Chapter 1 Resource Masters - ktlmathclass.weebly.comktlmathclass.weebly.com/.../5/7/6/25760552/alg_2_resource_ws_ch_1.pdf©Glencoe/McGraw-Hill v Glencoe Algebra 2 Assessment Options

Chapter 1Resource 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 1 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: 0-07-828004-4 Algebra 2Chapter 1 Resource Masters

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

Glencoe/McGraw-Hill

© Glencoe/McGraw-Hill iii Glencoe Algebra 2

Contents

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

Lesson 1-1Study Guide and Intervention . . . . . . . . . . . 1–2Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . . 3Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Reading to Learn Mathematics . . . . . . . . . . . . 5Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Lesson 1-2Study Guide and Intervention . . . . . . . . . . . 7–8Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . . 9Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Reading to Learn Mathematics . . . . . . . . . . . 11Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Lesson 1-3Study Guide and Intervention . . . . . . . . . 13–14Skills Practice . . . . . . . . . . . . . . . . . . . . . . . . 15Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Reading to Learn Mathematics . . . . . . . . . . . 17Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . 18




Chapter 1 AssessmentChapter 1 Test, Form 1 . . . . . . . . . . . . . . 37–38Chapter 1 Test, Form 2A . . . . . . . . . . . . . 39–40Chapter 1 Test, Form 2B . . . . . . . . . . . . . 41–42Chapter 1 Test, Form 2C . . . . . . . . . . . . . 43–44Chapter 1 Test, Form 2D . . . . . . . . . . . . . 45–46Chapter 1 Test, Form 3 . . . . . . . . . . . . . . 47–48Chapter 1 Open-Ended Assessment . . . . . . . 49Chapter 1 Vocabulary Test/Review . . . . . . . . 50Chapter 1 Quizzes 1 & 2 . . . . . . . . . . . . . . . . 51Chapter 1 Quizzes 3 & 4 . . . . . . . . . . . . . . . . 52Chapter 1 Mid-Chapter Test . . . . . . . . . . . . . 53Chapter 1 Cumulative Review . . . . . . . . . . . . 54Chapter 1 Standardized Test Practice . . . . 55–56

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

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A30

© Glencoe/McGraw-Hill iv Glencoe Algebra 2

Teacher’s Guide to Using theChapter 1 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 1 Resource Masters includes the core materials neededfor Chapter 1. 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 1-1.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 1Resource 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 52–53. 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 _____

11

© Glencoe/McGraw-Hill vii Glencoe Algebra 2

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 1.As you study the chapter, complete each term’s definition or description.Remember to add the page number where you found the term. Add these pages toyour Algebra Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

absolute value

algebraic expression

Associative Property

uh·SOH·shee·uh·tihv

Commutative Property

kuh·MYOO·tuh·tihv

compound inequality

Distributive Property

dih·STRIH·byuh·tihv

empty set

Identity Property

intersection

Inverse Property

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Algebra 2

Vocabulary Term Found on Page Definition/Description/Example

irrational numbers

open sentence

rational numbers

Reflexive Property

set-builder notation

Substitution Property

Symmetric Property

suh·MEH·trihk

Transitive Property

Trichotomy Property

try·KAH·tuh·mee

union

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

11

Study Guide and InterventionExpressions and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

© Glencoe/McGraw-Hill 1 Glencoe Algebra 2

Less

on

1-1

Order of Operations

1. Simplify the expressions inside grouping symbols.Order of 2. Evaluate all powers.Operations 3. Do all multiplications and divisions from left to right.

4. Do all additions and subtractions from left to right.

Evaluate [18 � (6 � 4)] � 2.

[18 � (6 � 4)] � 2 � [18 � 10] � 2� 8 � 2� 4

Evaluate 3x2 � x(y � 5)if x � 3 and y � 0.5.

Replace each variable with the given value.3x2 � x(y � 5) � 3 � (3)2 � 3(0.5 � 5)

� 3 � (9) � 3(�4.5)� 27 � 13.5� 13.5

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the value of each expression.

1. 14 � (6 � 2) 17 2. 11 � (3 � 2)2 �14 3. 2 � (4 � 2)3 � 6 4

4. 9(32 � 6) 135 5. (5 � 23)2 � 52 144 6. 52 � � 18 � 2 34.25

7. �6 8. (7 � 32)2 � 62 40 9. 20 � 22 � 6 11

10. 12 � 6 � 3 � 2(4) 6 11. 14 � (8 � 20 � 2) �7 12. 6(7) � 4 � 4 � 5 38

13. 8(42 � 8 � 32) �240 14. �24 15. 4

Evaluate each expression if a � 8.2, b � �3, c � 4, and d � � .

16. 49.2 17. 5(6c � 8b � 10d) 215 18. �6

19. ac � bd 31.3 20. (b � c)2 � 4a 81.8 21. � 6b � 5c �54.4

22. 3� � � b �21 23. cd � 4 24. d(a � c) �6.1

25. a � b � c 7.45 26. b � c � 4 � d �15 27. � d 8.7a�b � c

b�d

c�d

a�d

c2 � 1�b � d

ab�d

1�2

6 � 9 � 3 � 15��8 � 2

6 � 4 � 2��4 � 6 � 1

16 � 23 � 4��1 � 22

1�4


Formulas A formula is a mathematical sentence that uses variables to express therelationship between certain quantities. If you know the value of every variable except onein a formula, you can use substitution and the order of operations to find the value of theunknown variable.

To calculate the number of reams of paper needed to print n copies

of a booklet that is p pages long, you can use the formula r � , where r is the

number of reams needed. How many reams of paper must you buy to print 172 copies of a 25-page booklet?

Substitute n � 172 and p � 25 into the formula r � .

r �

�

� 8.6

You cannot buy 8.6 reams of paper. You will need to buy 9 reams to print 172 copies.

For Exercises 1–3, use the following information.

For a science experiment, Sarah counts the number of breaths needed for her to blow up abeach ball. She will then find the volume of the beach ball in cubic centimeters and divideby the number of breaths to find the average volume of air per breath.

1. Her beach ball has a radius of 9 inches. First she converts the radius to centimetersusing the formula C � 2.54I, where C is a length in centimeters and I is the same lengthin inches. How many centimeters are there in 9 inches? 22.86 cm

2. The volume of a sphere is given by the formula V � �r3, where V is the volume of the

sphere and r is its radius. What is the volume of the beach ball in cubic centimeters?(Use 3.14 for �.) 50,015 cm3

3. Sarah takes 40 breaths to blow up the beach ball. What is the average volume of air perbreath? about 1250 cm3

4. A person’s basal metabolic rate (or BMR) is the number of calories needed to support hisor her bodily functions for one day. The BMR of an 80-year-old man is given by theformula BMR � 12w � (0.02)(6)12w, where w is the man’s weight in pounds. What is theBMR of an 80-year-old man who weighs 170 pounds? 1795 calories

4�3

43,000�500

(172)(25)��500

np�500

np�500

Study Guide and Intervention (continued)

Expressions and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

ExampleExample

ExercisesExercises

Skills PracticeExpressions and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1


Less

on

1-1


1. 18 � 2 � 3 27 2. 9 � 6 � 2 � 1 13

3. (3 � 8)2(4) � 3 97 4. 5 � 3(2 � 12 � 2) �7

5. � [�9 � 10(3)] �7 6. 3

7. (168 � 7)32 � 43 152 8. [3(5) � 128 � 22]5 �85

Evaluate each expression if r � �1, s � 3, t � 12, v � 0, and w � � .

9. 6r � 2s 0 10. 2st � 4rs 84

11. w(s � r) �2 12. s � 2r � 16v 1

13. (4s)2 144 14. s2r � wt �3

15. 2(3r � w) �7 16. 4

17. �w[t � (t � r)] 18. 0

19. 9r2 � (s2 � 1)t 105 20. 7s � 2v � 22

21. TEMPERATURE The formula K � C � 273 gives the temperature in kelvins (K) for agiven temperature in degrees Celsius. What is the temperature in kelvins when thetemperature is 55 degrees Celsius? 328 K

22. TEMPERATURE The formula C � (F � 32) gives the temperature in degrees Celsius

for a given temperature in degrees Fahrenheit. What is the temperature in degreesCelsius when the temperature is 68 degrees Fahrenheit? 20�C

5�9

2w�r

rv3�s2

25�2

3v � t�5s � t

1�2

6(7 � 5)��4

1�3



1. 3(4 � 7) � 11 �20 2. 4(12 � 42) �16

3. 1 � 2 � 3(4) � 2 �3 4. 12 � [20 � 2(62 � 3 � 22)] 88

5. 20 � (5 � 3) � 52(3) 85 6. (�2)3 � (3)(8) � (5)(10) 18

7. 18 � {5 � [34 � (17 � 11)]} 41 8. [4(5 � 3) � 2(4 � 8)] � 16 1

9. [6 � 42] �5 10. [�5 � 5(�3)] �5

11. 32 12. � (�1)2 � 4(�9) �53

Evaluate each expression if a � , b � �8, c � �2, d � 3, and e � .

13. ab2 � d 45 14. (c � d)b �8

15. � d2 12 16. 12

17. (b � de)e2 �1 18. ac3 � b2de �70

19. �b[a � (c � d)2] 206 20. � 22

21. 9bc � 141 22. 2ab2 � (d3 � c) 67

23. TEMPERATURE The formula F � C � 32 gives the temperature in degrees

Fahrenheit for a given temperature in degrees Celsius. What is the temperature indegrees Fahrenheit when the temperature is �40 degrees Celsius? �40�F

24. PHYSICS The formula h � 120t � 16t2 gives the height h in feet of an object t secondsafter it is shot upward from Earth’s surface with an initial velocity of 120 feet persecond. What will the height of the object be after 6 seconds? 144 ft

25. AGRICULTURE Faith owns an organic apple orchard. From her experience the last fewseasons, she has developed the formula P � 20x � 0.01x2 � 240 to predict her profit P indollars this season if her trees produce x bushels of apples. What is Faith’s predictedprofit this season if her orchard produces 300 bushels of apples? $4860

9�5

1�e

c�e2

ac4�d

d(b � c)�ac

ab�c

1�3

3�4

(�8)2�5 � 9

�8(13 � 37)��6

1�4

1�2

Practice (Average)

Expressions and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

Reading to Learn MathematicsExpressions and Formulas

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1


Less

on

1-1

Pre-Activity How are formulas used by nurses?

Read the introduction to Lesson 1-1 at the top of page 6 in your textbook.

• Nurses use the formula F � to control the flow rate for IVs. Name

the quantity that each of the variables in this formula represents and theunits in which each is measured.

F represents the and is measured in per minute.

V represents the of solution and is measured in

.

d represents the and is measured in per milliliter.

t represents and is measured in .

• Write the expression that a nurse would use to calculate the flow rate of an IV if a doctor orders 1350 milliliters of IV saline to be given over 8 hours, with a drop factor of 20 drops per milliliter. Do not find the valueof this expression.

Reading the Lesson1. There is a customary order for grouping symbols. Brackets are used outside of

parentheses. Braces are used outside of brackets. Identify the innermost expression(s) ineach of the following expressions.

a. [(3 � 22) � 8] � 4 (3 � 22)b. 9 � [5(8 � 6) � 2(10 � 7)] (8 � 6) and (10 � 7)c. {14 � [8 � (3 � 12)2]} � (63 � 100) (3 � 12)

2. Read the following instructions. Then use grouping symbols to show how the instructionscan be put in the form of a mathematical expression.

Multiply the difference of 13 and 5 by the sum of 9 and 21. Add the result to 10. Thendivide what you get by 2. [(13 � 5)(9 � 21) � 10] � 2

3. Why is it important for everyone to use the same order of operations for evaluatingexpressions? Sample answer: If everyone did not use the same order ofoperations, different people might get different answers.

Helping You Remember4. Think of a phrase or sentence to help you remember the order of operations.

Sample answer: Please excuse my dear Aunt Sally. (parentheses;exponents; multiplication and division; addition and subtraction)

1350 � 20��

8 � 60

minutestime

dropsdrop factor

millilitersvolume

dropsflow rate

V � d�t


Significant DigitsAll measurements are approximations. The significant digits of an approximatenumber are those which indicate the results of a measurement. For example, themass of an object, measured to the nearest gram, is 210 grams. The measurement210– g has 3 significant digits. The mass of the same object, measured to thenearest 100 g, is 200 g. The measurement 200 g has one significant digit.

1. Nonzero digits and zeros between significant digits are significant. Forexample, the measurement 9.071 m has 4 significant digits, 9, 0, 7, and 1.

2. Zeros at the end of a decimal fraction are significant. The measurement 0.050 mm has 2 significant digits, 5 and 0.

3. Underlined zeros in whole numbers are significant. The measurement 104,00–0 km has 5 significant digits, 1, 0, 4, 0, and 0.

In general, a computation involving multiplication or division of measurementscannot be more accurate than the least accurate measurement in the computation.Thus, the result of computation involving multiplication or division ofmeasurements should be rounded to the number of significant digits in the leastaccurate measurement.

The mass of 37 quarters is 210– g. Find the mass of one quarter.

mass of 1 quarter � 210– g � 37 210– has 3 significant digits.

37 does not represent a measurement.

� 5.68 g Round the result to 3 significant digits.

Why?

Write the number of significant digits for each measurement.

1. 8314.20 m 2. 30.70 cm 3. 0.01 mm 4. 0.0605 mg

6 4 1 3

5. 370–,000 km 6. 370,00–0 km 7. 9.7 � 104 g 8. 3.20 � 10�2 g

3 5 2 3

Solve. Round each result to the correct number of significant digits.

9. 23 m � 1.54 m 10. 12,00–0 ft � 520 ft 11. 2.5 cm � 25

35 m2 23.1 63 cm

12. 11.01 mm � 11 13. 908 yd � 0.5 14. 38.6 m � 4.0 m

121.1 mm 1820 yd 150 m2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-11-1

ExampleExample

Study Guide and InterventionProperties of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2


Less

on

1-2

Real Numbers All real numbers can be classified as either rational or irrational. The setof rational numbers includes several subsets: natural numbers, whole numbers, andintegers.

R real numbers {all rationals and irrationals}

Q rational numbers {all numbers that can be represented in the form , where m and n are integers and n is not equal to 0}

I irrational numbers {all nonterminating, nonrepeating decimals}

N natural numbers {1, 2, 3, 4, 5, 6, 7, 8, 9, …}

W whole numbers {0, 1, 2, 3, 4, 5, 6, 7, 8, …}

Z integers {…, �3, �2, �1, 0, 1, 2, 3, …}

Name the sets of numbers to which each number belongs.

a. � rationals (Q), reals (R)

b. �25��25� � 5 naturals (N), wholes (W), integers (Z), rationals (Q), reals (R)


1. Q, R 2. ��81� Z, Q, R 3. 0 W, Z, Q, R 4. 192.0005 Q, R

5. 73 N, W, Z, Q, R 6. 34 Q, R 7. Q, R 8. 26.1 Q, R

9. � I, R 10. N, W, Z, Q, R 11. �4.1�7� Q, R

12. N, W, Z, Q, R 13. �1 Z, Q, R 14. �42� I, R

15. �11.2 Q, R 16. � Q, R 17. I, R

18. 33.3� Q, R 19. 894,000 N, W, Z, Q, R 20. �0.02 Q, R

�5��2

8�13

�25��5

15�3

�36��9

1�2

6�7

11�3

m�n

ExercisesExercises

ExampleExample


Properties of Real Numbers

Real Number Properties

For any real numbers a, b, and c

Property Addition Multiplication

Commutative a � b � b � a a � b � b � a

Associative (a � b) � c � a � (b � c) (a � b) � c � a � (b � c)

Identity a � 0 � a � 0 � a a � 1 � a � 1 � a

Inverse a � (�a) � 0 � (�a) � a If a is not zero, then a � � 1 � � a.

Distributive a(b � c) � ab � ac and (b � c)a � ba � ca

Simplify 9x � 3y � 12y � 0.9x.

9x � 3y � 12y � 0.9x � 9x � (� 0.9x) � 3y � 12y Commutative Property (�)

� (9 � (� 0.9))x � (3 � 12)y Distributive Property

� 8.1x � 15y Simplify.

Simplify each expression.

1. 8(3a � b) � 4(2b � a) 2. 40s � 18t � 5t � 11s 3. (4j � 2k �6j �3k)

20a 51s � 13t k � j

4. 10(6g � 3h) � 4(5g �h) 5. 12� � � 6. 8(2.4r � 3.1s) � 6(1.5r � 2.4s)

80g � 26h 4a � 3b 10.2r � 39.2s

7. 4(20 � 4p) � (4 � 16p) 8. 5.5j � 8.9k � 4.7k �10.9j 9. 1.2(7x � 5) � (10 � 4.3x)

77 � 4p 4.2k � 5.4j 12.7x � 16

10. 9(7e � 4f) � 0.6(e � 5f ) 11. 2.5m(12 � 8.5) 12. p � r � r � p

62.4e � 39f 8.75m p � r

13. 4(10g � 80h) � 20(10h � 5g) 14. 2(15 � 45c) � (12 � 18c)

140g � 120h 40 � 105c

15. (7 � 2.1x)3 � 2(3.5x � 6) 16. (18 � 6n � 12 � 3n)

0.7x � 9 20 � 2n

17. 14( j � 2) � 3j(4 � 7) 18. 50(3a � b) � 20(b � 2a)2j � 7 190a � 70b

2�3

5�6

4�5

1�4

1�2

3�5

1�5

3�4

3�4

b�4

a�3

2�5

1�5

1�a

1�a



NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2

ExampleExample

ExercisesExercises

Skills PracticeProperties of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2


Less

on

1-2


1. 34 N, W, Z, Q, R 2. �525 Z, Q, R

3. 0.875 Q, R 4. N, W, Z, Q, R

5. ��9� Z, Q, R 6. �30� I, R

Name the property illustrated by each equation.

7. 3 � x � x � 3 8. 3a � 0 � 3aComm. (�) Add. Iden.

9. 2(r � w) � 2r � 2w 10. 2r � (3r � 4r) � (2r � 3r) � 4rDistributive Assoc. (�)

11. 5y� � � 1 12. 15x(1) � 15x

Mult. Inv. Mult. Iden.

13. 0.6[25(0.5)] � [0.6(25)]0.5 14. (10b � 12b) � 7b � (12b � 10b) � 7bAssoc. (�) Comm. (�)

Name the additive inverse and multiplicative inverse for each number.

15. 15 �15, 16. 1.25 �1.25, 0.8

17. � , � 18. 3 �3 ,


19. 3x � 5 � 2x � 3 5x � 2 20. x � y � z � y � x � z 0

21. �(3g � 3h) � 5g � 10h 2g � 13h 22. a2 � a � 4a � 3a2 � 1 �2a2 � 3a � 1

23. 3(m � z) � 5(2m � z) 13m � 8z 24. 2x � 3y � (5x � 3y � 2z) �3x � 2z

25. 6(2 � v) � 4(2v � 1) 8 � 2v 26. (15d � 3) � (8 � 10d) 10d � 31�2

1�3

4�15

3�4

3�4

5�4

4�5

4�5

1�15

1�5y

12�3



1. 6425 2. �7� 3. 2� 4. 0N, W, Z, Q, R I, R I, R W, Z, Q, R

5. �� Q, R 6. ��16� Z, Q, R 7. �35 Z, Q, R 8. �31.8 Q, R

Name the property illustrated by each equation.

9. 5x � (4y � 3x) � 5x � (3x � 4y) 10. 7x � (9x � 8) � (7x � 9x) � 8

Comm. (�) Assoc. (�)

11. 5(3x � y) � 5(3x � 1y) 12. 7n � 2n � (7 � 2)n

Mult. Iden. Distributive

13. 3(2x)y � (3 � 2)(xy) 14. 3x � 2y � 3 � 2 � x � y 15. (6 � �6)y � 0y

Assoc. (�) Comm. (�) Add. Inv.

16. � 4y � 1y 17. 5(x � y) � 5x � 5y 18. 4n � 0 � 4n

Mult. Inv. Distributive Add. Iden.

Name the additive inverse and multiplicative inverse for each number.

19. 0.4 �0.4, 2.5 20. �1.6 1.6, �0.625

21. � , � 22. 5 �5 ,


23. 5x � 3y � 2x � 3y 3x 24. �11a � 13b � 7a � 3b �4a � 16b

25. 8x � 7y � (3 � 6y) 8x � y � 3 26. 4c � 2c � (4c � 2c) �4c

27. 3(r � 10s) � 4(7s � 2r) �5r � 58s 28. (10a � 15) � (8 � 4a) 4a � 1

29. 2(4 � 2x � y) � 4(5 � x � y) 30. � x � 12y� � (2x � 12y)

�12 � 8x � 6y 13y

31. TRAVEL Olivia drives her car at 60 miles per hour for t hours. Ian drives his car at 50 miles per hour for (t � 2) hours. Write a simplified expression for the sum of thedistances traveled by the two cars. (110t � 100) mi

32. NUMBER THEORY Use the properties of real numbers to tell whether the following

statement is true or false: If a b, it follows that a� � b� �. Explain your reasoning.

false; counterexample: 5� � � 4� �1�4

1�5

1�b

1�a

1�4

3�5

5�6

1�2

1�5

6�35

5�6

5�6

16�11

11�16

11�16

1�4

25�36

Practice (Average)


NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2

Reading to Learn MathematicsProperties of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2


Less

on

1-2

Pre-Activity How is the Distributive Property useful in calculating store savings?


• Why are all of the amounts listed on the register slip at the top of page11 followed by negative signs? Sample answer: The amount ofeach coupon is subtracted from the total amount ofpurchases so that you save money by using coupons.

• Describe two ways of calculating the amount of money you saved byusing coupons if your register slip is the one shown on page 11.Sample answer: Add all the individual coupon amounts oradd the amounts for the scanned coupons and multiply thesum by 2.

