbe

r w

ith

ver

tica

l li

nes

at

the

ex

clu

ded

valu

e an

d th

e so

luti

on t

o th

e eq

uat

ion

.

Tes

t n

��

1.T

est

n�

1.T

est

n�

3.

��

��

is t

rue.

�

is n

ottr

ue.

�

is t

rue.

Th

e so

luti

on i

s n

0

or n

2.

2.

Sol

ve e

ach

in

equ

alit

y.

1.

32.

4x

3.�

�

�1

a

�0

x�

�o

r 0

x

�0

p

4.�

�5.

�

26.

�1

�

�2

x

0

x

0 o

r

x

1x

�

1 o

r 0

x

1

or

x�

5o

r x

�2

1 � 2

2� x

�1

3� x2

�1

5 � x4

� x�

11 � 4

2 � x3 � 2x

39 � 201 � 2

1 � 2

2 � 34 � 5p

1 � 2p1 � x

3� a

�1

2 � 34 � 15

2 � 92 � 3

4 � 52 � 3

2 � 34 � 5

2 � 3

�3

�2

�1

01

22.2 3

2 � 34 � 5n

2 � 3n

2 � 34 � 5n

2 � 3n

2 � 34 � 5n

2 � 3n

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

So

lvin

g R

atio

nal

Eq

uat

ion

s an

d In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-6

9-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

So

lvin

g R

atio

nal

Eq

uat

ion

s an

d In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-6

9-6

©G

lenc

oe/M

cGra

w-H

ill54

9G

lenc

oe A

lgeb

ra 2

Lesson 9-6

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Ch

eck

you

r so

luti

ons.

1.�

�1

2.2

��

3.�

�1

4.3

�z

�1,

2

5.�

56.

��

5,8

7.�

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3,4

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810

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k�

0

11.2

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0

v

412

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n

�

3 o

r 0

n

3

13.

�

�0

m

1

14.

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10

x

15.

��

93

16.

�4

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17.2

��

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20.

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23.

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24.

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52

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34

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38

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92

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21

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122x

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1x

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2x

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2

4� w

2�

41

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21

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22

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35

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2 � 58z

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24 � z

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5 � 2q

b�

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3b�

2� b

�1

9x�

7� x

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15 � x

3 � 22 � x

1 � 2x5 � 2

3 � m1

� 2m

12 � n3 � n

5 � v3 � v

4 � 3k3 � k

x�

1� x

�10

x�

2� x

�4

12 � y3 � 2

2x�

3� x

�1

8 � ss

�3

�5

1� d

�2

2� d

� 1

2 � z�

6�2

9 � 3x

12 � 51 � 3

4 � n1 � 2

x� x

�1

©G

lenc

oe/M

cGra

w-H

ill55

0G

lenc

oe A

lgeb

ra 2

Sol

ve e

ach

eq

uat

ion

or

ineq

ual

ity.

Ch

eck

you

r so

luti

ons.

1.�

�16

2.�

1 �

�1,

2

3.�

�,4

4.�

s�

4

5.�

�1

all r

eals

exc

ept

56.

��

0

7.

t

�5

or

�

t

08.

��

9.�

�2

10.5

�

0

a

2

11.

�

0

x

712

.8 �

�y

0

or

y�

2

13.

�

p

0 o

r p

�14

.�

�

15.g

��

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16.b

��

1 �

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17.2

��

18.5

��

6

19.

��

20.

�4

��

�5 3� ,5

21.

��

722

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2

23.

��

024

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25.

��

26.3

��

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all r

eals

exc

ept

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and

4

27.B

ASK

ETB

ALL

Kia

na h

as m

ade

9 of

19

free

thr

ows

so f

ar t

his

seas

on.H

er g

oal

is t

o m

ake

60%

of

her

fre

e th

row

s.If

Kia

na

mak

es h

er n

ext

xfr

ee t

hro

ws

in a

row

,th

e fu

nct

ion

f(x)

�re

pres

ents

Kia

na’s

new

rat

io o

f fr

ee t

hrow

s m

ade.

How

man

y su

cces

sful

fre

e

thro

ws

in a

row

wil

l ra

ise

Kia

na’

s pe

rcen

t m

ade

to 6

0%?

6

28.O

PTIC

ST

he l

ens

equa

tion

�

�re

late

s th

e di

stan

ce p

of a

n ob

ject

fro

m a

len

s,th

e

dist

ance

qof

th

e im

age

of t

he

obje

ct f

rom

th

e le

ns,

and

the

foca

l le

ngt

h f

of t

he

len

s.W

hat

is

the

dist

ance

of

an o

bjec

t fr

om a

len

s if

th

e im

age

of t

he

obje

ct i

s 5

cen

tim

eter

sfr

om t

he

len

s an

d th

e fo

cal

len

gth

of

the

len

s is

4 c

enti

met

ers?

20 c

m

1 � f1 � q

1 � p

9 �

x� 19

�x

22� a

�5

6a�

1� 2a

�7

r2�

16� r2

�16

4� r

�4

r� r

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x� 2

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c�

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3 � 23

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14 � 3x

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� x�

6x

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3

b�

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�1

2b� b

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2� g

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g� g

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2� x

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4� x

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6� x

�1

65 � 31 � 5

1 � 3p4 � p

19 � y3 � y

3 � 2x1 � 10

4 � 5x

7 � a3 � a

�1

� w�

34

� w�

2

11 � 53

� h�

15 � h

1 � 2h1 � 2

9� 2t

�1

5 � t

5 � 85 � x

1� 3x

�2

y� y

�5

5� y

�5

5s�

8� s

�2

s� s

�2

2 � 34 � p

p�

10� p2

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x � 2x

� x�

13 � 2

3 � 412 � x

Pra

ctic

e (

Ave

rag

e)

So

lvin

g R

atio

nal

Eq

uat

ion

s an

d In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-6

9-6


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Rat

ion

al E

qu

atio

ns

and

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

9-6

9-6

©G

lenc

oe/M

cGra

w-H

ill55

1G

lenc

oe A

lgeb

ra 2

Lesson 9-6

Pre-

Act

ivit

yH

ow a

re r

atio

nal

eq

uat

ion

s u

sed

to

solv

e p

rob

lem

s in

volv

ing

un

itp

rice

?R

ead

the

intr

oduc

tion

to

Les

son

9-6

at t

he t

op o

f pa

ge 5

05 i

n yo

ur t

extb

ook.

•If

you

in

crea

se t

otal

nu

mbe

r of

min

ute

s of

lon

g-di

stan

ce c

alls

fro

m M

arch

to A

pril

,wil

l you

r lo

ng-d

ista

nce

phon

e bi

ll in

crea

se o

r de

crea

se?

incr

ease

•W

ill

you

r ac

tual

cos

t pe

r m

inu

te i

ncr

ease

or

decr

ease

?d

ecre

ase

Rea

din

g t

he

Less

on

1.W

hen

sol

vin

g a

rati

onal

equ

atio

n,a

ny

poss

ible

sol

uti

on t

hat

res

ult

s in

0 i

n t

he

den

omin

ator

mu

st b

e ex

clu

ded

from

th

e li

st o

f so

luti

ons.

2.S

upp

ose

that

on

a q

uiz

you

are

ask

ed t

o so

lve

the

rati

onal

in

equ

alit

y �

�0.

Com

plet

e th

e st

eps

of t

he

solu

tion

.

Ste

p 1

Th

e ex

clu

ded

valu

es a

re

and

.

Ste

p 2

Th

e re

late

d eq

uat

ion

is

��

0 .

To s

olve

thi

s eq

uati

on,m

ulti

ply

both

sid

es b

y th

e L

CD

,whi

ch i

s .

Sol

vin

g th

is e

quat

ion

wil

l sh

ow t

hat

th

e on

ly s

olu

tion

is

�4.

Ste

p 3

Div

ide

a n

um

ber

lin

e in

to

regi

ons

usi

ng

the

excl

ude

d va

lues

an

d th

eso

luti

on o

f th

e re

late

d eq

uat

ion

.Dra

w d

ash

ed v

erti

cal

lin

es o

n t

he

nu

mbe

r li

ne

belo

w t

o sh

ow t

hes

e re

gion

s.

Con

side

r th

e fo

llow

ing

valu

es o

f �

for

vari

ous

test

val

ues

of

z.

If z

��

5,�

�0.

2.If

z�

�3,

��

�1.

If z

��

1,�

�9.

If z

�1,

��

�5.

Usi

ng

this

in

form

atio

n a

nd

you

r n

um

ber

lin

e,w

rite

th

e so

luti

on o

f th

e in

equ

alit

y.

z

�4

or

�2

z

0

Hel

pin

g Y

ou

Rem

emb

er3.