Reading the Lesson1. Refer to the Key Concepts box on page 11. The numbers 2.5�7� and 0.010010001… both

involve decimals that “go on forever.” Explain why one of these numbers is rational andthe other is irrational. Sample answer: 2.5�7� � 2.5757… is a repeatingdecimal because there is a block of digits, 57, that repeats forever, sothis number is rational. The number 0.010010001… is a non-repeatingdecimal because, although the digits follow a pattern, there is no blockof digits that repeats. So this number is an irrational number.

2. Write the Associative Property of Addition in symbols. Then illustrate this property byfinding the sum 12 � 18 � 45 in two different ways. (a � b) � c � a � (b � c);Sample answer: (12 � 18) � 45 � 30 � 45 � 75;12 � (18 � 45) � 12 � 63 � 75

3. Consider the equations (a � b) � c � a � (b � c) and (a � b) � c � c � (a � b). One of theequations uses the Associative Property of Multiplication and one uses the CommutativeProperty of Multiplication. How can you tell which property is being used in eachequation? The first equation uses the Associative Property ofMultiplication. The quantities a, b, and c are used in the same order, butthey are grouped differently on the two sides of the equation. The secondequation uses the quantities in different orders on the two sides of theequation. So the second equation uses the Commutative Property ofMultiplication.

Helping You Remember4. How can the meanings of the words commuter and association help you to remember the

difference between the commutative and associative properties? Sample answer:A commuter is someone who travels back and forth to work or anotherplace, and the commutative property says you can switch the order whentwo numbers that are being added or multiplied. An association is agroup of people who are connected or united, and the associativeproperty says that you can switch the grouping when three numbers areadded or multiplied.


Properties of a GroupA set of numbers forms a group with respect to an operation if for that operationthe set has (1) the Closure Property, (2) the Associative Property, (3) a memberwhich is an identity, and (4) an inverse for each member of the set.

Does the set {0, 1, 2, 3, …} form a group with respect to addition?Closure Property: For all numbers in the set, is a � b in the set? 0 � 1 � 1, and 1 is

in the set; 0 � 2 � 2, and 2 is in the set; and so on. The set hasclosure for addition.

Associative Property: For all numbers in the set, does a � (b � c) � (a � b) � c? 0 � (1 � 2) � (0 � 1) � 2; 1 � (2 � 3) � (1 � 2) � 3; and so on.The set is associative for addition.

Identity: Is there some number, i, in the set such that i � a � a � a � ifor all a? 0 � 1 � 1 � 1 � 0; 0 � 2 � 2 � 2 � 0; and so on.The identity for addition is 0.

Inverse: Does each number, a, have an inverse, a , such that a � a � a � a � i? The integer inverse of 3 is �3 since �3 � 3 � 0, and 0 is the identity for addition. But the set does notcontain �3. Therefore, there is no inverse for 3.

The set is not a group with respect to addition because only three of the four properties hold.

Is the set {�1, 1} a group with respect to multiplication?Closure Property: (�1)(�1) � 1; (�1)(1) � �1; (1)(�1) � �1; (1)(1) � 1

The set has closure for multiplication.

Associative Property: (�1)[(�1)(�1)] � (�1)(1) � �1; and so onThe set is associative for multiplication.

Identity: 1(�1) � �1; 1(1) � 1The identity for multiplication is 1.

Inverse: �1 is the inverse of �1 since (�1)(�1) � 1, and 1 is the identity.1 is the inverse of 1 since (1)(1) � 1, and 1 is the identity.Each member has an inverse.

The set {�1, 1} is a group with respect to multiplication because all four properties hold.

Tell whether the set forms a group with respect to the given operation.

1. {integers}, addition yes 2. {integers}, multiplication no

3. ��12�, �

22�, �

32�, …�, addition no 4. {multiples of 5}, multiplication no

5. {x, x2, x3, x4, …} addition no 6. {�1�, �2�, �3�, …}, multiplication no

7. {irrational numbers}, addition no 8. {rational numbers}, addition yes

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-21-2

Example 1Example 1

Example 2Example 2

Study Guide and InterventionSolving Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3


Less

on

1-3

Verbal Expressions to Algebraic Expressions The chart suggests some ways tohelp you translate word expressions into algebraic expressions. Any letter can be used torepresent a number that is not known.

Word Expression Operation

and, plus, sum, increased by, more than addition

minus, difference, decreased by, less than subtraction

times, product, of (as in of a number) multiplication

divided by, quotient division

1�2

Write an algebraicexpression to represent 18 less thanthe quotient of a number and 3.

� 18n�3

Write a verbal sentence torepresent 6(n � 2) � 14.

Six times the difference of a number and twois equal to 14.


ExercisesExercises

Write an algebraic expression to represent each verbal expression.

1. the sum of six times a number and 25 6n � 25

2. four times the sum of a number and 3 4(n � 3)

3. 7 less than fifteen times a number 15n � 7

4. the difference of nine times a number and the quotient of 6 and the same number9n �

5. the sum of 100 and four times a number 100 � 4n

6. the product of 3 and the sum of 11 and a number 3(11 � n)

7. four times the square of a number increased by five times the same number 4n2 � 5n

8. 23 more than the product of 7 and a number 7n � 23

Write a verbal sentence to represent each equation. Sample answers are given.

9. 3n � 35 � 79 The difference of three times a number and 35 is equal to 79.

10. 2(n3 � 3n2) � 4n Twice the sum of the cube of a number and three times thesquare of the number is equal to four times the number.

11. �n5�n

3� � n � 8 The quotient of five times a number and the sum of thenumber and 3 is equal to the difference of the number and 8.

6�n


Properties of Equality You can solve equations by using addition, subtraction,multiplication, or division.

Addition and Subtraction For any real numbers a, b, and c, if a � b,Properties of Equality then a � c � b � c and a � c � b � c.

Multiplication and Division For any real numbers a, b, and c, if a � b,

Properties of Equality then a � c � b � c and, if c is not zero, � .b�c

a�c


Solving Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3

Solve 100 � 8x � 140.

100 � 8x � 140100 � 8x � 100 � 140 � 100

�8x � 40x � �5

Solve 4x � 5y � 100 for y.

4x � 5y � 1004x � 5y � 4x � 100 � 4x

5y � 100 � 4x

y � (100 � 4x)

y � 20 � x4�5

1�5


ExercisesExercises

Solve each equation. Check your solution.

1. 3s � 45 15 2. 17 � 9 � a �8 3. 5t � 1 � 6t � 5 4

4. m � 5. 7 � x � 3 8 6. �8 � �2(z � 7) �3

7. 0.2b � 10 50 8. 3x � 17 � 5x � 13 15 9. 5(4 � k) � �10k �4

10. 120 � y � 60 80 11. n � 98 � n 28 12. 4.5 � 2p � 8.7 2.1

13. 4n � 20 � 53 � 2n 5 14. 100 � 20 � 5r �16 15. 2x � 75 � 102 � x 9

Solve each equation or formula for the specified variable.

16. a � 3b � c, for b b � 17. � 10, for t t �

18. h � 12g � 1, for g g � 19. � 12, for p p �

20. 2xy � x � 7, for x x � 21. � � 6, for f f � 24 � 2d

22. 3(2j � k) � 108, for j j � 18 � 23. 3.5s � 42 � 14t, for s s � 4t � 12

24. � 5m � 20, for m m � 25. 4x � 3y � 10, for y y � x � 10�3

4�3

20n�5n � 1

m�n

k�2

f�4

d�2

7�2y � 1

4r�q

3pq�r

h � 1�

12

s�20

s�2t

a � c�

3

1�2

5�2

3�4

1�2

3�4

1�2

2�3

Skills PracticeSolving Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3


Less

on

1-3


1. 4 times a number, increased by 7 2. 8 less than 5 times a number

4n � 7 5n � 8

3. 6 times the sum of a number and 5 4. the product of 3 and a number, divided by 9

6(n � 5)

5. 3 times the difference of 4 and a number 3(4 � n)

6. the product of �11 and the square of a number �11n2

Write a verbal expression to represent each equation. 7–10. Sample answers

7. n � 8 � 16 8. 8 � 3x � 5are given.

The difference of a number The sum of 8 and 3 times a and 8 is 16. number is 5.

9. b2 � 3 � b 10. � 2 � 2y

Three added to the square of A number divided by 3 is thea number is the number. difference of 2 and twice the

number.

Name the property illustrated by each statement.

11. If a � 0.5b, and 0.5b � 10, then a � 10. 12. If d � 1 � f, then d � f � 1.Transitive (�) Subtraction (�)

13. If �7x � 14, then 14 � �7x. 14. If (8 � 7)r � 30, then 15r � 30.Symmetric (�) Substitution (�)


15. 4m � 2 � 18 4 16. x � 4 � 5x � 2

17. 3t � 2t � 5 5 18. �3b � 7 � �15 � 2b

19. �5x � 3x � 24 3 20. 4v � 20 � 6 � 34 5

21. a � � 3 5 22. 2.2n � 0.8n � 5 � 4n 5


23. I � prt, for p p � 24. y � x � 12, for x x � 4y � 48

25. A � , for y y � 2A � x 26. A � 2�r2 � 2�rh, for h h �A � 2r2��

2rx � y�2

1�4

I�rt

2a�5

22�5

1�2

y�3

3n�9



1. 2 more than the quotient of a number and 5 2. the sum of two consecutive integers

� 2 n � (n � 1)

3. 5 times the sum of a number and 1 4. 1 less than twice the square of a number5(m � 1) 2y2 � 1

Write a verbal expression to represent each equation. 5–8. Sample answers

5. 5 � 2x � 4 6. 3y � 4y3are given.

The difference of 5 and twice a Three times a number is 4 times number is 4. the cube of the number.

7. 3c � 2(c � 1) 8. � 3(2m � 1) The quotient

Three times a number is twice the of a number and 5 is 3 times the difference of the number and 1. sum of twice the number and 1.

Name the property illustrated by each statement.

9. If t � 13 � 52, then 52 � t � 13. 10. If 8(2q � 1) � 4, then 2(2q � 1) � 1.Symmetric (�) Division (�)

11. If h � 12 � 22, then h � 10. 12. If 4m � �15, then �12m � 45.Subtraction (�) Multiplication (�)


13. 14 � 8 � 6r �1 14. 9 � 4n � �59 �17

15. � n � 16. s � �

17. �1.6r � 5 � �7.8 8 18. 6x � 5 � 7 � 9x

19. 5(6 � 4v) � v � 21 20. 6y � 5 � �3(2y � 1)


21. E � mc2, for m m � 22. c � , for d d �

23. h � vt � gt2, for v v � 24. E � Iw2 � U, for I I �

Define a variable, write an equation, and solve the problem.

25. GEOMETRY The length of a rectangle is twice the width. Find the width if theperimeter is 60 centimeters. w � width; 2(2w) � 2w � 60; 10 cm

26. GOLF Luis and three friends went golfing. Two of the friends rented clubs for $6 each. Thetotal cost of the rented clubs and the green fees for each person was $76. What was the costof the green fees for each person? g � green fees per person; 6(2) � 4g � 76; $16

2(E � U )��

w21�2

h � gt2�

t

3c � 1�

22d � 1�3

E�c2

1�6

3�7

4�5

1�5

11�12

3�4

5�6

1�4

5�8

1�2

3�4

m�5

y�5

Practice (Average)

Solving Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3

Reading to Learn MathematicsSolving Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3


Less

on

1-3

Pre-Activity How can you find the most effective level of intensity for yourworkout?


• To find your target heart rate, what two pieces of information must yousupply? age (A) and desired intensity level (I )

• Write an equation that shows how to calculate your target heart rate.

P � or P � (220 � A) I � 6

Reading the Lesson

1. a. How are algebraic expressions and equations alike?Sample answer: Both contain variables, constants, and operationsigns.

b. How are algebraic expressions and equations different?Sample answer: Equations contain equal signs; expressions do not.

c. How are algebraic expressions and equations related?Sample answer: An equation is a statement that says that twoalgebraic expressions are equal.

Read the following problem and then write an equation that you could use tosolve it. Do not actually solve the equation. In your equation, let m be the numberof miles driven.

2. When Louisa rented a moving truck, she agreed to pay $28 per day plus $0.42 per mile.If she kept the truck for 3 days and the rental charges (without tax) were $153.72, howmany miles did Louisa drive the truck? 3(28) � 0.42m � 153.72

Helping You Remember

3. How can the words reflection and symmetry help you remember and distinguish betweenthe reflexive and symmetric properties of equality? Think about how these words areused in everyday life or in geometry.Sample answer: When you look at your reflection, you are looking atyourself. The reflexive property says that every number is equal to itself.In geometry, symmetry with respect to a line means that the parts of afigure on the two sides of a line are identical. The symmetric property ofequality allows you to interchange the two sides of an equation. Theequal sign is like the line of symmetry.

(220 � A) I��

6


Venn DiagramsRelationships among sets can be shown using Venn diagrams. Study thediagrams below. The circles represent sets A and B, which are subsets of set S.

The union of A and B consists of all elements in either A or B.The intersection of A and B consists of all elements in both A and B.The complement of A consists of all elements not in A.

You can combine the operations of union, intersection, and finding the complement.

Shade the region (A ∩ B)�.

(A � B) means the complement of the intersection of A and B.First find the intersection of A and B. Then find its complement.

Draw a Venn diagram and shade the region indicated. See students’ diagrams.

1. A � B 2. A � B

3. A � B 4. A � B

5. (A � B) 6. A � B

Draw a Venn diagram and three overlapping circles. Then shade the region indicated. See students’ diagrams.

7. (A � B) � C 8. (A � B) � C

9. A � (B � C) 10. (A � B) � C

11. Is the union operation associative? yes

12. Is the intersection operation associative? yes

A B

S

A B

S

A B

S

A

S

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-31-3

ExampleExample

Study Guide and InterventionSolving Absolute Value Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4


Less

on

1-4

Absolute Value Expressions The absolute value of a number is the number ofunits it is from 0 on a number line. The symbol x is used to represent the absolute valueof a number x.

• Words For any real number a, if a is positive or zero, the absolute value of a is a. Absolute Value If a is negative, the absolute value of a is the opposite of a.

• Symbols For any real number a, a � a, if a � 0, and a � �a, if a � 0.

Evaluate �4 � �2xif x � 6.

�4 � �2x � �4 � �2 � 6� �4 � �12� 4 � 12� �8

Evaluate 2x � 3yif x � �4 and y � 3.

2x � 3y � 2(�4) � 3(3)� �8 � 9� �17� 17


ExercisesExercises

Evaluate each expression if w � �4, x � 2, y � , and z � �6.

1. 2x � 8 4 2. 6 � z � �7 �7 3. 5 � w � z 15

4. x � 5 � 2w �1 5. x � y � z �4 6. 7 � x � 3x 11

7. w � 4x 12 8. wz � xy 23 9. z � 3 5yz �39

10. 5 w � 2 z � 2y 34 11. z � 4 2z � y �40 12. 10 � xw 2

13. 6y � z � yz 6 14. 3 wx � 4x � 8y 27 15. 7 yz � 30 �9

16. 14 � 2 w � xy 4 17. 2x � y � 5y 6 18. xyz � wxz 54

19. z z � x x �32 20. 12 � 10x � 10y �3 21. 5z � 8w 31

22. yz � 4w � w 17 23. wz � 8y 20 24. xz � xz �241�2

3�4

1�2

1�4

1�2

1�2


Absolute Value Equations Use the definition of absolute value to solve equationscontaining absolute value expressions.

For any real numbers a and b, where b � 0, if a � b then a � b or a � �b.

Always check your answers by substituting them into the original equation. Sometimescomputed solutions are not actual solutions.

Solve 2x � 3 � 17. Check your solutions.


Solving Absolute Value Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4

ExampleExample

Case 1 a � b2x � 3 � 17

2x � 3 � 3 � 17 � 32x � 20x � 10

CHECK 2x � 3 � 17 2(10) � 3 � 17

20 � 3 � 17 17 � 17

17 � 17 ✓

Case 2 a � �b2x � 3 � �17

2x � 3 � 3 � �17 � 32x � �14x � �7

CHECK 2(�7) � 3 � 17 �14 � 3 � 17

�17 � 1717 � 17 ✓

There are two solutions, 10 and �7.

Solve each equation. Check your solutions.

1. x � 15 � 37 {�52, 22} 2. t � 4 � 5 � 0 {�1, 9}

3. x � 5 � 45 {�40, 50} 4. m � 3 � 12 � 2m {3}

5. 5b � 9 � 16 � 2 � 6. 15 � 2k � 45 {�15, 30}

7. 5n � 24 � 8 � 3n {�2} 8. 8 � 5a � 14 � a �� , 1�9. 4p � 11 � p � 4 �23, � � 10. 3x � 1 � 2x � 11 {�2, 12}

11. x � 3 � �1 � 12. 40 � 4x � 2 3x � 10 {6, �10}

13. 5f � 3f � 4 � 20 {12} 14. 4b � 3 � 15 � 2b {2, �9}

15. 6 � 2x � 3x � 1 � � 16. 16 � 3x � 4x � 12 {4}1�2

1�2

1�3

1�7

1�3

11�2

ExercisesExercises

Skills PracticeSolving Absolute Value Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4


Less

on

1-4

Evaluate each expression if w � 0.4, x � 2, y � �3, and z � �10.

1. 5w 2 2. �9y 27

3. 9y � z 17 4. � 17z �170

5. � 10z � 31 �131 6. � 8x � 3y � 2y � 5x �21

7. 25 � 5z � 1 �24 8. 44 � �2x � y 45

9. 2 4w 3.2 10. 3 � 1 � 6w 1.6

11. �3x � 2y � 4 �4 12. 6.4 � w � 1 7


13. y � 3 � 2 {�5, �1} 14. 5a � 10 {�2, 2}

15. 3k � 6 � 2 � , � 16. 2g � 6 � 0 {�3}

17. 10 � 1 � c {�9, 11} 18. 2x � x � 9 {�3, 3}

19. p � 7 � �14 � 20. 2 3w � 12 {�2, 2}

21. 7x � 3x � 2 � 18 {�4, 4} 22. 4 7 � y � 1 � 11 {4, 10}

23. 3n � 2 � � , � 24. 8d � 4d � 5 � 13 {�2, 2}

25. �5 6a � 2 � �15 �� , � 26. k � 10 � 9 �1�6

5�6

5�6

1�2

1�2

8�3

4�3


Evaluate each expression if a � �1, b � �8, c � 5, and d � �1.4.

1. 6a 6 2. 2b � 4 12

3. � 10d � a �15 4. 17c � 3b � 5 114

5. �6 10a � 12 �132 6. 2b � 1 � �8b � 5 �52

7. 5a � 7 � 3c � 4 23 8. 1 � 7c � a 33

9. �3 0.5c � 2 � �0.5b �17.5 10. 4d � 5 � 2a 12.6

11. a � b � b � a 14 12. 2 � 2d � 3 b �19.2


13. n � 4 � 13 {�9, 17} 14. x � 13 � 2 {11, 15}

15. 2y � 3 � 29 {�13, 16} 16. 7 x � 3 � 42 {�9, 3}

17. 3u � 6 � 42 {�12, 16} 18. 5x � 4 � �6 �

19. �3 4x � 9 � 24 � 20. �6 5 � 2y � �9 � , �21. 8 � p � 2p � 3 {11} 22. 4w � 1 � 5w � 37 {�4}

23. 4 2y � 7 � 5 � 9 {3, 4} 24. �2 7 � 3y � 6 � �14 �1, �25. 2 4 � s � �3s {�8} 26. 5 � 3 2 � 2w � �7 {�3, 1}

27. 5 2r � 3 � 5 � 0 {�2, �1} 28. 3 � 5 2d � 3 � 4 �

29. WEATHER A thermometer comes with a guarantee that the stated temperature differsfrom the actual temperature by no more than 1.5 degrees Fahrenheit. Write and solve anequation to find the minimum and maximum actual temperatures when thethermometer states that the temperature is 87.4 degrees Fahrenheit.t � 87.4 � 1.5; minimum: 85.9�F, maximum: 88.9�F

30. OPINION POLLS Public opinion polls reported in newspapers are usually given with amargin of error. For example, a poll with a margin of error of 5% is considered accurateto within plus or minus 5% of the actual value. A poll with a stated margin of error of 3% predicts that candidate Tonwe will receive 51% of an upcoming vote. Write andsolve an equation describing the minimum and maximum percent of the vote thatcandidate Tonwe is expected to receive.x � 51 � 3; minimum: 48%, maximum: 54%

11�3

13�4

7�4

Practice (Average)

Solving Absolute Value Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4

Reading to Learn MathematicsSolving Absolute Value Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

1-41-4


Less

on

1-4

Pre-Activity How can an absolute value equation describe the magnitude of anearthquake?


• What is a seismologist and what does magnitude of an earthquake mean?a scientist who studies earthquakes; a number from 1 to 10that tells how strong an earthquake is

• Why is an absolute value equation rather than an equation withoutabsolute value used to find the extremes in the actual magnitude of anearthquake in relation to its measured value on the Richter scale?Sample answer: The actual magnitude can vary from themeasured magnitude by up to 0.3 unit in either direction, soan absolute value equation is needed.

• If the magnitude of an earthquake is estimated to be 6.9 on the Richter

scale, it might actually have a magnitude as low as or as high

as .

Reading the Lesson

1. Explain how �a could represent a positive number. Give an example. Sampleanswer: If a is negative, then �a is positive. Example: If a � �25, then �a � �(�25) � 25.

2. Explain why the absolute value of a number can never be negative. Sample answer:The absolute value is the number of units it is from 0 on the number line.The number of units is never negative.

3. What does the sentence b � 0 mean? Sample answer: The number b is 0 orgreater than 0.

4. What does the symbol � mean as a solution set? Sample answer: If a solution setis �, then there are no solutions.


5. How can the number line model for absolute value that is shown on page 28 of yourtextbook help you remember that many absolute value equations have two solutions?Sample answer: The number line shows that for every positive number,there are two numbers that have that number as their absolute value.

7.26.6


Considering All Cases in Absolute Value Equations You have learned that absolute value equations with one set of absolute valuesymbols have two cases that must be considered. For example, | x � 3 | � 5 mustbe broken into x � 3 � 5 or �(x � 3) � 5. For an equation with two sets ofabsolute value symbols, four cases must be considered.

Consider the problem | x � 2 | � 3 � | x � 6 |. First we must write the equationsfor the case where x � 6 � 0 and where x � 6 � 0. Here are the equations forthese two cases:

| x � 2 | � 3 � x � 6

| x � 2 | � 3 � �(x � 6)

Each of these equations also has two cases. By writing the equations for bothcases of each equation above, you end up with the following four equations:

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

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

Solve each of these equations and check your solutions in the original equation,

| x � 2 | � 3 � | x � 6 |. The only solution to this equation is ��52�.

Solve each absolute value equation. Check your solution.

1. | x � 4 | � | x � 7 | x � �1.5 2. |2x � 9 | � | x � 3 | x � �12, �2

3. |�3x � 6 | � |5x � 10 | x � �2 4. | x � 4 | � 6 � | x � 3 | x � 2.5

5. How many cases would there be for an absolute value equation containing three sets of absolute value symbols? 8

6. List each case and solve | x � 2 | � |2x � 4 | � | x � 3 |. Check your solution.

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

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

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

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

No solution

Enrichment

NAME ______________________________________________ DATE______________ PERIOD _____

1-41-4

Study Guide and InterventionSolving Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5


Less

on

1-5

Solve Inequalities The following properties can be used to solve inequalities.

Addition and Subtraction Properties for Inequalities Multiplication and Division Properties for Inequalities

For any real numbers a, b, and c: For any real numbers a, b, and c, with c � 0:1. If a � b, then a � c � b � c and a � c � b � c. 1. If c is positive and a � b, then ac � bc and � .2. If a b, then a � c b � c and a � c b � c.

2. If c is positive and a b, then ac bc and .

3. If c is negative and a � b, then ac bc and .

4. If c is negative and a b, then ac � bc and � .

These properties are also true for � and �.

b�c

a�c

b�c

a�c

b�c

a�c

b�c

a�c

Solve 2x � 4 36.Then graph the solution set on anumber line.

2x � 4 � 4 36 � 42x 32x 16

The solution set is {x x 16}.

212019181716151413

Solve 17 � 3w � 35. Thengraph the solution set on a number line.

17 � 3w � 3517 � 3w � 17 � 35 � 17

�3w � 18w � �6

The solution set is (��, �6].

�9 �8 �7 �6 �5 �4 �3 �2 �1


ExercisesExercises

Solve each inequality. Describe the solution set using set-builder or intervalnotation. Then graph the solution set on a number line.

1. 7(7a � 9) � 84 2. 3(9z � 4) 35z � 4 3. 5(12 � 3n) � 165

{aa � 3} or (�∞, 3] {zz � 2} or (�∞, 2) {nn �7} or (�7, �∞)

4. 18 � 4k � 2(k � 21) 5. 4(b � 7) � 6 � 22 6. 2 � 3(m � 5) � 4(m� 3)

{kk �4} or (�4, �∞) {bb � 11} or (�∞, 11) {mm � 5} or (�∞, 5]

7. 4x � 2 �7(4x � 2) 8. (2y � 3) y � 2 9. 2.5d � 15 � 75

�xx � or � , �∞� {yy � �9} or (�∞, �9) {dd � 24} or (�∞, 24]

21 2219 20 23 24 25 26 27�12�14 �10 �8 �6�4 �3 �2 �1 0 1 2 3 4

1�2

1�2

1�3

2 30 1 4 5 6 7 88 96 7 10 11 12 13 14�8 �7 �6 �5 �4 �3 �2 �1 0

�8 �7 �6 �5 �4 �3 �2 �1 0�2 �1�4 �3 0 1 2 3 4�2 �1�4 �3 0 1 2 3 4


Real-World Problems with Inequalities Many real-world problems involveinequalities. The chart below shows some common phrases that indicate inequalities.

� � �

is less than is greater than is at most is at leastis fewer than is more than is no more than is no less than

is less than or equal to is greater than or equal to

SPORTS The Vikings play 36 games this year. At midseason, theyhave won 16 games. How many of the remaining games must they win in order towin at least 80% of all their games this season?

Let x be the number of remaining games that the Vikings must win. The total number ofgames they will have won by the end of the season is 16 � x. They want to win at least 80%of their games. Write an inequality with �.16 � x � 0.8(36)

x � 0.8(36) � 16x � 12.8

Since they cannot win a fractional part of a game, the Vikings must win at least 13 of thegames remaining.

1. PARKING FEES The city parking lot charges $2.50 for the first hour and $0.25 for eachadditional hour. If the most you want to pay for parking is $6.50, solve the inequality2.50 � 0.25(x � 1) � 6.50 to determine for how many hours you can park your car.At most 17 hours

PLANNING For Exercises 2 and 3, use the following information.

Ethan is reading a 482-page book for a book report due on Monday. He has already read 80 pages. He wants to figure out how many pages per hour he needs to read in order tofinish the book in less than 6 hours.

2. Write an inequality to describe this situation. � 6 or 6n � 482 � 80

3. Solve the inequality and interpret the solution. Ethan must read at least 67 pagesper hour in order to finish the book in less than 6 hours.

BOWLING For Exercises 4 and 5, use the following information.

Four friends plan to spend Friday evening at the bowling alley. Three of the friends need torent shoes for $3.50 per person. A string (game) of bowling costs $1.50 per person. If thefriends pool their $40, how many strings can they afford to bowl?

4. Write an equation to describe this situation. 3(3.50) � 4(1.50)n � 40

5. Solve the inequality and interpret the solution. The friends can bowl at most 4 strings.

482 � 80��

n


Solving Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5

ExampleExample

ExercisesExercises

Skills PracticeSolving Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5


Less

on

1-5

Solve each inequality. Describe the solution set using set-builder or intervalnotation. Then, graph the solution set on a number line.

1. � 2 {zz � �8} or (�∞, �8] 2. 3a � 7 � 16 {aa � 3} or (�∞, 3]

3. 16 � 3q � 4 {qq 4} or (4, ∞) 4. 20 � 3s 7s {ss � 2} or (�∞, 2)

5. 3x � �9 {xx � �3} or [�3, ∞) 6. 4b � 9 � 7 {bb � 4} or (�∞, 4]

7. 2z � �9 � 5z {zz 3} or (3, ∞) 8. 7f � 9 3f � 1 {ff 2} or (2, ∞)

9. �3s � 8 � 5s {ss � �1} or [�1, ∞) 10. 7t � (t � 4) � 25 �tt � � or ��∞, �

11. 0.7m � 0.3m � 2m � 4 {mm � 4} 12. 4(5x � 7) � 13 �xx � � � oror (�∞, 4]

��∞, � �13. 1.7y � 0.78 5 {yy 3.4} 14. 4x � 9 2x � 1 {xx 5} or (5, ∞)

or (3.4, ∞)

Define a variable and write an inequality for each problem. Then solve.

15. Nineteen more than a number is less than 42. n � 19 � 42; n � 23

16. The difference of three times a number and 16 is at least 8. 3n � 16 � 8; n � 8

17. One half of a number is more than 6 less than the same number. n n � 6; n � 12

18. Five less than the product of 6 and a number is no more than twice that same number.

6n � 5 � 2n; n � 5�4

1�2

�1 0 1 2 3 4 5 6 7�1�2 0 1 2 3 4 5 6

3�4�2 �1�4 �3 0 1 2 3 4�2 �1 0 1 2 3 4 5 6

3�4

�2 �1�4 �3 0 1 2 3 4�1�2�3�4 0 1 2 3 4

7�2

7�2

�1�2�3�4 0 1 2 3 4�1�2 0 1 2 3 4 5 6

�2 �1 0 1 2 3 4 5 6�1�2�3�4 0 1 2 3 4

�2 �1�4 �3 0 1 2 3 4�1 0 1 2 3 4 5 6 7

�2 �1�4 �3 0 1 2 3 4�7 �6�9 �8 �5 �4 �3 �2 �1

z��4


Solve each inequality. Describe the solution set using set-builder or intervalnotation. Then, graph the solution set on a number line.

1. 8x � 6 � 10 {xx � 2} or [2, ∞) 2. 23 � 4u � 11 {uu 3} or (3, ∞)

3. �16 � 8r � 0 {rr � �2} or (�∞, �2] 4. 14s � 9s � 5 {ss � 1} or (�∞, 1)

5. 9x � 11 6x � 9 �xx � or � , ∞� 6. �3(4w � 1) 18 �ww � � �or ��∞, � �

7. 1 � 8u � 3u � 10 {uu � 1} or [1, ∞) 8. 17.5 � 19 � 2.5x {xx � 0.6} or (�∞, 0.6)

9. 9(2r � 5) � 3 � 7r � 4 {rr � 4} 10. 1 � 5(x � 8) � 2 � (x � 5) {xx � 6} or (�∞, 4) or (�∞, 6]

11. � �3.5 {xx � �1} or [�1, ∞) 12. q � 2(2 � q) � 0 �qq � � or ��∞, �

13. �36 � 2(w � 77) �4(2w � 52) 14. 4n � 5(n � 3) 3(n � 1) � 4 {ww �3} or (�3, ∞) {nn � 4} or (�∞, 4)

Define a variable and write an inequality for each problem. Then solve.

15. Twenty less than a number is more than twice the same number.n � 20 2n; n � �20

16. Four times the sum of twice a number and �3 is less than 5.5 times that same number.4[2n � (�3)] � 5.5n; n � 4.8

17. HOTELS The Lincoln’s hotel room costs $90 a night. An additional 10% tax is added.Hotel parking is $12 per day. The Lincoln’s expect to spend $30 in tips during their stay.Solve the inequality 90x � 90(0.1)x � 12x � 30 � 600 to find how many nights theLincoln’s can stay at the hotel without exceeding total hotel costs of $600. 5 nights

18. BANKING Jan’s account balance is $3800. Of this, $750 is for rent. Jan wants to keep abalance of at least $500. Write and solve an inequality describing how much she canwithdraw and still meet these conditions. 3800 � 750 � w � 500; w � $2550

��

��

4�3

4�3

4x � 3�2

� �

�2 �1�4 �3 0 1 2 3 4�1�2�3�4 0 1 2 3 4

5�4�2 �1�4 �3 0 1 2 3 4�1�2�3�4 0 1 2 3 4

5�4

2�3

2�3

�2 �1�4 �3 0 1 2 3 4�2 �1�4 �3 0 1 2 3 4

�1�2 0 1 2 3 4 5 6�1�2�3�4 0 1 2 3 4

Practice (Average)

Solving Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5

Reading to Learn MathematicsSolving Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5


Less

on

1-5

Pre-Activity How can inequalities be used to compare phone plans?


• Write an inequality comparing the number of minutes per monthincluded in the two phone plans. 150 � 400 or 400 150

• Suppose that in one month you use 230 minutes of airtime on yourwireless phone. Find your monthly cost with each plan.

Plan 1: Plan 2:

Which plan should you choose?

Reading the Lesson

1. There are several different ways to write or show inequalities. Write each of thefollowing in interval notation.

a. {x x � �3} (�∞, �3)

b. {x x � 5} [5, �∞)

c. (�∞, 2]

d. (�1, �∞)

2. Show how you can write an inequality symbol followed by a number to describe each ofthe following situations.

a. There are fewer than 600 students in the senior class. � 600

b. A student may enroll in no more than six courses each semester. � 6

c. To participate in a concert, you must be willing to attend at least ten rehearsals. � 10

d. There is space for at most 165 students in the high school band. � 165


3. One way to remember something is to explain it to another person. A common studenterror in solving inequalities is forgetting to reverse the inequality symbol whenmultiplying or dividing both sides of an inequality by a negative number. Suppose thatyour classmate is having trouble remembering this rule. How could you explain this ruleto your classmate? Sample answer: Draw a number line. Plot two positivenumbers, for example, 3 and 8. Then plot their additive inverses, �3 and�8. Write an inequality that compares the positive numbers and one thatcompares the negative numbers. Notice that 8 3, but �8 � �3. Theorder changes when you multiply by �1.

32 5410�1�2�3�4�5

�5 �4 �3 �2 �1 0 1 2 3 4 5

Plan 2$55$67


Equivalence RelationsA relation R on a set A is an equivalence relation if it has the following properties.

Reflexive Property For any element a of set A, a R a.

Symmetric Property For all elements a and b of set A, if a R b, then b R a.

Transitive Property For all elements a, b, and c of set A,if a R b and b R c, then a R c.

Equality on the set of all real numbers is reflexive, symmetric, and transitive.Therefore, it is an equivalence relation.

In each of the following, a relation and a set are given. Write yes if the relation is an equivalence relation on the given set. If it is not, tell which of the properties it fails to exhibit.

1. �, {all numbers} no; reflexive, symmetric

2. , {all triangles in a plane} yes

3. is the sister of, {all women in Tennessee} no; reflexive

4. �, {all numbers} no; symmetric

5. is a factor of, {all nonzero integers} no; symmetric

6. , {all polygons in a plane} yes

7. is the spouse of, {all people in Roanoke, Virginia} no; reflexive, transitive

8. ⊥ , {all lines in a plane} no; reflexive, transitive

9. is a multiple of, {all integers} no; symmetric

10. is the square of, {all numbers} no; reflexive, symmetric, transitive

11. ��, {all lines in a plane} no; reflexive

12. has the same color eyes as, {all members of the Cleveland Symphony Orchestra} yes

13. is the greatest integer not greater than, {all numbers}no; reflexive, symmetric, transitive

14. is the greatest integer not greater than, {all integers} yes

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-51-5

Study Guide and InterventionSolving Compound and Absolute Value Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6


Less

on

1-6

Compound Inequalities A compound inequality consists of two inequalities joined bythe word and or the word or. To solve a compound inequality, you must solve each partseparately.

Example: x �4 and x � 3 The graph is the intersection of solution sets of two inequalities.

Example: x � �3 or x 1 The graph is the union of solution sets of two inequalities.

�5 �4 �3 �2 �1 0 1 2 3 4 5

OrCompoundInequalities

�3 �2�5 �4 �1 0 1 2 3 4 5

AndCompoundInequalities

Solve �3 � 2x � 5 � 19.Graph the solution set on a number line.

�3 � 2x � 5 and 2x � 5 � 19�8 � 2x 2x � 14�4 � x x � 7

�4 � x � 7

�4 �2�8 �6 0 2 4 6 8

Solve 3y �2 � 7 or 2y � 1 � �9. Graph the solution seton a number line.

3y � 2 � 7 or 2y � 1 � �93y � 9 or 2y � �8y � 3 or y � �4

�8 �6 �4 �2 0 2 4 6 8


ExercisesExercises

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

1. �10 � 3x � 2 � 14 2. 3a � 8 � 23 or a � 6 7

{x�4 � x � 4} {aa � 5 or a 52}

3. 18 � 4x � 10 � 50 4. 5k � 2 � �13 or 8k � 1 19

{x7 � x � 15} {kk � �3 or k 2.5}

5. 100 � 5y � 45 � 225 6. b � 2 10 or b � 5 � �4

{y29 � y � 54} {bb � �12 or b 18}

7. 22 � 6w �2 � 82 8. 4d � 1 �9 or 2d � 5 � 11

{w4 � w � 14} {all real numbers}

0�1�2�3�4 1 2 3 40 2 4 6 8 10 12 14 16

�24 �12 0 12 240 10 20 30 40 50 60 70 80

3�4

2�3

�4 �3 �2 �1 0 1 2 3 43 5 7 9 11 13 15 17 19

�10 0 10 20 30 40 50 60 70�8 �6 �4 �2 0 2 4 6 8

1�4


Absolute Value Inequalities Use the definition of absolute value to rewrite anabsolute value inequality as a compound inequality.

For all real numbers a and b, b 0, the following statements are true.

1. If a � b, then �b � a � b.2. If a b, then a b or a � �b.

These statements are also true for � and �.


Solving Compound and Absolute Value Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6

Solve x � 2 4. Graphthe solution set on a number line.

By statement 2 above, if x � 2 4, then x � 2 4 or x � 2 � �4. Subtracting 2from both sides of each inequality gives x 2 or x � �6.

�8 �6 �4 �2 0 2 4 6 8

Solve 2x � 1 � 5.Graph the solution set on a number line.

By statement 1 above, if 2x � 1 � 5, then�5 � 2x � 1 � 5. Adding 1 to all three partsof the inequality gives �4 � 2x � 6.Dividing by 2 gives �2 � x � 3.

�4 �2�8 �6 0 2 4 6 8


ExercisesExercises


1. 3x � 4 � 8 �x�4 � x � � 2. 4s � 1 27 {ss � �6.5 or s 6.5}

3. � 3 � 5 {c�4 � c � 16} 4. a � 9 � 30 {aa � �39 or a � 21}

5. 2f � 11 9 {ff � 1 or f 10} 6. 5w � 2 � 28 {w�6 � w � 5.2}

7. 10 � 2k � 2 {k4 � k � 6} 8. � 5 � 2 10 {xx � �6 or x 26}

9. 4b � 11 � 17 �b� � b � 7� 10. 100 � 3m 20 �mm � 26 or m 40�0 10 20 305 15 25 35 40�4 0 4 8�2 2 6 10 12

2�3

3�2

�10 0 10 20�5 5 15 25 300 2 4 61 3 5 7 8

x�2

�8 �4 0 4�6 �2 2 6 8�4 0 4 8�2 2 6 10 12

�40 �20 0 20 40�8 0 8 16�4 4 12 20 24

c�2

�8 �4 0 4�6 �2 2 6 8�5 �4 �3 �2 �1 0 1 2 3

4�3

Skills PracticeSolving Compound and Absolute Value Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6


Less

on

1-6

Write an absolute value inequality for each of the following. Then graph thesolution set on a number line.

1. all numbers greater than or equal to 2 2. all numbers less than 5 and greater or less than or equal to �2 n � 2 than �5 n � 5

3. all numbers less than �1 or greater 4. all numbers between �6 and 6 n � 6than 1 n 1

Write an absolute value inequality for each graph.

5. n � 1 6. n � 4

7. n � 3 8. n 2.5


9. 2c � 1 5 or c � 0 {cc 2 10. �11 � 4y � 3 � 1 {y�2 � y � 1}or c � 0}

11. 10 �5x 5 {x�2 � x � �1} 12. 4a � �8 or a � �3 {aa � �2or a � �3}

13. 8 � 3x � 2 � 23 {x2 � x � 7} 14. w � 4 � 10 or �2w � 6 all realnumbers

15. t � 3 {tt � �3 or t � 3} 16. 6x � 12 {x�2 � x � 2}

17. �7r 14 {rr � �2 or r 2} 18. p � 2 � �2 �

19. n � 5 � 7 {n�2 � n � 12} 20. h � 1 � 5 {hh � �6 or h � 4}

�8 �6 �4 �2 0 2 4 6 8�4 �2 0 2 4 6 8 10 12

0�1�2�3�4 1 2 3 4�4 �3 �2 �1 0 1 2 3 4

�4 �3 �2 �1 0 1 2 3 4�4 �3 �2 �1 0 1 2 3 4

0�1�2�3�4 1 2 3 40 1 2 3 4 5 6 7 8

�4 �3 �2 �1 0 1 2 3 4�4 �3 �2 �1 0 1 2 3 4

�4 �3 �2 �1 0 1 2 3 4�4 �3 �2 �1 0 1 2 3 4

�4 �3 �2 �1 0 1 2 3 4�2 �1�4 �3 0 1 2 3 4

�4 �3 �2 �1 0 1 2 3 4�2 �1�4 �3 0 1 2 3 4

�8 �6 �4 �2 0 2 4 6 8�4 �3 �2 �1 0 1 2 3 4

�8 �6 �4 �2 0 2 4 6 8�4 �3 �2 �1 0 1 2 3 4


Write an absolute value inequality for each of the following. Then graph thesolution set on a number line.

1. all numbers greater than 4 or less than �4 n 4

2. all numbers between �1.5 and 1.5, including �1.5 and 1.5 n � 1.5

Write an absolute value inequality for each graph.

3. n � 10 4. n �


5. �8 � 3y � 20 � 52 {y4 � y � 24} 6. 3(5x � 2) � 24 or 6x � 4 4 � 5x{xx � 2 or x 8}

7. 2x � 3 15 or 3 � 7x � 17 {xx �2} 8. 15 � 5x � 0 and 5x � 6 � �14 {xx � 3}

9. 2w � 5 �ww � � or w � � 10. y � 5 � 2 {x�7 � x � �3}

11. x � 8 � 3 {xx � 5 or x � 11} 12. 2z � 2 � 3 �z� � z � �

13. 2x � 2 � 7 � �5 {x�2 � x � 0} 14. x x � 1 all real numbers

15. 3b � 5 � �2 � 16. 3n � 2 � 2 � 1 �n� � n � �

17. RAINFALL In 90% of the last 30 years, the rainfall at Shell Beach has varied no morethan 6.5 inches from its mean value of 24 inches. Write and solve an absolute valueinequality to describe the rainfall in the other 10% of the last 30 years.r � 24 6.5; {rr � 17.5 or r 30.5}

18. MANUFACTURING A company’s guidelines call for each can of soup produced not to varyfrom its stated volume of 14.5 fluid ounces by more than 0.08 ounces. Write and solve anabsolute value inequality to describe acceptable can volumes.v � 14.5 � 0.08; {v14.42 � v � 14.58}

5�3

1�3

5�2

1�2

�8 �7 �6 �5 �4 �3 �2 �1 0�4 �3 �2 �1 0 1 2 3 4

5�2

5�2

�1�2�3�4 0 1 2 3 4�1�2�3�4 0 1 2 3 4

�2 0 2 4 6 8 10 12 140 4 8 12 16 20 24 28 32

4�3�4 �3 �2 �1 0 1 2 3 4�20 �10 0 10 20

�8 �6 �4 �2 0 2 4 6 8

Practice (Average)

Solving Compound and Absolute Value Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6

Reading to Learn MathematicsSolving Compound and Absolute Value Inequalities

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6


Less

on

1-6

Pre-Activity How are compound inequalities used in medicine?