How

are

th

e pr

oces

ses

of a

ddin

g ra

tion

al e

xpre

ssio

ns

wit

h d

iffe

ren

t de

nom

inat

ors

and

ofso

lvin

g ra

tion

al e

xpre

ssio

ns

alik

e,an

d h

ow a

re t

hey

dif

fere

nt?

Sam

ple

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.

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5 6n�

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mn

→∞�2

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lim

n→

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9-6

9-6

Exam

ple

Exam

ple


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10. B

A

D

B

C

B

A

D

B

C

asymptote: x � 0;hole: x � 3

A

A

C

D

B

A

B

C

B

A

D

C

C

A

B

D

B

A

A

C

Chapter 9 Assessment Answer Key Form 1 Form 2APage 553 Page 554 Page 555

(continued on the next page)


11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:�x �

x1

�

B

A

B

C

B

D

C

B

A

B

A

C

A

D

B

C

D

A

D

B

�x �

x1

�

B

C

A

B

D

A

D

C

C

A

Chapter 9 Assessment Answer Key Form 2A (continued) Form 2BPage 556 Page 557 Page 558

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: �

4.8 h

m � 0 or m � 5

1

direct variation

square root

P � �Ak

�

1

192 customers

15

hole: x � �2

asymptote: x � 3

(n � 2)(n � 4)(n � 6)

36m4n4

�3m

7� 1�

�x �

22

�

�9(m

8� 5)�

�b �

25

�

�x �

x8

�

��32

�, 3

Chapter 9 Assessment Answer Key Form 2CPage 559 Page 560

xO

f (x) f (x) � x � 3x � 2


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: �

48 min

��13

�

inverse variation

constant

R � �Uk

�

9

1050 permits

63

xO

f (x)f (x) � x

x � 2

asymptotes: x � �5, x � �2

asymptote: x � �4;hole: x � 6

(n � 1)(n � 5)(n � 2)

42s3t4

�2n

5� 1�

�x �

33

�

�2(y

3� 2)�

�m

6�m

1�

�x

x�

2

5�

�2, �52

�

Chapter 9 Assessment Answer Key Form 2DPage 561 Page 562

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

��x2

3(2xx��

23)

�;

x � ��23

�, 0, �32

�

�1255�

z � �1 or �1� z � 1

x � 0 or x � �32

�

�

10

inverse variation

rational

I � �7R.2�; 40

271 mi

0.02

�110�

hole: x � �2

asymptote: x � 3

(c � d)(c � d)2

�1

0

�44mm

��

33nn

�

�g �

53

�

�3x((23xx

��

35))

�

��13

�, 0, �52

�

Chapter 9 Assessment Answer Key Form 3Page 563 Page 564

xO

f (x) �

f (x)

�2 (x � 3)2

xO

f (x) � x2�42x � 4

f (x)

Chapter 9 Assessment Answer KeyPage 565, Open-Ended Assessment

Scoring Rubric


Score General Description Specific Criteria

• Shows thorough understanding of the concepts ofsimplifying rational expressions, determining verticalasymptotes and point discontinuity of rational functions,solving joint variation problems, identifying equations asdifferent types of functions, and solving rational equationsand inequalities.

• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Goes beyond requirements of some or all problems.

• Shows an understanding of the concepts of simplifyingrational expressions, determining vertical asymptotes andpoint discontinuity of rational functions, solving jointvariation problems, identifying equations as different typesof functions, and solving rational equations andinequalities.

• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Satisfies all requirements of problems.

• Shows an understanding of most of the concepts ofsimplifying rational expressions, determining verticalasymptotes and point discontinuity of rational functions,solving joint variation problems, identifying equations asdifferent types of functions, and solving rational equationsand inequalities.

• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Satisfies the requirements of most of the problems.

• Final computation is correct.• No written explanations or work is shown to substantiate

the final computation.• Satisfies minimal requirements of some of the problems.

• Shows little or no understanding of most of the concepts ofsimplifying rational expressions, determining verticalasymptotes and point discontinuity of rational functions,solving joint variation problems, identifying equations asdifferent types of functions, and solving rational equationsand inequalities.

• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Does not satisfy requirements of problems.• No answer may be given.

0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given

1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation

2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem

3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation

4 SuperiorA correct solution that is supported by well-developed, accurateexplanations

An

swer

s


Chapter 9 Assessment Answer Key Page 565, Open-Ended Assessment

Sample Answers

1. Each student response must includethree expressions which, when

simplified, “reduce to” �a �a

5�.