• Five patients arrive at a medical laboratory at 11:30 A.M. for a glucosetolerance test. Each of them is asked when they last had something toeat or drink. Some of the patients are given the test and others are toldthat they must come back another day. Each of the patients is listedbelow with the times when they started to fast. (The P.M. times refer tothe night before.) Which of the patients were accepted for the test?

Ora 5:00 A.M. Juanita 11:30 P.M. Jason and JuanitaJason 1:30 A.M. Samir 5:00 P.M.

Reading the Lesson

1. a. Write a compound inequality that says, “x is greater than �3 and x is less than orequal to 4.” �3 � x � 4

b. Graph the inequality that you wrote in part a on a number line.

2. Use a compound inequality and set-builder notation to describe the following graph.{xx � �1 or x 3}

3. Write a statement equivalent to 4x � 5 2 that does not use the absolute valuesymbol. 4x � 5 2 or 4x � 5 � �2

4. Write a statement equivalent to 3x � 7 � 8 that does not use the absolute valuesymbol. �8 � 3x � 7 � 8


5. Many students have trouble knowing whether an absolute value inequality should betranslated into an and or an or compound inequality. Describe a way to remember whichof these applies to an absolute value inequality. Also describe how to recognize thedifference from a number line graph. Sample answer: If the absolute valuequantity is followed by a � or � symbol, the expression inside theabsolute value bars must be between two numbers, so this becomes anand inequality. The number line graph will show a single interval betweentwo numbers. If the absolute value quantity is followed by a or �symbol, it becomes an or inequality, and the graph will show twodisconnected intervals with arrows going in opposite directions.

�4�5 �3 �2 �1 0 1 2 3 4 5

�4 �3 �2 �1 0 1 2 3 5�5 4


Conjunctions and DisjunctionsAn absolute value inequality may be solved as a compound sentence.

Solve �2x � � 10.

�2 x � � 10 means 2x � 10 and 2x �10.

Solve each inequality. x � 5 and x �5.

Every solution for �2x � � 10 is a replacement for x that makes both x � 5 and x �5 true.

A compound sentence that combines two statements by the word and is a conjunction.

Solve �3x � 7� � 11.�3x � 7 � � 11 means 3x � 7 � 11 or 3x � 7 � �11.

Solve each inequality. 3x � 18 or 3x � �4

x � 6 or x � ��43�

Every solution for the inequality is a replacement for x that makes either

x � 6 or x � ��43� true.

A compound sentence that combines two statements by the word or is a disjunction.

Solve each inequality. Then write whether the solution is a conjunction ordisjunction.

1. �4x � 24 2. �x � 7 � � 8

x 6 or x � �6; disjunction x � 15 and x � �1; conjunction

3. �2x � 5 � � 1 4. �x � 1 � � 1

x � �2 and x �3; conjunction x � 2 or x � 0; disjunction

5. �3x � 1 � � x 6. 7 � �2x � 5

x � �12

� and x � �14

�; conjunction x � 1 and x �1; conjunction

7. � �2x� � 1 � � 7 8. ��x �

34

� � � 4

x � 12 or x � �16; disjunction x � 16 and x �8; conjunction

9. �8 � x � 2 10. �5 � 2x � � 3

x � 6 or x 10; disjunction x � 1 and x � 4; conjunction

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6

Example 1Example 1

Example 2Example 2

Chapter 1 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. Find the value of 4 � 5[14 � (8 � 3)].A. 27 B. 19 C. 49 D. �46 1.

2. Evaluate (a � y)2 � 2y if a � 5 and y � �3.A. 58 B. �2 C. 70 D. 10 2.

3. Evaluate � �2b � if b � 8.A. �16 B. 6 C. 10 D. 16 3.

4. The formula S � �n(n

2� 1)� can be used to find the sum of the first n natural

numbers. Find the sum of the first 20 natural numbers.A. 210 B. 20 C. 21 D. 190 4.

5. Name the sets of numbers to which �35� belongs.

A. rationals B. naturals, realsC. rationals, reals D. integers, rationals, reals 5.

6. Simplify 2(x � 3) � 5(2x � 1).A. 12x � 1 B. 12x � 11 C. 12x � 2 D. 9x � 1 6.

7. Select the algebraic expression that represents the verbal expression:the product of nine and a number

A. �n9

� B. 9n C. 9 � n D. 9 � n 7.

For Questions 8–11, solve each equation.

8. �12�y � 8

A. 16 B. 4 C. �14� D. 10 8.

9. 4(2x � 9) � 3x � 4

A. �32 B. ��

532� C. �

430� D. 8 9.

10. � x � 5 � � 4A. {9} B. {1} C. {9, 1} D. � 10.

11. 4� x � 3 � � 20A. {2} B. {�8} C. {2, �8} D. � 11.

11


Chapter 1 Test, Form 1 (continued)

12. Which equation could be used to solve the following problem?The sum of 4 times a number and 7 is 31. Find the number.A. 4(n � 7) � 31 B. 4n � 7 � 31C. 4n � 7 � 31 D. 4n � 7 � 31 12.

13. Amar is five years older than his sister. The sum of their ages is 39.Find Amar’s age.A. 17 B. 22 C. 34 D. 29 13.

For Questions 14–18, solve each inequality.

14. �8w � 4 � 12A. {w � w � �1} B. {w � w � �1}C. {w � w � �2} D. {w � w � �2} 14.

15. 2x � 1 � 5 or 7 � x � 1A. {x � 3 � x � 6} B. {x � x � 3 or x 6}C. {x � x � 6} D. � 15.

16. �3 � 2y � 1 � 9

A. �y � ��32� � y � 4� B. all real numbers

C. �y � �2 � y � �92�� D. {y � � 2 � y � 4} 16.

17. � m � 8 � 3A. {m � �11 � m � �5} B. {m � m � �5 or m 5}C. {m � m � �11 or m �5} D. � 17.

18. � 2x � 5 � � 9A. {x � �4 � x � 14} B. {x � �2 � x � 7}C. {x � x � �2 or x � 7} D. all real numbers 18.

19. Identify the graph of the solution set of 9 3 � 2x.A. B.

C. D. 19.

20. A parking garage charges $2 for the first hour and $1 for each additional hour. Fran has $7.50 to spend for parking. What is the greatest number of hours Fran can park?A. 3 B. 5 C. 6 D. 7 20.

Bonus Solve 11 � 7 � x � �5. B:

�1�2 0 1 2 4 5 63�1�2 0 1 2 4 5 63

�1�2 0 1 2 4 5 63�1�2 0 1 2 4 5 63

NAME DATE PERIOD

11

Chapter 1 Test, Form 2A

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Find the value of 5 � 4 3 � 6 � 1.

A. �72� B. �

257� C. 6 D. �

157� 1.

2. Evaluate 2b(4a � c2) if a � 5, b � �32�, and c � 11.

A. �303 B. 423 C. �6 D. ��3023

� 2.

3. Evaluate �� 3c � d � if c � �1 and d � 5.A. 8 B. 2 C. �7 D. �8 3.

4. The formula for the surface area of a sphere is A � 4�r2, where r is the length of the radius. Find the surface area of a sphere with a radius of

14 feet. Use �272� for �.

A. 7248 ft2 B. 7744 ft2 C. 2464 ft2 D. 704 ft2 4.

5. Name the sets of numbers to which ��13� belongs.

A. naturals, rationals B. rational, realsC. integers, rationals D. integers, rationals, reals 5.

6. Simplify �13�(15x � 9) � �

15�(25x � 5).

A. 10x � 2 B. �634�x � �

3125�

C. 5x � 2 D. �15�(40x � 4) 6.

7. Name the property illustrated by 5(x � y) � 5(y � x).A. Commutative Property of MultiplicationB. Distributive PropertyC. Commutative Property of AdditionD. Associative Property of Addition 7.


8. 23 � 5 � �23�m

A. �42 B. �12 C. �27 D. 42 8.

9. 18 � 3 � 4x � 10 �A. {1, �1} B. {1, 4} C. {4, �4} D. {4} 9.

10. 5(2x � 6) � 7x � 3A. �9 B. 9 C. 11 D. � 10.

11. � x � 3 � � 10 � 2A. {�5} B. {�5, 11} C. {11} D. � 11.

11


Chapter 1 Test, Form 2A (continued)

12. Jamie is 4 years younger than her brother. Five years from now, the sum of their ages will be 32. Find Jamie’s present age.A. 9 B. 10 C. 13 D. 14 12.

13. One side of a triangle is four centimeters longer than the shortest side. The third side of the triangle is twice as long as the shortest side. Find the length of the longest side of the triangle if its perimeter is 40 centimeters.A. 9 cm B. 13 cm C. 24 cm D. 18 cm 13.


14. 0.38 �2x

5� 7�

A. {x � x � 4.45} B. {x � x � 98.5} C. {x � x � 13} D. {x � x � 3.69} 14.

15. 9 � 7 � x � �1A. {x � �2 � x � 8} B. �

C. {x � x � �2 or x � 8} D. {x � x � �2} 15.

16. 5x � 4 � 26 or 29 � 3x 2A. {x � 6 � x � 9} B. {x � x � 6 or x 9}C. all real numbers D. {x � x 9} 16.

17. � 2x � 3 � � 7A. {x � x � 5} B. {x � �5 � x � 5}C. {x � �2 � x � 5} D. all real numbers 17.

18. 2�m � 7 � 8A. {m � �11 � m � �3} B. all real numbersC. {m � m � �13 or m �1} D. {m � m � �11 or m �3} 18.

19. Identify the graph of the solution set of �2.3 � 4 � 0.9y.A. B.

C. D. 19.

20. One number is four times a second number. If you take one-half of the second number and increase it by the first number, the result is at least 45.Find the least possible value for the second number.A. 10 B. 9 C. 11 D. 12 20.

Bonus Carlos expects the grade on his next Algebra test to be between 75 and 85. Using g to represent Carlos’ test grade,write an absolute value inequality to describe this situation. B:

�1�2�3�4 0 1 2 430 1 2 4 5 6 7 83

0 1 2 4 5 6 7 83�1�2�3�4�5�6�7 0 1

NAME DATE PERIOD

11

Chapter 1 Test, Form 2B

NAME DATE PERIOD

SCORE


Ass

essm

ent


1. Find the value of 5 � 8 2 � 4 � 11.

A. ��243� B. ��

121� C. �3 D. �2 1.

2. Evaluate (a � y)2 � 2y3 if a � 2 and y � �3.A. �29 B. 43 C. 79 D. �53 2.

3. Evaluate �� a � 3b � if a � �2 and b � 6.A. 20 B. �16 C. �20 D. �36 3.

4. The formula A � �180(n

n� 2)� relates the measure A of an interior angle of

a regular polygon to the number of sides n. If an interior angle measures 120�, find the number of sides.A. 5 B. 6 C. 8 D. 10 4.

5. Name the sets of numbers to which �28 belongs.A. integers B. naturals, integers, realsC. integers, rationals D. integers, rationals, reals 5.

6. Simplify �13�(6x � 3) � 4(3x � 2).

A. �10x � 9 B. �9x � 9 C. �10x � 1 D. �10x � 7 6.

7. Name the property illustrated by 7 (9 � 1) � (9 � 1) 7.A. Distributive PropertyB. Commutative Property of MultiplicationC. Associative Property of MultiplicationD. Commutative Property of Addition 7.


8. �52y�

� �134�

A. �2185�

B. �335� C. �3

35�

D. �1258�

8.

9. 3� x � 5 � � 12A. {9} B. {1} C. {1, 9} D. � 9.

10. 3(5x � 1) � 3x � 3

A. �12� B. 2 C. �2 D. ��

12� 10.

11. � y � 8 � � 6 � 15A. {17} B. {�1} C. {17, �1} D. � 11.

11


Chapter 1 Test, Form 2B (continued)

12. Yoshi is 12 years older than his sister. Six years from now, the sum of their ages will be 32. Find Yoshi’s present age.A. 10 B. 18 C. 4 D. 16 12.

13. Two sides of a triangle are equal in length. The length of the third side is three meters less than the sum of the lengths of the other two sides. Find the length of the longest side of the triangle if its perimeter is 29 meters.

A. 8 m B. 13 m C. �535� m D. 10 m 13.


14. �3(r � 11) � 15 � 9A. {r � r � 13} B. {r � r � 13} C. {r � r � �13} D. {r � r � �13} 14.

15. � 2 � 4z � 10 � 12A. {z � �3 � z � 2} B. {z � �3 � z � 3}

C. �z � �3 � z � �12�� D. �z � ��

12� � z � �

12�� 15.

16. 2x � 5 � 10 or 33 � 4x � 5

A. �x � x � �125� or x � 7� B. �x � 7 � x � �

125��

C. all real numbers D. � 16.

17. 3� m � 4 � 6A. {m � 2 � m � 6} B. {m � m � 2 or m 6}C. {m � m � 1 or m 7} D. all real numbers 17.

18. � 3w � 7 � � 2

A. �w � �53� � w � 3� B. {w � �3 � w � 3}

C. {w � w � 3} D. all real numbers 18.

19. Identify the graph of the solution set of 8.5 6.1 � 0.6y.A. B.

C. D. 19.

20. One number is two less than a second number. If you take one-half of the first number and increase it by the second number, the result is at least 41.Find the least possible value for the second number.

A. 30 B. 28 C. �832� D. 15 20.

Bonus Solve � x � � x 0. B:

�1�2 0 1 2 4 5 63�1�2 0 1 2 4 5 63

�1�2 0 1 2 4 5 63�1�2 0 1 2 4 5 63

NAME DATE PERIOD

11

Chapter 1 Test, Form 2C


1. Find the value of 6 � 82 � 4 � 2. 1.

2. Evaluate �3a2

c�2

2b� if a � 1, b � 2, and c � 3. 2.

For Questions 3 and 4, evaluate each expression if a � 2.5 and b � �8.

3. �� b � 2a � 3.

4. 3� b � 6 � � � a � 4.

5. Use I = prt, the formula for simple interest over t years, 5.to find I when p = $2500, r = 8.5%, and t = 30 months.


6. 1.82 6.

7. �25� 7.

8. �56� 8.

For Questions 9 and 10, name each property illustrated by each equation.

9. ��151��2�

15�� 1 9.

10. �ab � 0 � �ab 10.

11. Simplify �14�(12v � 8) � 2(6v � 1). 11.

12. Write an algebraic expression to represent the verbal expression 12.ten less than the cube of a number.

Solve each equation.

13. 4x � 18 13.

14. 5x � 2 � 3x � 24 14.

15. � 2x � 3 � � 7 15.

16. 4� x � 2 � � 24 16.

NAME DATE PERIOD

SCORE 11

Ass

essm

ent


Chapter 1 Test, Form 2C (continued)


17. The sum of twice a number and 6 is 28. What is the number? 17.

18. Lana ordered concert tickets that cost $7.50 for children 18.and $12.00 for adults. She ordered 8 more children’s tickets than adults’ tickets. Her total bill was $138.How many of each type of ticket did she order?

For Questions 19–24, solve each inequality. Describe the solution set using set builder or interval notation. Then,graph the solution set on a number line.

19. 3t � 5 31 19.

20. 2(x � 3) � 54 20.

21. �5 � 6n � 17 � 13 21.

22. 7v � 6 � �22 or 11 � v � 19 22.

23. � x � 2 � 4 23.

24. � 2x � 3 � � 5 24.

25. Define a variable and write an inequality. Then solve the 25.resulting inequality. The Braves play 162 games in a season.So far, they have won 56 and lost 40. To win at least 60% of all games, how many more games must they win?

Bonus Find the value of k so that the equation below has the B:solution set {�5}.4(x � 3) � x(3 � k)

�1�2�3�4�5�6 0 1 2

�1�2 0 1 2 4 5 63

�1�2�3�4 0 1 2 43

1 2 4 5 6 7 83

222120 23 24 26 27 2825

642 8 10 14 16 1812

NAME DATE PERIOD

11

Chapter 1 Test, Form 2D


1. Find the value of 4 � 62 � 9 � 3. 1.

2. Evaluate �5a3�c

b2� if a � 4, b � 3, and c � 2. 2.

For Questions 3 and 4, evaluate each expression if a � 3.5 and b � �10.

3. �� b � 2a � 3.

4. � �3 � a � � � �b2� � 4.

5. Use I � prt, the formula for simple interest over t years, to 5.find I when p = $2000, r = 6%, and t = 18 months.


6. �16� 6.

7. �2.5 7.

8. �79� 8.

For Questions 9 and 10, name the property illustrated by each equation.

9. 3ab � (�3ab) � 0 9.

10. 1xyz � xyz 10.

11. Simplify �15�(10x � 15) � 4(2x � 5). 11.

12. Write an algebraic expression to represent the verbal expression 12.five times the sum of seven and a number.


13. 5n � 3 � 12 13.

14. 7x � 10 � 4x � 11 14.

15. � 6w � 3 � � 9 15.

16. � x � 4 � � 5 � �2 16.

NAME DATE PERIOD

SCORE 11

Ass

essm

ent


Chapter 1 Test, Form 2D (continued)


17. The sum of 3 times a number and 1 is 25. Find the number. 17.

18. The length of a rectangular garden is 7 feet longer than its 18.width. The perimeter of the garden is 38 feet. Find the width and length of the garden.


19. 10t � 14 � 6 19.

20. 3(4x � 2) � 7x � 19 20.

21. �7 � 9x � 2 � 11 21.

22. 5n � 7 � 2 or 17 � 2n � 11 22.

23. � x � 5 � 3 23.

24. � 2x � 1 � � 9 24.

25. Define a variable and write an inequality. Then solve the 25.resulting inequality. The 25 coins in Danielle’s piggy bank have a value of at least $1.44. The bank contains only nickels and dimes. What is the fewest number of dimes that could be in the bank?

Bonus Find the value of k so that the equation below has the B:solution set {�3}.3(2x � 1) � x(2 � k)

�4�8 0 4 8

6420 8 10

�1�2 0 1 2 4 5 63

�1�2�3 0 1 2 43

�1�2 0 1 2 4 5 63

�1�2 0 1 2 4 5 63

NAME DATE PERIOD

11

Chapter 1 Test, Form 3


1. Find the value of 8 � 2 32 � 6 � 14. 1.

2. Evaluate (n � v)2 � 3v3 if n � 5 and v � �2. 2.

3. Determine whether the statement is sometimes, always, or 3.never true. Explain your reasoning.If a and b are real numbers, then �� a � 2b � is negative.

4. The formula for the volume of a cylinder is V � �r2h, where r 4.is the radius of the base and h is the height of the cylinder.Find the volume of a cylinder with a radius of 1.2 inches and a height of 3 inches. Use 3.14 for π.

5. Name the sets of numbers to which each number belongs.

a. �4 b. �15� c. 0 d. �34� e. 2

6. Simplify �38�(16x � 8) � �

23�(15y � 12). 6.

7. Write a verbal expression to represent the algebraic expression 7.4(n3 � 2n).


8. �6(n � 8) � 4(12 � 5n) � 14n 8.

9. 2� 3x � 5 � � 149.

10. A � �12�h(a � b), for a

10.

11. � y � 8 � � 7 � 3 11.

12. Define a variable, write an equation, and solve the problem. 12.The width of a rectangle is 3 meters more than one-fourth its length. The perimeter is 10 meters more than twice its length.Find the length and width.

NAME DATE PERIOD

SCORE 11

Ass

essm

ent

5. a.

b.

c.

d.

e.


Chapter 1 Test, Form 3 (continued)

13. The formula for the area of a triangle is 13.

A � �12�bh, where b represents the base length,

and h represents the height. The perimeter of the triangle shown is 28 inches. Write an equation for the area A of this triangle in terms of its base length b.


14.14. 2.8 � �

4x5� 3�

15. �3(5y � 4) � 17 15.

16. 5x � 2 � �18 or 2x � 1 � 21 16.

17. �343� � 3w � 9 � 12 17.

18. � x � 3 � � 5 18.

19. � 3w � 7 � � 2 19.

20. Define a variable and write an inequality. Then solve the resulting inequality. Mr. Brooks plans to invest part of $5000 in a stock that pays 8% interest annually. The rest will be invested in a savings account that pays 6% interest annually. Mr. Brooks wants to make at least $350 on the investment for the first year. What is the least amount that should be invested in the stock? 20.

Bonus A jet is flying from Hawaii to San Francisco, a distance B:of 2400 miles. In still air, the jet flies at 600 mph, but there is now a 40-mph tailwind. In case of emergency,how many hours after takeoff will it be faster for the jet to go on to San Francisco rather than to return to Hawaii?

NAME DATE PERIOD

11

b inches

10 inches

Chapter 1 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 solution in more thanone way or investigate beyond the requirements of the problem.

1. a. State the property of real numbers or the property of equalitythat justifies each step in the solution of the equation given.

3x � 5 � 8x Given3x � 5 � (�3x) � 8x � (�3x) ___________________

3x � [(�3x) � 5] � 8x � (�3x) ___________________[3x � (�3x)] � 5 � 8x � (�3x) ___________________

0 � 5 � 8x � (�3x) ___________________5 � 8x � (�3x) ___________________5 � [8 � (�3)]x ___________________5 � 5 x Substitution

�15� 5 � �

15�(5x) ___________________

�15� 5 � ��

15� 5�x ___________________

1 � 1 x ___________________1 � x ___________________x � 1 ___________________

b. Write your own solution of the equation 6(7 � x) � 3 � 9x as youwould write it on a test. Compare your solution to the solutionabove. Did you use all of the same properties as you listed aboveto solve your equation? Explain.