Sample answer: �3a3�a

15�, �a2a�

2

5a�,

�(a �a(a

5)�(a

1�)

1)�.

2a. Students should explain that the heightcan be found by dividing the volume bythe product of the length and width ofthe box.

2b. (x � 3) in.2c. Sample answer: Substitute a value for x

in each of the given expressions for thelength, width, and volume, and the samevalue for x in the expression found for h,and then check that V � �wh.CHECK For x � 5,length � (5) � 10 � 15 in.width � 2(5) � 10 in.volume � 2(5)3 � 26(5)2 � 60(5)

� 1200 in3

height � (5) � 3 � 8 in.Verify V � �wh: 1200 � (15)(10)(8) ✓

3. Each student response must include twopolynomials in which 3, y � 2, and y � 2each appears as a factor of at least oneof those polynomials, but which have noother factor. Sample answer:y2 � 4, 3(y � 2).

4. Student responses should indicate thatthe graph of f(x) has a hole at x � �2,but no vertical asymptote. Its graph is astraight line with a hole in it at(�2, �5). The graph of g(x) also has ahole at x � �2, but has a verticalasymptote at x � 0. Its graph is not astraight line, but two curves having a

hole in the graph at ��2, �52��.

5a. d � 0.10hr5b. joint variation; the amount deducted

varies directly as the product of two quantities, the hourly wage and thenumber of hours worked.

5c. Students should indicate that theyshould substitute r � 9.50 and h � 36 in the formula they wrote inpart a.The amount deducted was $34.20.

6a. Students should conclude that C(x)is a rational function since it is of

the form y � �pq(

(xx))

�, where

p(x) � 60x � 17,000 and q(x) � x � 50are polynomial functions.

6b. Students should indicate that R(x) is aconstant function since it is of the formy � a, where a is any number.

6c. 80 � �60x

x��

1570,000

�

6d. x � 1050; The company must produceand sell at least 1050 CD players inorder to ensure that the revenue fromeach one is greater than the averagecost of producing each one.

In addition to the scoring rubric found on page A25, the following sample answers may be used as guidance in evaluating open-ended assessment items.


1. inverse variation

2. rational function

3. asymptote

4. least commondenominator

5. joint variation

6. continuity

7. rational inequality

8. inverselyproportional

9. constant ofvariation

10. point discontinuity

11. Sample answer: Arational expressionis the ratio of twopolynomials. Thedenominator cannotbe 0.

12. Sample answer: Acomplex fraction isa fraction in whichthe numerator,denominator, orboth, containfractions.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Quiz (Lesson 9–3)

Page 567

1.

2.

3.4.

1.

2.

3.

4.

5.

Quiz (Lesson 9–6)

Page 568

1.

2.

3.

4.

5. 18

�5 � m � �2

t � 0 or t � 2

9

10

absolute value

greatest integer

quadratic

12

inverse; 30

hole: x � �4

asymptote: x � 1;hole: x � �3

asymptotes: x � �2, x � 1

�y

1�2

3�

�35

5m�

22nm

�

(t � 1)(2t � 1)(t � 4)

15(x � 2)(x � 2)60a2b3

E

�p5

�

(2x � 3)(x � 6)

�x �

33

�

�8ax2

5�

Chapter 9 Assessment Answer Key Vocabulary Test/Review Quiz (Lessons 9–1 and 9–2) Quiz (Lessons 9–4 and 9–5)

Page 566 Page 567 Page 568

An

swer

s

xO

f (x)f (x) � 4

x � 3

y

xO

y � 3�x � � 2


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. 11.2 mL

20

3(y � 2)

circle

�(x �

251)2� � �

(y �9

2)2� � 1

x2 � (y � 3)2 � 125

1, 2, 3, 6, �12

�, �32

�

23

y

xO

�1 �i

w�125�

yes

asymptote: x � 4;hole: x � �3

21(3c � 5)

72s3t4

B

C

A

A

D

Chapter 9 Assessment Answer Key Mid-Chapter Test Cumulative ReviewPage 569 Page 570

xO

f (x) � 4(x � 2)2

f (x)

x2 � 5x � 3��(x � 3)(x � 2)(x � 4)


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 12.

13. 14.

15.

16.

17.

18. DCBA

DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 0 4

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 3

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

3 0/ 2

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

2 5

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

An

swer

s

Chapter 9 Assessment Answer KeyStandardized Test Practice

Page 571 Page 572

chapter 9 resource masters - ktl math classes · chapter 9 test, form 3 . . . . . . . . . . ....

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