2. Given the inequality � x � 3 � � k, find a value of k, if possible, thatsatisfies each condition. In each case, explain your choice.a. Find a value of k for which the inequality has no solution.b. Find a value of k for which the inequality has exactly one solution.c. Find a value of k for which a solution exists but for which the

solution set does not include 5.

3. a. Write a word problem for the inequality 2 � �14�x � 10.

b. Solve your problem and explain the meaning of your answer.

c. Graph the solution of the inequality 2 � �14�x � 10. Does the graph

have meaning for your word problem? Why or why not?

NAME DATE PERIOD

SCORE 11

Ass

essm

ent


Chapter 1 Vocabulary Test/Review

Choose from the terms above to complete each sentence.

1. The of addition says that adding 0 to anynumber does not change its value.

2. The are the numbers that can be written asratios of two integers, with the integer in the denominator not being 0.

3. The property that allows you to switch the two sides of an equation

is the .

4. 3x � 3x is an example of the .

5. The graph of a compound inequality containing the word and is the

of the graphs of the two separateinequalities.

6. {x � x � 6.3} describes a set by using .

7. The of Multiplication says that you canreverse the order of two factors without changing the value of theirproduct.

8. If 2y � 6 � 3 and 3 � 4y � 21, then 2y � 6 � 4y � 21. This is an

example of the .

9. Two inequalities combined by the word and or the word or form a

10. The of a number is the number of unitsbetween that number and 0 on a number line.

In your own words—Define each term.

11. irrational number

12. Trichotomy Property

absolute valueAddition Propertyalgebraic expressionAssociative PropertyCommutative Propertycompound inequalitycounterexampleDistributive Property

Division Property empty setequationformulaIdentity Propertyintersectioninterval notationInverse Property

irrational numbersMultiplication Propertyopen sentenceorder of operationsrational numbersreal numbersReflexive Propertyset-builder notation

solutionSubstitution PropertySubtraction PropertySymmetric PropertyTransitive PropertyTrichotomy Propertyunionvariable

NAME DATE PERIOD

SCORE 11

Chapter 1 Quiz (Lessons 1–1 and 1–2)

11


1. Find the value of 40 � 62 � 4 3. 1.

2. Evaluate 3n2 � 2an if a � �3 and n � 4. 2.

3. The formula for the perimeter P of a rectangle is 3.P � 2(� � w), where � represents the length, and wrepresents the width of the rectangle. Find the perimeter of a rectangle with a length of 19.2 meters and a width of 4.7 meters.

4. Name the sets of numbers to which �5� belongs. 4.

5. Simplify �13�(6v � 1) � �

34�(8v � 2). 5.

NAME DATE PERIOD

SCORE

Chapter 1 Quiz (Lesson 1–3)

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

1. Standardized Test Practice If 7n � 3 � �43�, what is the

value of 7n � 5?

A. ��53� B. �

130� C. �

133� D. ��2

51�

1.

For Questions 2 and 3, solve each equation. Check your solution.

2. �34� � �

23x� � �

56x� � �

12� 2.

3. 8 � 7w � 3w � 9 3.

4. Solve y � mx � b for x. 4.

5. Define a variable, write an equation, and solve the problem. 5.Carla began a running program to prepare for track team try-outs. On her first day she ran 3 miles, and on her second day she ran 5 miles. Since then, Carla has run 7 miles each day. If her log book shows that Carla has run a total of 99 miles, for how many days has Carla been running 7 miles?

NAME DATE PERIOD

SCORE 11

Ass

essm

ent


1. Evaluate � a � 8b � if a � �3 and b � �14�. 1.

For Questions 2 and 3, solve each equation.

2. �4� 3x � 1 � � �20 2.

3. � 5 � 2x � � x � 5 3.

4. Solve 7 � 3x � 2x � 6, and graph its solution set on a 4.number line.

5. Define a variable and write an inequality. Then solve.The Boston Celtics play an 82-game schedule. If they have won 41 of their first 50 games, how many more games must they win to win at least 70% of all 82 games? 5.

0 1� 15

15

25

35

45

65

Chapter 1 Quiz (Lesson 1–6)

Solve each inequality. Describe the solution set using set builder or interval notation. Then, graph the solution set on a number line.

1. 3x � 5 4 or 9 � 2x 5 1.

2. �6 � 5m � 1 � 39 2.

3. � x � 7 � 4 3.

4. � 2x � 7 � � 5 4.

5. � 4x � 9 � � �2 5.

�1�2�3�4 0 1 2 43

0 1 2 4 5 6 73

�3�11

�1 8

�1 0 1 2 4 5 63

NAME DATE PERIOD

SCORE

Chapter 1 Quiz (Lessons 1–4 and 1–5)

11

NAME DATE PERIOD

SCORE

11

Chapter 1 Mid-Chapter Test (Lessons 1–1 through 1–5)


NAME DATE PERIOD

SCORE 11

Ass

essm

ent

For Questions 1–5, write the letter for the correct answer in the blank at the right of each question.

1. Find the value of (9 � 2)8 � 6 � 2.A. 11 B. 41 C. 22 D. 85 1.

2. Name the sets of numbers to which �7 belongs.A. integers, rationalsB. integers, rationals, realsC. whole numbers, integers, realsD. integers, reals 2.

3. Name the property illustrated by �ab � ab � 0.A. Additive Inverse B. Additive IdentityC. Multiplicative Inverse D. Multiplicative Identity 3.

4. Solve 6(x � 5) � x � 5.A. 2 B. 0 C. 7 D. 5 4.

5. Simplify �12�(8y � 10) � 3(y � 1).

A. y � 8 B. 7y � 2 C. y � 9 D. y � 13 5.

6. Write an algebraic expression to represent the verbal 6.expression the difference of three times a number x and 7.

7. Given the formula C � �5(F �

932)

�, find the value of C if F is 68. 7.

8. Define a variable, write an equation, and solve the problem. 8.Adults’ tickets to a play cost $5 and students’ tickets cost $2.If 295 tickets were sold and a total of $950 was collected,how many students’ tickets were sold?

9. Evaluate m � np2 if m � 0.5, n � �3, and p � �2. 9.

10. Solve h � ��2a

b� for b. 10.

11. Find the value of 17 � [6 � (23 � 1)]. 11.

Part II

Part I


Chapter 1 Cumulative Review (Chapter 1)

1. Simplify ��7�15��

15� 2. Evaluate (�0.7)2.

(Prerequisite Skill) (Prerequisite Skill)

For Questions 3 and 4, find the value of each expression.

3. 4 6 � 3 � 12 4. 19 � [(6 � 24) � 7 22](Lesson 1–1) (Lesson 1–1)

5. Use the formula F � �95�C � 32 to find the value of F if C � 25. 5.

(Lesson 1–1)

6. Name the sets of numbers to which the number 13 belongs. 6.(Lesson 1–2)

7. Simplify �14�(16x � 12) � �

13�(9x � 3). (Lesson 1–2) 7.

8. Write an algebraic expression to represent the verbal 8.expression the square of a number increased by the cube of the same number. (Lesson 1–3)


9. 12x � 51 � 3(x � 7) 10. � 2y � 1 � � 4 � 13(Lesson 1–3) (Lesson 1–4)

11. �5(m � 5) � 3(10 � 2m) � m (Lesson 1–3) 11.

Solve each inequality. Graph the solution set.

12. 4(t � 5) � 5 � t (Lesson 1–5) 12.

13. 3x � 5 � �10 or 12 � x � 20 (Lesson 1–6) 13.

14. � x � 3 � � 4 (Lesson 1–6) 14.


16. Forty-eight decreased by three times a number is thirty-six. 16.Find the number. (Lesson 1–3)

Define a variable and write an inequality. Then solve.

17. The Cincinnati Reds play 162 games in a season. So far 17.they have won 57 games. How many more games must they win in order to win at least 65% of all games for the season?(Lesson 1–5)

�1�2�3�4�5�6�7 0 1

�1�2�3�4 0 1 2 43

0 1 2 4 5 6 7 83

NAME DATE PERIOD

11

9.

10.

1.

2.

3.

4.

Standardized Test Practice (Chapter 1)


1. If 2x � 6 is an even integer, what is the next consecutive even integer?A. 2x � 5 B. 2x � 7 C. 2x � 4 D. 2x � 8 1.

2. 9 is 18% of what number?E. 200 F. 50 G. 1.62 H. 50% 2.

3. Which number is least?

A. �35� B. �1

56�

C. �1499�

D. �59� 3.

4. The radius of a circle is tripled. What happens to the area of the circle?E. area is tripled F. area is multiplied by 6

G. area is multiplied by 9 H. area is multiplied by �13� 4.

5. Which number is not a solution of 2x � 3 � 5?A. 7 B. 2 C. 4 D. 6 5.

6. Which represents a rational number?E. �17� F. �36� G. �50� H. �101� 6.

7. In the figure shown, the length of X�Y� is �13�

of the perimeter of �ABC. What is the length of X�Y�?A. 16 B. 24 C. 96 D. 32 7.

8. Which number is not prime?E. 73 F. 79 G. 91 H. 97 8.

9. If x � 0, which of the following is negative?

A. �x B. x2 C. x3 D. ��1x�

9.

10. If a � b is defined as ba, what is the value of 2 � 3?E. 9 F. 8 G. 6 H. 3 10.

11. 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 11. DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

NAME DATE PERIOD

11

Ass

essm

ent

ZX

Y

40

24

Part 1: Multiple Choice

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


Standardized Test Practice (continued)

12. The average of 8, 6, 9, 12, and 4x is x. 12. 13.What is the value of x?

13. B�C� � C�D�B�C� � A�B� AB � 3BC � 6CD � 5What is the length of the shortest path from A to D?

14. If �130�

� �0x.3�

, what is the value of x? 14. 15.

15. Simplify �19 �19

191�9

19�.

Column A Column B

16. 16.

17. 17.

18. 18.

19. 19. DCBAg3g

2xx � 20

DCBA

(x + 20)˚

2x˚

x˚

DCBA2b � 3�4b

2� 6�

DCBA�0.002�5�(0.05)2

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.

�1�2 0 1 2 3�1�2 0 1 2 3

�1�2 0 1 2 3�1�2 0 1 2 3

NAME DATE PERIOD

11

NAME DATE PERIOD

A B

C D

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

D

C

B

Standardized Test PracticeStudent Record Sheet (Use with pages 52–53 of the Student Edition.)

© Glencoe/McGraw-Hill A1 Glencoe Algebra 2

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

1 4 7 9

2 5 8 10

3 6

Solve the problem and write your answer in the blank.

For Questions 13–18, also enter your answer by writing each number or symbol ina box. Then fill in the corresponding oval for that number or symbol.

11 13 15 17

12

14 16 18

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

19 21 23

20 22 DCBADCBA

DCBADCBADCBA

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

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

DCBADCBA

DCBADCBADCBADCBA

DCBADCBADCBADCBA

NAME DATE PERIOD

11

An

swer

s

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 1-1)

Stu

dy G

uid

e a

nd I

nte

rven

tion

Exp

ress

ion

s an

d F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-1

1-1

©G

lenc

oe/M

cGra

w-H

ill1

Gle

ncoe

Alg

ebra

2

Lesson 1-1

Ord

er o

f O

per

atio

ns

1.S

impl

ify t

he e

xpre

ssio

ns in

side

gro

upin

g sy

mbo

ls.

Ord

er o

f2.

Eva

luat

e al

l pow

ers.

Op

erat

ion

s3.

Do

all m

ultip

licat

ions

and

div

isio

ns f

rom

left

to r

ight

.4.

Do

all a

dditi

ons

and

subt

ract

ions

fro

m le

ft to

rig

ht.

Eva

luat

e [1

8 �

(6 �

4)]

�2.

[18

�(6

�4)

] �

2 �

[18

�10

] �

2�

8 �

2�

4

Eva

luat

e 3x

2�

x(y

�5)

if x

�3

and

y�

0.5.

Rep

lace

eac

h va

riab

le w

ith

the

give

n va

lue.

3x2

�x(

y�

5) �

3 �

(3)2

�3(

0.5

�5)

�3

�(9

) �

3(�

4.5)

�27

�13

.5�

13.5

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

valu

e of

eac

h e

xpre

ssio

n.

1.14

�(6

�2)

172.

11 �

(3 �

2)2

�14

3.2

�(4

�2)

3�

64

4.9(

32�

6)13

55.

(5 �

23)2

�52

144

6.52

��

18 �

234

.25

7.�

68.

(7 �

32)2

�62

409.

20 �

22�

611

10.1

2 �

6 �

3 �

2(4)

611

.14

�(8

�20

�2)

�7

12.6

(7)

�4

�4

�5

38

13.8

(42

�8

�32

)�

240

14.

�24

15.

4

Eva

luat

e ea

ch e

xpre

ssio

n i

f a

�8.

2,b

��

3,c

�4,

and

d�

�.

16.

49.2

17.5

(6c

�8b

�10

d)

215

18.

�6

19.a

c�

bd31

.320

.(b

�c)

2�

4a81

.821

.�

6b�

5c�

54.4

22.3

��

b�

2123

.cd

�4

24.d

(a�

c)�

6.1

25.a

�b

�c

7.45

26.b

�c

�4

�d

�15

27.

�d

8.7

a� b

�c

b � dc � d

a � dc2�

1� b

�d

ab � d

1 � 26 �

9 �

3 �

15�

�8

�2

6 �

4 �

2�

�4

�6

�1

16 �

23�

4�

�1

�22

1 � 4

©G

lenc

oe/M

cGra

w-H

ill2

Gle

ncoe

Alg

ebra

2

Form

ula

sA

for

mu

lais

a m

ath

emat

ical

sen

ten

ce t

hat

use

s va

riab

les

to e

xpre

ss t

he

rela

tion

ship

bet

wee

n c

erta

in q

uan

titi

es.I

f yo

u k

now

th

e va

lue

of e

very

var

iabl

e ex

cept

on

ein

a f

orm

ula

,you

can

use

su

bsti

tuti

on a

nd

the

orde

r of

ope

rati

ons

to f

ind

the

valu

e of

th

eu

nkn

own

var

iabl

e. To

calc

ula

te t

he

nu

mb

er o

f re

ams

of p

aper

nee

ded

to

pri

nt

nco

pie

s

of a

boo

kle

t th

at i

s p

pag

es l

ong,

you

can

use

th

e fo

rmu

la r

�,w

her

e r

is t

he

nu

mb

er o

f re

ams

nee

ded

.How

man

y re

ams

of p

aper

mu

st y

ou b

uy

to p

rin

t 17

2 co

pie

s of

a 2

5-p

age

boo

kle

t?

Su

bsti

tute

n�

172

and

p�

25 i

nto

th

e fo

rmu

la r

�.

r� � �

8.6

You

can

not

bu

y 8.

6 re

ams

of p

aper

.You

wil

l n

eed

to b

uy

9 re

ams

to p

rin

t 17

2 co

pies

.

For

Exe

rcis

es 1

–3,u

se t

he

foll

owin

g in

form

atio

n.

For

a s

cien

ce e

xper

imen

t,S

arah

cou

nts

th

e n

um

ber

of b

reat

hs

nee

ded

for

her

to

blow

up

abe

ach

bal

l.S

he

wil

l th

en f

ind

the

volu

me

of t

he

beac

h b

all

in c

ubi

c ce

nti

met

ers

and

divi

deby

th

e n

um

ber

of b

reat

hs

to f

ind

the

aver

age

volu

me

of a

ir p

er b

reat

h.

1.H

er b

each

bal

l h

as a

rad

ius

of 9

in

ches

.Fir

st s

he

con

vert

s th

e ra

diu

s to

cen

tim

eter

su

sin

g th

e fo

rmu

la C

�2.

54I,

wh

ere

Cis

a l

engt

h i

n c

enti

met

ers

and

Iis

th

e sa

me

len

gth

in i

nch

es.H

ow m

any

cen

tim

eter

s ar

e th

ere

in 9

in

ches

?22

.86

cm

2.T

he

volu

me

of a

sph

ere

is g

iven

by

the

form

ula

V�

�r3

,wh

ere

Vis

th

e vo

lum

e of

th

e

sph

ere

and

ris

its

rad

ius.

Wh

at i

s th

e vo

lum

e of

th

e be

ach

bal

l in

cu

bic

cen

tim

eter

s?(U

se 3

.14

for

�.)

50,0

15 c

m3

3.S

arah

tak

es 4

0 br

eath

s to

blo

w u

p th

e be

ach

bal

l.W

hat

is

the

aver

age

volu

me

of a

ir p

erbr

eath

?ab

ou

t 12

50 c

m3

4.A

per

son

’s b

asal

met

abol

ic r

ate

(or

BM

R)

is t

he

nu

mbe

r of

cal

orie

s n

eede

d to

su

ppor

t h

isor

her

bod

ily

fun

ctio

ns

for

one

day.

Th

e B

MR

of

an 8

0-ye

ar-o

ld m

an i

s gi

ven

by

the

form

ula

BM

R �

12w

�(0

.02)

(6)1

2w,w

her

e w

is t

he

man

’s w

eigh

t in

pou

nds

.Wh

at i

s th

eB

MR

of

an 8

0-ye

ar-o

ld m

an w

ho

wei

ghs

170

pou

nds

?17

95 c

alo

ries

4 � 3

43,0

00�

500

(172

)(25

)�

�50

0

np

� 500

np

� 500

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Exp

ress

ion

s an

d F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-1

1-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Exp

ress

ion

s an

d F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-1

1-1

©G

lenc

oe/M

cGra

w-H

ill3

Gle

ncoe

Alg

ebra

2

Lesson 1-1

Fin

d t

he

valu

e of

eac

h e

xpre

ssio

n.

1.18

�2

�3

272.

9 �

6 �

2 �

113

3.(3

�8)

2 (4)

�3

974.

5 �

3(2

�12

�2)

�7

5.�

[�9

�10

(3)]

�7

6.3

7.(1

68 �

7)32

�43

152

8.[3

(5)

�12

8 �

22]5

�85

Eva

luat

e ea

ch e

xpre

ssio

n i

f r

��

1,s

�3,

t�

12,v

�0,

and

w�

�.

9.6r

�2s

010

.2st

�4r

s84

11.w

(s�

r)�

212

.s�

2r�

16v

1

13.(

4s)2

144

14.s

2 r�

wt

�3

15.2

(3r

�w

) �

716

.4

17.�

w[t

�(t

�r)

]18

.0

19.9

r2�

(s2

�1)

t10

520

.7s

�2v

�22

21.T

EMPE

RA

TUR

ET

he

form

ula

K�

C�

273

give

s th

e te

mpe

ratu

re i

n k

elvi

ns

(K)

for

agi

ven

tem

pera

ture

in

deg

rees

Cel

siu

s.W

hat

is

the

tem

pera

ture

in

kel

vin

s w

hen

th

ete

mpe

ratu

re i

s 55

deg

rees

Cel

siu

s?32

8 K

22.T

EMPE

RA

TUR

ET

he

form

ula

C�

(F�

32)

give

s th

e te

mpe

ratu

re i

n d

egre

es C

elsi

us

for

a gi

ven

tem

pera

ture

in

deg

rees

Fah

ren

hei

t.W

hat

is

the

tem

pera

ture

in

deg

rees

Cel

siu

s w

hen

th

e te

mpe

ratu

re i

s 68

deg

rees

Fah

ren

hei

t?20

�C

5 � 9

2w �r

rv3

� s225 � 2

3v�

t� 5s

�t

1 � 2

6(7

�5)

�� 4

1 � 3

©G

lenc

oe/M

cGra

w-H

ill4

Gle

ncoe

Alg

ebra

2

Fin

d t

he

valu

e of

eac

h e

xpre

ssio

n.

1.3(

4 �

7) �

11�

202.

4(12

�42

)�

16

3.1

�2

�3(

4) �

2�

34.

12 �

[20

�2(

62�

3 �

22)]

88

5.20

�(5

�3)

�52

(3)

856.

(�2)

3�

(3)(

8) �

(5)(

10)

18

7.18

�{5

�[3

4 �

(17

�11

)]}

418.

[4(5

�3)

�2(

4 �

8)]

�16

1

9.[6

�42

]�

510

.[�

5 �

5(�

3)]

�5

11.

3212

.�

(�1)

2�

4(�

9)�

53

Eva

luat

e ea

ch e

xpre

ssio

n i

f a

�,b

��

8,c

��

2,d

�3,

and

e�

.

13.a

b2�

d45

14.(

c�

d)b

�8

15.

�d

212

16.

12

17.(

b�

de)

e2�

118

.ac3

�b2

de

�70

19.�

b[a

�(c

�d

)2]

206

20.

�22

21.9

bc�

141

22.2

ab2

�(d

3�

c)67

23.T

EMPE

RA

TUR

ET

he

form

ula

F�

C�

32 g

ives

th

e te

mpe

ratu

re i

n d

egre

es

Fah

ren

hei

t fo

r a

give

n t

empe

ratu

re i

n d

egre

es C

elsi

us.

Wh

at i

s th

e te

mpe

ratu

re i

nde

gree

s Fa

hre

nh

eit

wh

en t

he

tem

pera

ture

is

�40

deg

rees

Cel

siu

s?�

40�F

24.P

HY

SIC

ST

he

form

ula

h�

120t

�16

t2gi

ves

the

hei

ght

hin

fee

t of

an

obj

ect

tse

con

dsaf

ter

it i

s sh

ot u

pwar

d fr

om E

arth

’s s

urf

ace

wit

h a

n i

nit

ial

velo

city

of

120

feet

per

seco

nd.

Wh

at w

ill

the

hei

ght

of t

he

obje

ct b

e af

ter

6 se

con

ds?

144

ft

25.A

GR

ICU

LTU

RE

Fait

h o

wn

s an

org

anic

app

le o

rch

ard.

Fro

m h

er e

xper

ien

ce t

he

last

few

seas

ons,

she

has

dev

elop

ed t

he

form

ula

P�

20x

�0.

01x2

�24

0 to

pre

dict

her

pro

fit

Pin

doll

ars

this

sea

son

if

her

tre

es p

rodu

ce x

bush

els

of a

pple

s.W

hat

is

Fait

h’s

pre

dict

edpr

ofit

th

is s

easo

n i

f h

er o

rch

ard

prod

uce

s 30

0 bu

shel

s of

app

les?

$486

0

9 � 5

1 � e

c � e2ac

4�dd(b

� c

)�

acab � c

1 � 33 � 4

(�8)

2� 5

�9

�8(

13 �

37)

�� 6

1 � 41 � 2

Pra

ctic

e (

Ave

rag

e)

Exp

ress

ion

s an

d F

orm

ula

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-1

1-1



Readin

g t

o L

earn

Math

em

ati

csE

xpre

ssio

ns

and

Fo

rmu

las

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-1

1-1

©G

lenc

oe/M

cGra

w-H

ill5

Gle

ncoe

Alg

ebra

2

Lesson 1-1

Pre-

Act

ivit

yH

ow a

re f

orm

ula

s u

sed

by

nu

rses

?

Rea

d th

e in

trod

uct

ion

to

Les

son

1-1

at

the

top

of p

age

6 in

you

r te

xtbo

ok.

•N

urs

es u

se t

he

form

ula

F�

to c

ontr

ol t

he

flow

rat

e fo

r IV

s.N

ame

the

quan

tity

th

at e

ach

of

the

vari

able

s in

th

is f

orm

ula

rep

rese

nts

an

d th

eu

nit

s in

wh

ich

eac

h i

s m

easu

red.

Fre

pres

ents

th

e an

d is

mea

sure

d in

pe

r m

inu

te.

Vre

pres

ents

th

e of

sol

uti

on a

nd

is m

easu

red

in

.

dre

pres

ents

th

e an

d is

mea

sure

d in

pe

r m

illi

lite

r.

tre

pres

ents

an

d is

mea

sure

d in

.

•W

rite

th

e ex

pres

sion

th

at a

nu

rse

wou

ld u

se t

o ca

lcu

late

th

e fl

ow r

ate

of a

n I

V i

f a

doct

or o

rder

s 13

50 m

illi

lite

rs o

f IV

sal

ine

to b

e gi

ven

ove

r 8

hou

rs,w

ith

a d

rop

fact

or o

f 20

dro

ps p

er m

illi

lite

r.D

o n

ot f

ind

the

valu

eof

th

is e

xpre

ssio

n.

Rea

din

g t

he

Less

on

1.T

her

e is

a c

ust

omar

y or

der

for

grou

pin

g sy

mbo

ls.B

rack

ets

are

use

d ou

tsid

e of

pare

nth

eses

.Bra

ces

are

use

d ou

tsid

e of

bra

cket

s.Id

enti

fy t

he

inn

erm

ost

expr

essi

on(s

) in

each

of

the

foll

owin

g ex

pres

sion

s.

a.[(

3 �

22)

�8]

�4

(3 �

22)

b.

9 �

[5(8

�6)

�2(

10 �

7)]

(8 �

6) a

nd

(10

�7)

c.{1

4 �

[8 �

(3 �

12)2

]} �

(63

�10

0)(3

�12

)

2.R

ead

the

foll

owin

g in

stru

ctio

ns.

Th

en u

se g

rou

pin

g sy

mbo

ls t

o sh

ow h

ow t

he

inst

ruct

ion

sca

n b

e pu

t in

th

e fo

rm o

f a

mat

hem

atic

al e

xpre

ssio

n.

Mu

ltip

ly t

he

diff

eren

ce o

f 13

an

d 5

by t

he

sum

of

9 an

d 21

.Add

th

e re

sult

to

10.T

hen

divi

de w

hat

you

get

by

2.[(

13 �

5)(9

�21

) �

10]

�2

3.W

hy

is i

t im

port

ant

for

ever

yon

e to

use

th

e sa

me

orde

r of

ope

rati

ons

for

eval

uat

ing

expr

essi

ons?

Sam

ple

an

swer

:If

eve

ryo

ne

did

no

t u

se t

he

sam

e o

rder

of

op

erat

ion

s,d

iffe

ren

t p

eop

le m

igh

t g

et d

iffe

ren

t an

swer

s.

Hel

pin

g Y

ou

Rem

emb

er4.

Th

ink

of a

ph

rase

or

sen

ten

ce t

o h

elp

you

rem

embe

r th

e or

der

of o

pera

tion

s.S

amp

le a

nsw

er:

Ple

ase

excu

se m

y d

ear

Au

nt

Sal

ly.(

par

enth

eses

;ex

po

nen

ts;

mu

ltip

licat

ion

an

d d

ivis

ion

;ad

dit

ion

an

d s

ub

trac

tio

n)

1350

�20

��

8 �

60

min

ute

sti

me

dro

ps

dro

p f

acto

r

mill

ilite

rsvo

lum

e

dro

ps

flo

w r

ateV

�d

�t

©G

lenc

oe/M

cGra

w-H

ill6

Gle

ncoe

Alg

ebra

2

Sig

nif

ican

t D

igit

sA

ll m

easu

rem

ents

are

app

roxi

mat

ion

s.T

he

sign

ific

ant

dig

its

of a

n a

ppro

xim

ate

nu

mbe

r ar

e th

ose

wh

ich

in

dica

te t

he

resu

lts

of a

mea

sure

men

t.F

or e

xam

ple,

the

mas

s of

an

obj

ect,

mea

sure

d to

th

e n

eare

st g

ram

,is

210

gram

s.T

he

mea

sure

men

t21

0 –g

has

3 s

ign

ific

ant

digi

ts.T

he

mas

s of

th

e sa

me

obje

ct,m

easu

red

to t

he

nea

rest

100

g,i

s 20

0 g.

Th

e m

easu

rem

ent

200

g h

as o

ne

sign

ific

ant

digi

t.

1.N

onze

ro d

igit

s an

d ze

ros

betw

een

sig

nif

ican

t di

gits

are

sig

nif

ican

t.F

orex

ampl

e,th

e m

easu

rem

ent

9.07

1 m

has

4 s

ign

ific

ant

digi

ts,9

,0,7

,an

d 1.

2.Z

eros

at

the

end

of a

dec

imal

fra

ctio

n a

re s

ign

ific

ant.

Th

e m

easu

rem

ent

0.05

0 m

m h

as 2

sig

nif

ican

t di

gits

,5 a

nd

0.

3.U

nde

rlin

ed z

eros

in

wh

ole

nu

mbe

rs a

re s

ign

ific

ant.

Th

e m

easu

rem

ent

104,

00 –0 km

has

5 s

ign

ific

ant

digi

ts,1

,0,4

,0,a

nd

0.

In g

ener

al,a

com

puta

tion

in

volv

ing

mu

ltip

lica

tion

or

divi

sion

of

mea

sure

men

tsca

nn

otbe

mor

e ac

cura

te t

han

the

leas

t ac

cura

te m

easu

rem

ent

in t

he c

ompu

tati

on.

Th

us,

the

resu

lt o

f co

mpu

tati

on i

nvo

lvin

g m

ult

ipli

cati

on o

r di

visi

on o

fm

easu

rem

ents

sh

ould

be

rou

nde

d to

th

e n

um

ber

of s

ign

ific

ant

digi

ts i

n t

he

leas

tac

cura

te m

easu

rem

ent.

Th

e m

ass

of 3

7 q

uar

ters

is

210 –

g.F

ind

th

e m

ass

of o

ne

qu

arte

r.

mas

s of

1 q

uar

ter

�21

0 –g

�37

210 –

has

3 si

gnifi

cant

dig

its.

37 d

oes

not

repr

esen

t a

mea

sure

men

t.

�5.

68 g

Rou

nd t

he r

esul

t to

3 s

igni

fican

t di

gits

.

Why

?

Wri

te t

he

nu

mb

er o

f si

gnif

ican

t d

igit

s fo

r ea

ch m

easu

rem

ent.

1.83

14.2

0 m

2.30

.70

cm3.

0.01

mm

4.0.

0605

mg

64

13

5.37

0 –,000

km

6.37

0,00 –0

km7.

9.7

�10

4g

8.3.

20 �

10�

2g

35

23

Sol

ve.R

oun

d e

ach

res

ult

to

the

corr

ect

nu

mb

er o

f si

gnif

ican

t d

igit

s.

9.23

m �

1.54

m10

.12,

00 –0 ft

�52

0ft

11.

2.5

cm �

25

35 m

223

63 c

m

12.1

1.01

mm

�11

13.9

08 y

d �

0.5

14.3

8.6

m �

4.0

m

121.

1 m

m18

20 y

d15

0 m

2

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-1

1-1

Exam

ple

Exam

ple


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

Pro

per

ties

of

Rea

l Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-2

1-2

©G

lenc

oe/M

cGra

w-H

ill7

Gle

ncoe

Alg

ebra

2

Lesson 1-2

Rea

l Nu

mb

ers

All

rea

l n

um

bers

can

be

clas

sifi

ed a

s ei

ther

rat

ion

al o

r ir

rati

onal

.Th

e se

tof

rat

ion

al n

um

bers

in

clu

des

seve

ral

subs

ets:

nat

ura

l n

um

bers

,wh

ole

nu

mbe

rs,a

nd

inte

gers

.

Rre

al n

umbe

rs{a

ll ra

tiona

ls a

nd ir

ratio

nals

}

Qra

tiona

l num

bers

{all

num

bers

tha

t ca

n be

rep

rese

nted

in t

he f

orm

,

whe

re m

and

nar

e in

tege

rs a

nd

nis

not

equ

al t

o 0}

Iirr

atio

nal n

umbe

rs{a

ll no

nter

min

atin

g, n

onre

peat

ing

deci

mal

s}

Nna

tura

l num

bers

{1,

2, 3

, 4,

5,

6, 7

, 8,

9,

…}

Ww

hole

num

bers

{0,

1, 2

, 3,

4,

5, 6

, 7,

8,

…}

Zin

tege

rs{…

, �

3, �

2, �

1, 0

, 1,

2,

3, …

}

Nam

e th

e se

ts o

f n

um

ber

s to

wh

ich

eac

h n

um

ber

bel

ongs

.

a.�

rati

onal

s (Q

),re

als

(R)

b.

�25�

�25�

�5

nat

ura

ls (

N),

wh

oles

(W

),in

tege

rs (

Z),

rati

onal

s (Q

),re

als

(R)

Nam

e th

e se

ts o

f n

um

ber

s to

wh

ich

eac

h n

um

ber

bel

ongs

.

1.Q

,R2.

��

81�Z

,Q,R

3.0

W,Z

,Q,R

4.19

2.00

05Q

,R

5.73

N,W

,Z,Q

,R

6.34

Q,R

7.Q

,R8.

26.1

Q,R

9.�

I,R

10.

N,W

,Z,Q

,R11

.�4.

1�7�Q

,R

12.

N,W

,Z,Q

,R13

.�1

Z,Q

,R14

.�42�

I,R

15.�

11.2

Q,R

16.�

Q,R

17.

I,R

18.3

3.3�

Q,R

19.8

94,0

00N

,W,Z

,Q,R

20.�

0.02

Q,R

�5�

�2

8 � 13

�25�

�5

15 � 3

�36�

�9

1 � 2

6 � 7

11 � 3

m � n

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill8

Gle

ncoe

Alg

ebra

2

Pro

per

ties

of

Rea

l Nu

mb

ers

Rea

l Nu

mb

er P

rop

erti

es

For

any

rea

l num

bers

a,

b, a

nd c

Pro

per

tyA

dd

itio

nM

ult

iplic

atio

n

Com

mut

ativ

ea

�b

�b

�a

a�

b�

b�

a

Ass

ocia

tive

(a�

b) �

c�

a�

(b�

c)(a

�b)

�c

�a

�(b

�c)

Iden

tity

a�

0 �

a�

0 �

aa

�1

�a

�1

�a

Inve

rse

a�

(�a)

�0

�(�

a) �

aIf

ais

not

zer

o, t

hen

a�

�1

��

a.

Dis

trib

utiv

ea(

b�

c) �

ab�

acan

d (b

�c)

a�

ba�

ca

Sim

pli

fy 9

x�

3y�

12y

�0.

9x.

9x�

3y�

12y

�0.

9x�

9x�

(�0.

9x)

�3y

�12

yC

omm

utat

ive

Pro

pert

y (�

)

�(9

�(�

0.9)

)x�

(3 �

12)y

Dis

trib

utiv

e P

rope

rty

�8.

1x�

15y

Sim

plify

.

Sim

pli

fy e

ach

exp

ress

ion

.

1.8(

3a�

b) �

4(2b

�a)

2.40

s�

18t

�5t

�11

s3.

(4j

�2k

�6j

�3k

)

20a

51s

�13

tk

�j

4.10

(6g

�3h

) �

4(5g

�h

)5.

12�

��

6.8(

2.4r

�3.

1s)

�6(

1.5r

�2.

4s)

80g

�26

h4a

� 3

b10

.2r

�39

.2s

7.4(

20 �

4p)

�(4

�16

p)8.

5.5j

�8.

9k�

4.7k

�10

.9j

9.1.

2(7x

�5)

�(1

0 �

4.3x

)

77 �

4p4.

2k�

5.4j

12.7

x�

16

10.9

(7e

�4f

) �

0.6(

e�

5f)

11.2

.5m

(12

�8.

5)12

.p

�r

�r

�p

62.4

e�

39f

8.75

mp

�r

13.4

(10g

�80

h)

�20

(10h

�5g

)14

.2(1

5 �

45c)

�(1

2 �

18c)

140g

�12

0h40

�10

5c

15.(

7 �

2.1x

)3 �

2(3.

5x�

6)16

.(1

8 �

6n�

12 �

3n)

0.7x

�9

20 �

2n

17.1

4(j

�2)

�3j

(4 �

7)18

.50(

3a�

b) �

20(b

�2a

)2j

�7

190a

�70

b

2 � 3

5 � 6

4 � 51 � 4

1 � 23 � 5

1 � 53 � 4

3 � 4

b � 4a � 3

2 � 5

1 � 51 � a1 � a

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

Pro

per

ties

of

Rea

l Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-2

1-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Pro

per

ties

of

Rea

l Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-2

1-2

©G

lenc

oe/M

cGra

w-H

ill9

Gle

ncoe

Alg

ebra

2

Lesson 1-2

Nam

e th

e se

ts o

f n

um

ber

s to

wh

ich

eac

h n

um

ber

bel

ongs

.

1.34

N,W

,Z,Q

,R2.

�52

5Z

,Q,R

3.0.

875

Q,R

4.N

,W,Z

,Q,R

5.�

�9�

Z,Q

,R6.

�30�

I,R

Nam

e th

e p

rop

erty

ill

ust

rate

d b

y ea

ch e

qu

atio

n.

7.3

�x

�x

�3

8.3a

�0

�3a

Co

mm

.(�

)A

dd

.Id

en.

9.2(

r�

w)

�2r

�2w

10.2

r�

(3r

�4r

) �

(2r

�3r

) �

4rD

istr

ibu

tive

Ass

oc.

(�)

11.5

y ��

112

.15x

(1)

�15

x

Mu

lt.I

nv.

Mu

lt.I

den

.

13.0

.6[2

5(0.

5)]

�[0

.6(2

5)]0

.514

.(10

b�

12b)

�7b

�(1

2b�

10b)

�7b

Ass

oc.

(�)

Co

mm

.(�

)

Nam

e th

e ad

dit

ive

inve

rse

and

mu

ltip

lica

tive

in

vers

e fo

r ea

ch n

um

ber

.

15.1

5 �

15,

16.1

.25

�1.

25,0

.8

17.�

,�18

.3�

3,

Sim

pli

fy e

ach

exp

ress

ion

.

19.3

x�

5 �

2x�

35x

�2

20.x

�y

�z

�y

�x

�z

0

21.�

(3g

�3h

) �

5g�

10h

2g�

13h

22.a

2�

a�

4a�

3a2

�1

�2a

2�

3a�

1

23.3

(m�

z) �

5(2m

�z)

13m

�8z

24.2

x�

3y�

(5x

�3y

�2z

)�

3x�

2z

25.6

(2 �

v) �

4(2v

�1)

8 �

2v26

.(1

5d�

3) �

(8 �

10d

)10

d�

31 � 2

1 � 3

4 � 153 � 4

3 � 45 � 4

4 � 54 � 5

1 � 15

1 � 5y

12 � 3

©G

lenc

oe/M

cGra

w-H

ill10

Gle

ncoe

Alg

ebra

2

Nam

e th

e se

ts o

f n

um

ber

s to

wh

ich

eac

h n

um

ber

bel

ongs

.

1.64

252.

�7�

3.2�

4.0

N,W

,Z,Q

,RI,

RI,

RW

,Z,Q

,R

5.��

Q,R

6.�

�16�

Z,Q

,R7.

�35

Z,Q

,R8.

�31

.8Q

,R

Nam

e th

e p

rop

erty

ill

ust

rate

d b

y ea

ch e

qu

atio

n.

9.5x

�(4

y�

3x)

�5x

�(3

x�

4y)

10.7

x�

(9x

�8)

�(7

x�

9x)

�8

Co

mm

.(�

)A

sso

c.(�

)

11.5

(3x

�y)

�5(

3x�

1y)

12.7

n�

2n�

(7 �

2)n

Mu

lt.I

den

.D

istr

ibu

tive

13.3

(2x)

y�

(3 �

2)(x

y)14

.3x

�2y

�3

�2

�x

�y

15.(

6 �

�6)

y�

0y

Ass

oc.

(�)

Co

mm

.(�

)A

dd

.Inv

.

16.

�4y

�1y

17.5

(x�

y) �

5x�

5y18

.4n

�0

�4n

Mu

lt.I

nv.

Dis

trib

uti

veA

dd

.Id

en.

Nam

e th

e ad

dit

ive

inve

rse

and

mu

ltip

lica

tive

in

vers

e fo

r ea

ch n

um

ber

.

19.0

.4�

0.4,

2.5

20.�

1.6

1.6,

�0.

625

21.�

,�22

.5�

5,

Sim

pli

fy e

ach

exp

ress

ion

.

23.5

x�

3y�

2x�

3y3x

24.�

11a

�13

b�

7a�

3b�

4a�

16b

25.8

x�

7y�

(3 �

6y)

8x�

y�

326

.4c

�2c

�(4

c�

2c)

�4c

27.3

(r�

10s)

�4(

7s�

2r)

�5r

�58

s28

.(1

0a�

15)

�(8

�4a

)4a

�1

29.2

(4 �

2x�

y) �

4(5

�x

�y)

30.

�x

�12

y ��

(2x

�12

y)

�12

�8x

�6y

13y

31.T

RA

VEL

Oli

via

driv

es h

er c

ar a

t 60

mil

es p

er h

our

for

th

ours

.Ian

dri

ves

his

car

at

50 m

iles

per

hou

r fo

r (t

�2)

hou

rs.W

rite

a s

impl

ifie

d ex

pres

sion

for

th

e su

m o

f th

edi

stan

ces

trav

eled

by

the

two

cars

.(1

10t

�10

0) m

i

32.N

UM

BER

TH

EORY

Use

th

e pr

oper

ties

of

real

nu

mbe

rs t

o te

ll w

het

her

th

e fo

llow

ing

stat

emen

t is

tru

e or

fal

se:I

f a

b,

it f

ollo

ws

that

a�

�b �

�.Exp

lain

you

r re

ason

ing.

fals

e;co

un

tere

xam

ple

:5�

��4�

�1 � 4

1 � 5

1 � b1 � a

1 � 43 � 5

5 � 6

1 � 21 � 5

6 � 355 � 6

5 � 616 � 11

11 � 1611 � 16

1 � 4

25 � 36

Pra

ctic

e (

Ave

rag

e)

Pro

per

ties

of

Rea

l Nu

mb

ers

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-2

1-2


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csP

rop

erti

es o

f R

eal N

um

ber

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-2

1-2

©G

lenc

oe/M

cGra

w-H

ill11

Gle

ncoe

Alg

ebra

2

Lesson 1-2

Pre-

Act

ivit

yH

ow i

s th

e D

istr

ibu

tive

Pro

per

ty u

sefu

l in

cal

cula

tin

g st

ore

savi

ngs

?

Rea

d th

e in

trod

uct

ion

to

Les

son

1-2

at

the

top

of p

age

11 i

n y

our

text

book

.

•W

hy

are

all

of t

he

amou

nts

lis

ted

on t

he

regi

ster

sli

p at

th

e to

p of

pag

e11

fol

low

ed b

y n

egat

ive

sign

s?S

amp

le a

nsw

er:T

he

amo

un

t o

fea

ch c

ou

po

n is

su

btr

acte

d f

rom

th

e to

tal a

mo

un

t o

fp

urc

has

es s

o t

hat

yo

u s

ave

mo

ney

by

usi

ng

co

up

on

s.•

Des

crib

e tw

o w

ays

of c

alcu

lati

ng

the

amou

nt

of m

oney

you

sav

ed b

yu

sin

g co

upo

ns

if y

our

regi

ster

sli

p is

th

e on

e sh

own

on

pag

e 11

.S

amp

le a

nsw

er:

Ad

d a

ll th

e in

div

idu

al c

ou

po

n a

mo

un

ts o

rad

d t

he

amo

un

ts f

or

the

scan

ned

co

up

on

s an

d m

ult

iply

th

esu

m b

y 2.

Rea

din

g t

he

Less

on

1.R

efer

to

the

Key

Con

cept

s bo

x on

pag

e 11

.Th

e n

um

bers

2.5�

7�an

d 0.

0100

1000

1… b

oth

invo

lve

deci

mal

s th

at “

go o

n f

orev

er.”

Exp

lain

wh

y on

e of

th

ese

nu

mbe

rs i

s ra

tion

al a

nd

the

oth

er i

s ir

rati

onal

.S

amp

le a

nsw

er:

2.5�7�

�2.

5757

… is

a r

epea

tin

gd

ecim

al b

ecau

se t

her

e is

a b

lock

of

dig

its,

57,t

hat

rep

eats

fo

reve

r,so

this

nu

mb

er is

rat

ion

al.T

he

nu

mb

er 0

.010

0100

01…

is a

no

n-r

epea

tin

gd

ecim

al b

ecau

se,a

lth

ou

gh

th

e d

igit

s fo

llow

a p

atte

rn,t

her

e is

no

blo

cko

f d

igit

s th

at r

epea

ts.S

o t

his

nu

mb

er is

an

irra

tio

nal

nu

mb

er.

2.W

rite

th

e A

ssoc

iati

ve P

rope

rty

of A

ddit

ion

in

sym

bols

.Th

en i

llu

stra

te t

his

pro

pert

y by

fin

din

g th

e su

m 1

2 �

18 �

45 i

n t

wo

diff

eren

t w

ays.

(a�

b)

�c

�a

�(b

�c)

;S

amp

le a

nsw

er:

(12

�18

) �

45 �

30 �

45 �

75;

12 �

(18

�45

) �

12 �

63 �

75

3.C

onsi

der

the

equ

atio

ns

(a�

b) �

c�

a�

(b�

c) a

nd

(a�

b) �

c�

c�

(a�

b).O

ne

of t

he

equ

atio

ns

use

s th

e A

ssoc

iati

ve P

rope

rty

of M

ult

ipli

cati

on a

nd

one

use

s th

e C

omm

uta

tive

Pro

pert

y of

Mu

ltip

lica

tion

.How

can

you

tel

l w

hic

h p

rope

rty

is b

ein

g u

sed

in e

ach

equ

atio

n?

Th

e fi

rst

equ

atio

n u

ses

the

Ass

oci

ativ

e P

rop

erty

of

Mu

ltip

licat

ion

.Th

e q

uan

titi

es a

,b,a

nd

car

e u

sed

in t

he

sam

e o

rder

,bu

tth

ey a

re g

rou

ped

dif

fere

ntl

y o

n t

he

two

sid

es o

f th

e eq

uat

ion

.Th

e se

con

deq

uat

ion

use

s th

e q

uan

titi

es in

dif

fere

nt

ord

ers

on

th

e tw

o s

ides

of

the

equ

atio

n.S

o t

he

seco

nd

eq

uat

ion

use

s th

e C

om

mu

tati

ve P

rop

erty

of

Mu

ltip

licat

ion

.

Hel

pin

g Y

ou

Rem

emb

er4.

How

can

th

e m

ean

ings

of

the

wor

ds c

omm

ute

ran

d as

soci

atio

nh

elp

you

to

rem

embe

r th

edi

ffer

ence

bet

wee

n t

he

com

mu

tati

ve a

nd

asso

ciat

ive

prop

erti

es?

Sam

ple

an

swer

:A

co

mm

ute

r is

so

meo

ne

wh

o t

rave

ls b

ack

and

fo

rth

to

wo

rk o

r an

oth

erp

lace

,an

d t

he

com

mu

tati

ve p

rop

erty

say

s yo

u c

an s

wit

ch t

he

ord

erw

hen

two

nu

mb

ers

that

are

bei

ng

ad

ded

or

mu

ltip

lied

.An

ass

oci

atio

n is

ag

rou

p o

f p

eop

le w

ho

are

co

nn

ecte

d o

r u

nit

ed,a

nd

th

e as

soci

ativ

ep

rop

erty

say

s th

at y

ou

can

sw

itch

th

e g

rou

pin

gw

hen

th

ree

nu

mb

ers

are

add

ed o

r m

ult

iplie

d.

©G

lenc

oe/M

cGra

w-H

ill12

Gle

ncoe

Alg

ebra

2

Pro

per

ties

of

a G

rou

pA

set

of

nu

mbe

rs f

orm

s a

grou

p w

ith

res

pect

to

an o

pera

tion

if

for

that

ope

rati

onth

e se

t h

as (

1) t

he

Clo

sure

Pro

pert

y,(2

) th

e A

ssoc

iati

ve P

rope

rty,

(3)

a m

embe

rw

hich

is a

n id

enti

ty,a

nd (

4) a

n in

vers

e fo

r ea

ch m

embe

r of

the

set

.

Doe

s th

e se

t {0

,1,2

,3,…

} fo

rm a

gro

up

wit

h r

esp

ect

to a

dd

itio

n?

Clo

sure

Pro

per

ty:

For

all

nu

mbe

rs i

n t

he

set,

is a

�b

in t

he

set?

0 �

1 �

1,an

d 1

isin

th

e se

t;0

�2

�2,

and

2 is

in

th

e se

t;an

d so

on

.Th

e se

t h

ascl

osu

re f

or a

ddit

ion

.

Ass

ocia

tive

Pro

per

ty:

For

all

nu

mbe

rs i

n t

he

set,

does

a�

(b�

c) �

(a�

b) �

c?

0 �

(1 �

2) �

(0 �

1) �

2;1

�(2

�3)

�(1

�2)

�3;

and

so o

n.

Th

e se

t is

ass

ocia

tive

for

add

itio

n.

Iden

tity

:Is

th

ere

som

e n

um

ber,

i,in

th

e se

t su

ch t

hat

i�

a�

a�

a�

ifo

r al

la?

0 �

1 �

1 �

1 �

0;0

�2

�2

�2

�0;

and

so o

n.

Th

e id

enti

ty f

or a

ddit

ion

is

0.

Inve

rse:

Doe

s ea

ch n

um

ber,

a,h

ave

an i

nve

rse,

a ,s

uch

th

at

a �

a�

a�

a�

i? T

he

inte

ger

inve

rse

of 3

is

�3

sin

ce

�3

�3

�0,

and

0 is

th

e id

enti

ty f

or a

ddit

ion

.Bu

t th

e se

t do

es n

otco

nta

in �

3.T

her

efor

e,th

ere

is n

o in

vers

e fo

r 3.

The

set

is

not

a gr

oup

wit

h re

spec

t to

add

itio

n be

caus

e on

ly t

hree

of

the

four

pro

pert

ies

hold

.

Is t

he

set

{�1,

1} a

gro

up

wit

h r

esp

ect

to m

ult

ipli

cati

on?

Clo

sure

Pro

per

ty:

(�1)

(�1)

�1;

(�1)

(1)

��

1;(1

)(�

1) �

�1;

(1)(

1) �

1T

he

set

has

clo

sure

for

mu

ltip

lica

tion

.

Ass

ocia

tive

Pro

per

ty:

(�1)

[(�

1)(�

1)]

�(�

1)(1

) �

�1;

and

so o

nT

he

set

is a

ssoc

iati

ve f

or m

ult

ipli

cati

on.

Iden

tity

:1(

�1)

��

1;1(

1) �

1T

he

iden

tity

for

mu

ltip

lica

tion

is

1.

Inve

rse:

�1

is t

he

inve

rse

of �

1 si

nce

(�

1)(�

1) �

1,an

d 1

is t

he

iden

tity

.1

is t

he

inve

rse

of 1

sin

ce (

1)(1

) �

1,an

d 1

is t

he

iden

tity

.E

ach

mem

ber

has

an

in

vers

e.

Th

e se

t {�

1,1}

is

a gr

oup

wit

h r

espe

ct t

o m

ult

ipli

cati

on b

ecau

se a

ll f

our

prop

erti

es h

old.

Tel

l w

het

her

th

e se

t fo

rms

a gr

oup

wit

h r

esp

ect

to t

he

give

n o

per

atio

n.

1.{i

nte

gers

},ad

diti

onye

s2.

{in

tege

rs},

mu

ltip

lica

tion

no

3.��1 2� ,

�2 2� ,�3 2� ,

…�,a

ddit

ion

no

4.{m

ult

iple

s of

5},

mu

ltip

lica

tion

no

5.{x

,x2 ,

x3,x

4 ,…

} ad

diti

onn

o6.

{�1�,

�2�,

�3�,

…},

mu

ltip

lica

tion

no

7.{i

rrat

ion

al n

um

bers

},ad

diti

onn

o8.

{rat

ion

al n

um

bers

},ad

diti

onye

s

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-2

1-2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2



Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-3

1-3

©G

lenc

oe/M

cGra

w-H

ill13

Gle

ncoe

Alg

ebra

2

Lesson 1-3

Ver

bal

Exp

ress

ion

s to

Alg

ebra

ic E

xpre

ssio

ns

Th

e ch

art

sugg

ests

som

e w

ays

toh

elp

you

tra

nsl

ate

wor

d ex

pres

sion

s in

to a

lgeb

raic

exp

ress

ion

s.A

ny

lett

er c

an b

e u

sed

tore

pres

ent

a n

um

ber

that

is

not

kn

own

.

Wo

rd E

xpre

ssio

nO

per

atio

n

and,

plu

s, s

um,

incr

ease

d by

, m

ore

than

addi

tion

min

us,

diffe

renc

e, d

ecre

ased

by,

less

tha

nsu

btra

ctio

n

times

, pr

oduc

t, of

(as

in

of a

num

ber)

mul

tiplic

atio

n

divi

ded

by,

quot

ient

divi

sion

1 � 2

Wri

te a

n a

lgeb

raic

exp

ress

ion

to

rep

rese

nt

18 l

ess

than

the

qu

otie

nt

of a

nu

mb

er a

nd

3.

�18

n � 3

Wri

te a

ver

bal

sen

ten

ce t

ore

pre

sen

t 6(

n�

2) �

14.

Six

tim

es t

he

diff

eren

ce o

f a

nu

mbe

r an

d tw

ois

equ

al t

o 14

.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Wri

te a

n a

lgeb

raic

exp

ress

ion

to

rep

rese

nt

each

ver

bal

exp

ress

ion

.

1.th

e su

m o

f si

x ti

mes

a n

um

ber

and

256n

�25

2.fo

ur

tim

es t

he

sum

of

a n

um

ber

and

34(

n�

3)

3.7

less

th

an f

ifte

en t

imes

a n

um

ber

15n

� 7

4.th

e di

ffer

ence

of

nin

e ti

mes

a n

um

ber

and

the

quot

ien

t of

6 a

nd

the

sam

e n

um

ber9n

�

5.th

e su

m o

f 10

0 an

d fo

ur

tim

es a

nu

mbe

r10

0 �

4n

6.th

e pr

odu

ct o

f 3

and

the

sum

of

11 a

nd

a n

um

ber

3(11

�n

)

7.fo

ur

tim

es t

he

squ

are

of a

nu

mbe

r in

crea

sed

by f

ive

tim

es t

he

sam

e n

um

ber

4n2

�5n

8.23

mor

e th

an t

he

prod

uct

of

7 an

d a

nu

mbe

r7n

�23

Wri

te a

ver

bal

sen

ten

ce t

o re

pre

sen

t ea

ch e

qu

atio

n.

Sam

ple

an

swer

s ar

e g

iven

.

9.3n

�35

�79

Th

e d

iffe

ren

ce o

f th

ree

tim

es a

nu

mb

er a

nd

35

is e

qu

al t

o 7

9.

10.2

(n3

�3n

2 ) �

4nTw

ice

the

sum

of

the

cub

e o

f a

nu

mb

er a

nd

th

ree

tim

es t

he

squ

are

of

the

nu

mb

er is

eq

ual

to

fo

ur

tim

es t

he

nu

mb

er.

11.�

n5 �n

3�

�n

�8

Th

e q

uo

tien

t o

f fi

ve t

imes

a n

um

ber

an

d t

he

sum

of

the

nu

mb

eran

d 3

is e

qu

al t

o t

he

dif

fere

nce

of

the

nu

mb

er a

nd

8.6 � n

©G

lenc

oe/M

cGra

w-H

ill14

Gle

ncoe

Alg

ebra

2

Pro

per

ties

of

Equ

alit

yYo

u c

an s

olve

equ

atio

ns

by u

sin

g ad

diti

on,s

ubt

ract

ion

,m

ult

ipli

cati

on,o

r di

visi

on.

Ad

dit

ion

an

d S

ub

trac

tio

nF

or a

ny r

eal n

umbe

rs a

, b,

and

c,

if a

�b,

Pro

per

ties

of

Eq

ual

ity

then

a�

c�

b�

can

d a

�c

�b

�c.

Mu

ltip

licat

ion

an

d D

ivis

ion

For

any

rea

l num

bers

a,

b, a

nd c

,if

a�

b,

Pro

per

ties

of

Eq

ual

ity

then

a�

c�

b�

can

d, if

cis

not

zer

o,

�.

b � ca � c

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

So

lvin

g E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-3

1-3

Sol

ve 1

00 �

8x�

140.

100

�8x

�14

010

0 �

8x�

100

�14

0 �

100

�8x

�40

x�

�5

Sol

ve 4

x�

5y�

100

for

y.

4x�

5y�

100

4x�

5y�

4x�

100

�4x

5y�

100

�4x

y�

(100

�4x

)

y�

20 �

x4 � 5

1 � 5

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sol

ve e

ach

eq

uat

ion

.Ch

eck

you

r so

luti

on.

1.3s

�45

152.

17 �

9 �

a�

83.

5t�

1 �

6t�

54

4.m

�5.

7 �

x�

38

6.�

8 �

�2(

z�

7)�

3

7.0.

2b�

1050

8.3x

�17

�5x

�13

159.

5(4

�k)

��

10k

�4

10.1

20 �

y�

6080

11.

n�

98 �

n28

12.4

.5 �

2p�

8.7

2.1

13.4

n�

20 �

53 �

2n5

14.1

00 �

20 �

5r�

1615

.2x

�75

�10

2 �

x9

Sol

ve e

ach

eq

uat

ion

or

form

ula

for

th

e sp

ecif

ied

var

iab

le.

16.a

�3b

�c,

for

bb

�17

.�

10,f

or t

t�

18.h

�12

g�

1,fo

r g

g�

19.

�12

,for

pp

�

20.2

xy�

x�

7,fo

r x

x�

21.

��

6,fo

r f

f�

24 �

2d

22.3

(2j

�k)

�10

8,fo

r j

j�18

�23

.3.5

s�

42 �

14t,

for

ss

�4t

�12

24.

�5m

�20

,for

mm

�25

.4x

�3y

�10

,for

yy

�x

�10 � 3

4 � 320

n� 5n

�1

m � n

k � 2

f � 4d � 2

7� 2y

�1

4r � q3p

q�

rh

�1

�12

s � 20s � 2t

a�

c�

3

1 � 2

5 � 23 � 4

1 � 23 � 4

1 � 22 � 3


An

swer

s


Skil

ls P

ract

ice

So

lvin

g E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-3

1-3

©G

lenc

oe/M

cGra

w-H

ill15

Gle

ncoe

Alg

ebra

2

Lesson 1-3

Wri

te a

n a

lgeb

raic

exp

ress

ion

to

rep

rese

nt

each

ver

bal

exp

ress

ion

.

1.4

tim

es a

nu

mbe

r,in

crea

sed

by 7

2.8

less

th

an 5

tim

es a

nu

mbe

r

4n�

75n

�8

3.6

tim

es t

he

sum

of

a n

um

ber

and

54.

the

prod

uct

of 3

and

a n

umbe

r,di

vide

d by

9

6(n

�5)

5.3

tim

es t

he

diff

eren

ce o

f 4

and

a n

um

ber

3(4

�n

)

6.th

e pr

odu

ct o

f �

11 a

nd

the

squ

are

of a

nu

mbe

r�

11n

2

Wri

te a

ver

bal

exp

ress

ion

to

rep

rese

nt

each

eq

uat

ion

.7–

10.S

amp

le a

nsw

ers

7.n

�8

�16

8.8

�3x

�5

are

giv

en.

Th

e d

iffe

ren

ce o

f a

nu

mb

er

Th

e su

m o

f 8

and

3 t

imes

a

and

8 is

16.

nu

mb

er is

5.

9.b2

�3

�b

10.

�2

�2y

Th

ree

add

ed t

o t

he

squ

are

of

A n

um

ber

div

ided

by

3 is

th

ea

nu

mb

er is

th

e n

um

ber

.d

iffe

ren

ce o

f 2

and

tw

ice

the

nu

mb

er.

Nam

e th

e p

rop

erty

ill

ust

rate

d b

y ea

ch s

tate

men

t.

11.I

f a

�0.

5b,a

nd

0.5b

�10

,th

en a

�10

.12

.If

d�

1 �

f,th

en d

�f

�1.

Tran

siti

ve (

�)

Su

btr

acti

on

(�

)

13.I

f �

7x�

14,t

hen

14

��

7x.

14.I

f (8

�7)

r�

30,t

hen

15r

�30

.S

ymm

etri

c (�

)S

ub

stit

uti

on

(�

)

Sol

ve e

ach

eq

uat

ion

.Ch

eck

you

r so

luti

on.

15.4

m�

2 �

184

16.x

�4

�5x

�2

17.3

t�

2t�

55

18.�

3b�

7 �

�15

�2b

19.�

5x�

3x�

243

20.4

v�

20 �

6 �

345

21.a

��

35

22.2

.2n

�0.

8n�

5 �

4n5

Sol

ve e

ach

eq

uat

ion

or

form

ula

for

th

e sp

ecif

ied

var

iab

le.

23.I

�pr

t,fo

r p

p�

24.y

�x

�12

,for

xx

�4y

�48

25.A

�,f

or y

y�

2A�

x26

.A�

2�r2

�2�

rh,f

or h

h�

A�

2r2

��

2r

x�

y�

2

1 � 4I � rt

2a � 5

22 � 5

1 � 2

y � 33n � 9

©G

lenc

oe/M

cGra

w-H

ill16

Gle

ncoe

Alg

ebra

2

Wri

te a

n a

lgeb

raic

exp

ress

ion

to

rep

rese

nt

each

ver

bal

exp

ress

ion

.

1.2

mor

e th

an t

he q

uoti

ent

of a

num

ber

and

52.

the

sum

of

two

con

secu

tive

in

tege

rs

�2

n�

(n�

1)

3.5

tim

es t

he

sum

of

a n

um

ber

and

14.

1 le

ss t

han

tw

ice

the

squ

are

of a

nu

mbe

r5(

m�

1)2y

2�

1

Wri

te a

ver

bal

exp

ress

ion

to

rep

rese

nt

each

eq

uat

ion

.5–

8.S

amp

le a

nsw

ers

5.5

�2x

�4

6.3y

�4y

3ar

e g

iven

.

Th

e d

iffe

ren

ce o

f 5

and

tw

ice

a T

hre

e ti

mes

a n

um

ber

is 4

tim

es

nu

mb

er is

4.

the

cub

e o

f th

e n

um

ber

.

7.3c

�2(

c�

1)8.

�3(

2m�

1)T

he

qu

oti

ent

Th

ree

tim

es a

nu

mb

er is

tw

ice

the

of

a n

um

ber

an

d 5

is 3

tim

es t

he

dif

fere

nce

of

the

nu

mb

er a

nd

1.

sum

of

twic

e th

e n

um

ber

an

d 1

.

Nam

e th

e p

rop

erty

ill

ust

rate

d b

y ea

ch s

tate

men

t.

9.If

t�

13 �

52,t

hen

52

�t

�13

.10

.If

8(2q

�1)

�4,

then

2(2

q�

1) �

1.S

ymm

etri

c (�

)D

ivis

ion

(�

)

11.I

f h

�12

�22

,th

en h

�10

.12

.If

4m�

�15

,th

en �

12m

�45

.S

ub

trac

tio

n (

�)

Mu

ltip

licat

ion

(�

)

Sol

ve e

ach

eq

uat

ion

.Ch

eck

you

r so

luti

on.

13.1

4 �

8 �

6r�

114

.9 �

4n�

�59

�17

15.

�n

�16

.s

��

17.�

1.6r

�5

��

7.8

818

.6x

�5

�7

�9x

19.5

(6 �

4v)

�v

�21

20.6

y�

5 �

�3(

2y�

1)

Sol

ve e

ach

eq

uat

ion

or

form

ula

for

th

e sp

ecif

ied

var

iab

le.

21.E

�m

c2,f

or m

m�

22.c

�,f

or d

d�

23.h

�vt

�gt

2 ,fo

r v

v�

24.E

�Iw

2�

U,f

or I

I�

Def

ine

a va

riab

le,w

rite

an

eq

uat

ion

,an

d s

olve

th

e p

rob

lem

.

25.G

EOM

ETRY

Th

e le

ngt

h o

f a

rect

angl

e is

tw

ice

the

wid

th.F

ind

the

wid

th i

f th

epe

rim

eter

is

60 c

enti

met

ers.

w�

wid

th;

2(2w

) �

2w�

60;

10 c

m

26.G

OLF

Lui

s an

d th

ree

frie

nds

wen

t go

lfin

g.T

wo

of t

he f

rien

ds r

ente

d cl

ubs

for

$6 e

ach.

The

tota

l cos

t of

the

ren

ted

club

s an

d th

e gr

een

fees

for

eac

h pe

rson

was

$76

.Wha

t w

as t

he c

ost

of t

he g

reen

fee

s fo

r ea

ch p

erso

n?g

�g

reen

fee

s p

er p

erso

n;6

(2)

�4g

�76

;$16

2(E

�U

)�

�w

21 � 2

h�

gt2

�t

3c�

1�

22d

�1

�3

E � c2

1 � 63 � 7

4 � 5

1 � 511 � 12

3 � 45 � 6

1 � 45 � 8

1 � 23 � 4

m � 5

y � 5

Pra

ctic

e (

Ave

rag

e)

So

lvin

g E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-3

1-3



Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-3

1-3

©G

lenc

oe/M

cGra

w-H

ill17

Gle

ncoe

Alg

ebra

2

Lesson 1-3

Pre-

Act

ivit

yH

ow c

an y

ou f

ind

th

e m

ost

effe

ctiv

e le

vel

of i

nte

nsi

ty f

or y

our

wor

kou

t?

Rea

d th

e in

trod

uct

ion

to

Les

son

1-3

at

the

top

of p

age

20 i

n y

our

text

book

.

•T

o fi

nd

you

r ta

rget

hea

rt r

ate,

wh

at t

wo

piec

es o

f in

form

atio

n m

ust

you

supp

ly?

age

(A)

and

des

ired

inte

nsi

ty le

vel (

I)•

Wri

te a

n e

quat

ion

th

at s

how

s h

ow t

o ca

lcu

late

you

r ta

rget

hea

rt r

ate.

P�

or

P�

(220

�A

)

I�6

Rea

din

g t

he

Less

on

1.a.

How

are

alg

ebra

ic e

xpre

ssio

ns

and

equ

atio

ns

alik

e?S

amp

le a

nsw

er:

Bo

th c

on

tain

var

iab

les,

con

stan

ts,a

nd

op

erat

ion

sig

ns.

b.

How

are

alg

ebra

ic e

xpre

ssio

ns

and

equ

atio

ns

diff

eren

t?S

amp

le a

nsw

er:

Eq

uat

ion

s co

nta

in e

qu

al s

ign

s;ex

pre

ssio

ns

do

no

t.

c.H

ow a

re a

lgeb

raic

exp

ress

ion

s an

d eq

uat

ion

s re

late

d?S

amp

le a

nsw

er:

An

eq

uat

ion

is a

sta

tem

ent

that

say

s th

at t

wo

alg

ebra

ic e

xpre

ssio

ns

are

equ

al.

Rea

d t

he

foll

owin

g p

rob

lem

an

d t

hen

wri

te a

n e

qu

atio

n t

hat

you

cou

ld u

se t

oso

lve

it.D

o n

ot a

ctu

ally

sol

ve t

he

equ

atio

n.I

n y

our

equ

atio

n,l

et m

be

the

nu

mb

erof

mil

es d

rive

n.

2.W

hen

Lou

isa

ren

ted

a m

ovin

g tr

uck

,sh

e ag

reed

to

pay

$28

per

day

plu

s $0

.42

per

mil

e.If

sh

e ke

pt t

he

tru

ck f

or 3

day

s an

d th

e re

nta

l ch

arge

s (w

ith

out

tax)

wer

e $1

53.7

2,h

owm

any

mil

es d

id L

ouis

a dr

ive

the

tru

ck?

3(28

) �

0.42

m�

153.

72

Hel

pin

g Y

ou

Rem

emb

er

3.H

ow c

an t

he

wor

ds r

efle

ctio

nan

d sy

mm

etry

hel

p yo

u r

emem

ber

and

dist

ingu

ish

bet

wee

nth

e re

flex

ive

and

sym

met

ric

prop

erti

es o

f eq

ual

ity?

Th

ink

abou

t h

ow t

hes

e w

ords

are

use

d in

eve

ryda

y li

fe o

r in

geo

met

ry.

Sam

ple

an

swer

:Wh

en y

ou

loo

k at

yo

ur

refl

ecti

on

,yo

u a

re lo

oki

ng

at

you

rsel

f.T

he

refl

exiv

e p

rop

erty

say

s th

at e

very

nu

mb

er is

eq

ual

to

itse

lf.

In g

eom

etry

,sym

met

ry w

ith

res

pec

t to

a li

ne

mea

ns

that

th

e p

arts

of

afi

gu

re o

n t

he

two

sid

es o

f a

line

are

iden

tica

l.T

he

sym

met

ric

pro

per

ty o

feq

ual

ity

allo

ws

you

to

inte

rch

ang

e th

e tw

o s

ides

of

an e

qu

atio

n.T

he

equ

al s

ign

is li

ke t

he

line

of

sym

met

ry.

(220

�A

)

I�

� 6

©G

lenc

oe/M

cGra

w-H

ill18

Gle

ncoe

Alg

ebra

2

Ven

n D

iag

ram

sR

elat

ion

ship

s am

ong

sets

can

be

show

n u

sin

g V

enn

dia

gram

s.S

tudy

th

edi

agra

ms

belo

w.T

he

circ

les

repr

esen

t se

ts A

and

B,w

hic

h a

re s

ubs

ets

of s

et S

.

Th

e u

nio

n o

f A

and

Bco

nsi

sts

of a

ll e

lem

ents

in

eit

her

Aor

B.

Th

e in

ters

ecti

on o

f A

and

Bco

nsi

sts

of a

ll e

lem

ents

in

bot

h A

and

B.

Th

e co

mpl

emen

t of

Aco

nsi

sts

of a

ll e

lem

ents

not

in A

.

You

can

com

bin

e th

e op

erat

ion

s of

un

ion

,in

ters

ecti

on,a

nd

fin

din

g th

e co

mpl

emen

t.

Sh

ade

the

regi

on (

A∩

B)�

.

(A �

B)

mea

ns

the

com

plem

ent

of t

he

inte

rsec

tion

of

Aan

d B

.F

irst

fin

d th

e in

ters

ecti

on o

f A

and

B.T

hen

fin

d it

s co

mpl

emen

t.

Dra

w a

Ven

n d

iagr

am a

nd

sh

ade

the

regi

on i

nd

icat

ed.

See

stu

den

ts’d

iag

ram

s.

1.A

� B

2.A

� B

3.A

� B

4.

A�

B

5.(A

� B

)6.

A �

B

Dra

w a

Ven

n d

iagr

am a

nd

th

ree

over

lap

pin

g ci

rcle

s.T

hen

sh

ade

the

regi

on i

nd

icat

ed.

See

stu

den

ts’d

iag

ram

s.

7.(A

� B

) �

C

8.(A

� B

)�

C

9.A

� (

B�

C)

10.(

A �

B)

�C

11.I

s th

e u

nio

n o

pera

tion

ass

ocia

tive

?ye

s

12.I

s th

e in

ters

ecti

on o

pera

tion

ass

ocia

tive

?ye

s

AB

S

AB

S

AB

S

A

S

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-3

1-3

Exam

ple

Exam

ple


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g A

bso

lute

Val

ue

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-4

1-4

©G

lenc

oe/M

cGra

w-H

ill19

Gle

ncoe

Alg

ebra

2

Lesson 1-4

Ab

solu

te V

alu

e Ex

pre

ssio

ns

Th

e ab

solu

te v

alu

eof

a n

um

ber

is t

he

nu

mbe

r of

un

its

it i

s fr

om 0

on

a n

um

ber

lin

e.T

he

sym

bol

x

is u

sed

to r

epre

sen

t th

e ab

solu

te v

alu

eof

a n

um

ber

x.

•W

ord

sF

or a

ny r

eal n

umbe

r a

, if

ais

pos

itive

or

zero

, th

e ab

solu

te v

alue

of

ais

a.

Ab

solu

te V

alu

eIf

ais

neg

ativ

e, t

he a

bsol

ute

valu

e of

ais

the

opp

osite

of

a.

•S

ymb

ols

For

any

rea

l num

ber

a,

a

�a,

if a

�0,

and

a

��

a, if

a�

0.

Eva

luat

e �

4�

�2x

if

x�

6.

�4

��

2x

��

4�

�2

�6

��

4�

�12

�

4 �

12�

�8

Eva

luat

e 2

x�

3y

if x

��

4 an

d y

�3.

2x

�3y

�

2(�

4) �

3(3)

�

�8

�9

��

17

�17

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Eva

luat

e ea

ch e

xpre

ssio

n i

f w

��

4,x

�2,

y�

,an

d z

��

6.

1.2

x�

84

2.6

�z

��

7�

73.

5 �

w�

z15

4.x

�5

�2

w

�1

5.x

�

y

�z

�

46.

7 �

x�

3x

11

7.w

�4x

12

8.w

z�

xy

239.

z

�3

5yz

�39

10.5

w

�2

z�

2y

3411

.z

�4

2z�

y�

4012

.10

�x

w

2

13.

6y�

z�

yz

614

.3w

x�

4x

�8y

27

15.7

yz

�30

�9

16.1

4 �

2w

�xy

4

17.

2x�

y�

5y6

18.

xyz

�w

xz

54

19.z

z

�x

x�

3220

.12

�1

0x�

10y

�3

21.

5z

�8w

31

22.

yz�

4w

� w

1723

.w

z�

8y

2024

.xz

�x

z�

241 � 2

3 � 4

1 � 2

1 � 4

1 � 2

1 � 2

©G

lenc

oe/M

cGra

w-H

ill20

Gle

ncoe

Alg

ebra

2

Ab

solu

te V

alu

e Eq

uat

ion

sU

se t

he

defi

nit

ion

of

abso

lute

val

ue

to s

olve

equ

atio

ns

con

tain

ing

abso

lute

val

ue

expr

essi

ons.

For

any

rea

l num

bers

aan

d b,

whe

re b

�0,

if

a�

bth

en a

�b

or a

��

b.

Alw

ays

chec

k yo

ur

answ

ers

by s

ubs

titu

tin

g th

em i

nto

th

e or

igin

al e

quat

ion

.Som

etim

esco

mpu

ted

solu

tion

s ar

e n

ot a

ctu

al s

olu

tion

s.

Sol

ve

2x�

3�

17.C

hec

k y

our

solu

tion

s.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

So

lvin

g A

bso

lute

Val

ue

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-4

1-4

Exam

ple

Exam

ple

Cas

e 1

a�

b2x

�3

�17

2x�

3 �

3 �

17 �

32x

�20

x�

10

CH

ECK

2x

�3

�17

2(1

0) �

3�

172

0 �

3�

171

7�

1717

�17

✓

Cas

e 2

a�

�b

2x�

3 �

�17

2x�

3 �

3 �

�17

�3

2x�

�14

x�

�7

CH

ECK

2(�

7) �

3�

17�

14 �

3�

17�

17

�17

17 �

17 ✓

Th

ere

are

two

solu

tion

s,10

an

d �

7.

Sol

ve e

ach

eq

uat

ion

.Ch

eck

you

r so

luti

ons.

1.x

�15

�

37

{�52

,22}

2.t

�4

�5

�0

{�1,

9}

3.x

�5

�45

{�

40,5

0}4.

m�

3�

12 �

2m{3

}

5.5

b�

9�

16 �

2�

6.1

5 �

2k

�45

{�15

,30}

7.5n

�24

�8

�3n

{�

2}8.

8 �

5a

�14

�a

��,1

�9.

4p

�11

�

p�

4�23

,��

10.

3x�

1�

2x�

11{�

2,12

}

11.

x�

3 ��

1�

12.4

0 �

4x�

23x

�10

{6

,�10

}

13.5

f�

3f

�4

�20

{12}

14.

4b�

3�

15 �

2b{2

,�9}

15.

6 �

2x

�3x

�1

��

16.

16 �

3x

�4x

�12

{4}

1 � 21 � 2

1 � 3

1 � 71 � 3

11 � 2

Exer

cises

Exer

cises



Skil

ls P

ract

ice

So

lvin

g A

bso

lute

Val

ue

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-4

1-4

©G

lenc

oe/M

cGra

w-H

ill21

Gle

ncoe

Alg

ebra

2

Lesson 1-4

Eva

luat

e ea

ch e

xpre

ssio

n i

f w

�0.

4,x

�2,

y�

�3,

and

z�

�10

.

1.5

w

22.

�9y

27

3.9

y�

z17

4.�

17z

�

170

5.�

10z

�31

�

131

6.�

8x

�3y

�

2y

�5x

�

21

7.25

�5

z�

1�

248.

44 �

�2x

�y

45

9.2

4w

3.2

10.3

�1

�6w

1.

6

11.

�3x

�2y

�

4�

412

.6.4

�w

�1

7

Sol

ve e

ach

eq

uat

ion

.Ch

eck

you

r so

luti

ons.

13.

y�

3�

2{�

5,�

1}14

.5a

�

10{�

2,2}

15.

3k�

6�

2�

,�

16.

2g�

6�

0{�

3}

17.1

0 �

1 �

c{�

9,11

}18

.2x

�x

�9

{�3,

3}

19.

p�

7�

�14

�20

.23

w

�12

{�2,

2}

21.

7x�

3x

�2

�18

{�4,

4}22

.47

�y

�1

�11

{4,1

0}

23.

3n�

2�

�,

�24

.8d

�4d

�

5 �

13{�

2,2}

25.�

56a

�2

��

15��

,�

26.

k�

10 �

9�

1 � 65 � 6

5 � 61 � 2

1 � 2

8 � 34 � 3

©G

lenc

oe/M

cGra

w-H

ill22

Gle

ncoe

Alg

ebra

2

Eva

luat

e ea

ch e

xpre

ssio

n i

f a

��

1,b

��

8,c

�5,

and

d�

�1.

4.

1.6

a6

2.2

b�

412

3.�

10d

�a

�15

4.1

7c

�3

b�

511

4

5.�

610

a�

12

�13

26.

2b

�1

��

8b�

5�

52

7.5

a�

7�

3c

�4

238.

1 �

7c

�a

33

9.�

30.

5c�

2�

�0.

5b

�17

.510

.4d

�

5 �

2a

12.6

11.

a�

b�

b�

a14

12.

2 �

2d

�3

b�

19.2

Sol

ve e

ach

eq

uat

ion

.Ch

eck

you

r so

luti

ons.

13.

n�

4�

13{�

9,17

}14

.x

�13

�

2{1

1,15

}

15.

2y�

3�

29{�

13,1

6}16

.7x

�3

�42

{�9,

3}

17.

3u�

6�

42{�

12,1

6}18

.5x

�4

��

6�

19.�

34x

�9

�24

�20

.�6

5 �

2y

��

9�1.

75,3

.25 �

21.

8 �

p�

2p�

3{1

1}22

.4w

�1

�5w

�37

{�38

}

23.4

2y

�7

�5

�9

{3,4

}24

.�2

7 �

3y

�6

��

14�1,

3�

25.2

4 �

s�

�3s

{�8}

26.5

�3

2 �

2w

��

7{�

3,1}

27.5

2r

�3

�5

�0

{�2,

�1}

28.3

�5

2d�

3�

4�

29.W

EATH

ERA

th

erm

omet

er c

omes

wit

h a

gu

aran

tee

that

th

e st

ated

tem

pera

ture

dif

fers

from

th

e ac

tual

tem

pera

ture

by

no

mor

e th

an 1

.5 d

egre

es F

ahre

nh

eit.

Wri

te a

nd

solv

e an

equ

atio

n t

o fi

nd

the

min

imu

m a

nd

max

imu

m a

ctu

al t

empe

ratu

res

wh

en t

he

ther

mom

eter

sta

tes

that

th

e te

mpe

ratu

re i

s 87

.4 d

egre

es F

ahre

nh

eit.

x�

87.4

�

1.5;

or

85.9

�x

� 8

8.9

30.O

PIN

ION

PO

LLS

Pu

blic

opi

nio

n p

olls

rep

orte

d in

new

spap

ers

are

usu

ally

giv

en w

ith

am

argi

n o

f er

ror.

For

exa

mpl

e,a

poll

wit

h a

mar

gin

of

erro

r of

�5%

is

con

side

red

accu

rate

to w

ith

in p

lus

or m

inu

s 5%

of

the

actu

al v

alu

e.A

pol

l w

ith

a s

tate

d m

argi

n o

f er

ror

of�

3% p

redi

cts

that

can

dida

te T

onw

e w

ill

rece

ive

51%

of

an u

pcom

ing

vote

.Wri

te a

nd

solv

e an

equ

atio

n d

escr

ibin

g th

e m

inim

um

an

d m

axim

um

per

cen

t of

th

e vo

te t

hat

can

dida

te T

onw

e is

exp

ecte

d to

rec

eive

.x

�51

�

3 o

r 48

�x

� 5

4

2 � 3

Pra

ctic

e (

Ave

rag

e)

So

lvin

g A

bso

lute

Val

ue

Eq

uat

ion

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-4

1-4


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Ab

solu

te V

alu

e E

qu

atio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-4

1-4

©G

lenc

oe/M

cGra

w-H

ill23

Gle

ncoe

Alg

ebra

2

Lesson 1-4

Pre-

Act

ivit

yH

ow c

an a

n a

bso

lute

val

ue

equ

atio

n d

escr

ibe

the

mag

nit

ud

e of

an

eart

hq

uak

e?

Rea

d th

e in

trod

uct

ion

to

Les

son

1-4

at

the

top

of p

age

28 i

n y

our

text

book

.

•W

hat

is

a se

ism

olog

ist

and

wh

at d

oes

mag

nit

ude

of

an e

arth

quak

e m

ean

?a

scie

nti

st w

ho

stu

die

s ea

rth

qu

akes

;a

nu

mb

er f

rom

1 t

o 1

0th

at t

ells

ho

w s

tro

ng

an

ear

thq

uak

e is

•W

hy

is a

n a

bsol

ute

val

ue

equ

atio

n r

ath

er t

han

an

equ

atio

n w

ith

out

abso

lute

val

ue

use

d to

fin

d th

e ex

trem

es i

n t

he

actu

al m

agn

itu

de o

f an

eart

hqu

ake

in r

elat

ion

to

its

mea

sure

d va

lue

on t

he

Ric

hte

r sc

ale?

Sam

ple

an

swer

:Th

e ac

tual

mag

nit

ud

e ca

n v

ary

fro

m t

he

mea

sure

d m

agn

itu

de

by u

p t

o 0

.3 u

nit

in e

ith

er d

irec

tio

n,s

oan

ab

solu

te v

alu

e eq

uat

ion

is n

eed

ed.

•If

th

e m

agn

itu

de o

f an

ear

thqu

ake

is e

stim

ated

to

be 6

.9 o

n t

he

Ric

hte

r

scal

e,it

mig

ht

actu

ally

hav

e a

mag

nit

ude

as

low

as

or a

s h

igh

as

.

Rea

din

g t

he

Less

on

1.E

xpla

in h

ow �

aco

uld

rep

rese

nt

a po

siti

ve n

um

ber.

Giv

e an

exa

mpl

e.S

amp

lean

swer

:If

ais

neg

ativ

e,th

en �

ais

po

siti

ve.E

xam

ple

:If

a�

�25

,th

en

�a

��

(�25

) �

25.

2.E

xpla

in w

hy

the

abso

lute

val

ue

of a

nu

mbe

r ca

n n

ever

be

neg

ativ

e.S

amp

le a

nsw

er:

Th

e ab

solu

te v

alu

e is

th

e n

um

ber

of

un

its

it is

fro

m 0

on

th

e n

um

ber

lin

e.T

he

nu

mb

er o

f u

nit

s is

nev

er n

egat

ive.

3.W

hat

doe

s th

e se

nte

nce

b�

0 m

ean

?S

amp

le a

nsw

er:T

he

nu

mb

er b

is 0

or

gre

ater

th

an 0

.

4.W

hat

doe

s th

e sy

mbo

l �

mea

n a

s a

solu

tion

set

?S

amp

le a

nsw

er:

If a

so

luti

on

set

is �

,th

en t

her

e ar

e n

o s

olu

tio

ns.

Hel

pin

g Y

ou

Rem

emb

er

5.H

ow c

an t

he

nu

mbe

r li

ne

mod

el f

or a

bsol

ute

val

ue

that

is

show

n o

n p

age

28 o

f yo

ur

text

book

hel

p yo

u r

emem

ber

that

man

y ab

solu

te v

alu

e eq

uat

ion

s h

ave

two

solu

tion

s?S

amp

le a

nsw

er:T

he

nu

mb

er li

ne

sho

ws

that

fo

r ev

ery

po

siti

ve n

um

ber

,th

ere

are

two

nu

mb

ers

that

hav

e th

at n

um

ber

as

thei

r ab

solu

te v

alu

e.

7.2

6.6

©G

lenc

oe/M

cGra

w-H

ill24

Gle

ncoe

Alg

ebra

2

Co

nsi

der

ing

All

Cas

es in

Ab

solu

te V

alu

e E

qu

atio

ns

You

hav

e le

arn

ed t

hat

abs

olu

te v

alu

e eq

uat

ion

s w

ith

on

e se

t of

abs

olu

te v

alu

esy

mbo

ls h

ave

two

case

s th

at m

ust

be

con

side

red.

For

exa

mpl

e,|x

�3

|�5

mu

stbe

bro

ken

in

tox

�3

�5

or �

(x�

3) �

5.F

or a

n e

quat

ion

wit

h t

wo

sets

of

abso

lute

val

ue

sym

bols

,fou

r ca

ses

mu

st b

e co

nsi

dere

d.

Con

side

r th

e pr

oble

m |x

�2

|�3

�|x

�6

|.Fir

st w

e m

ust

wri

te t

he

equ

atio

ns

for

the

case

wh

ere

x�

6 �

0 an

d w

her

ex

�6

0.

Her

e ar

e th

e eq

uat

ion

s fo

rth

ese

two

case

s:

|x�

2|�

3 �

x�

6

|x�

2|�

3 �

�(x

�6)

Eac

h o

f th

ese

equ

atio

ns

also

has

tw

o ca

ses.

By

wri

tin

g th

e eq

uat

ion

s fo

r bo

thca

ses

of e

ach

equ

atio

n a

bove

,you

en

d u

p w

ith

th

e fo

llow

ing

fou

r eq

uat

ion

s:

x�

2 �

3 �

x�

6x

�2

�3

��

(x�

6)

�(x

�2)

�3

�x

�6

�x

�2

�3

��

(x�

6)

Sol

ve e

ach

of

thes

e eq

uat

ion

s an

d ch

eck

you

r so

luti

ons

in t

he

orig

inal

equ

atio

n,

|x�

2|�

3 �

|x�

6|.T

he

only

sol

uti

on t

o th

is e

quat

ion

is

��5 2� .

Sol

ve e

ach

ab

solu

te v

alu

e eq

uat

ion

.Ch

eck

you

r so

luti

on.

1.|x

�4

|�|x

�7

|x�

�1.

52.

|2x�

9|�

|x�

3|x

��

12,�

2

3.|�

3x�

6|�

|5x�

10|x

��

24.

|x�

4|�

6 �

|x�

3|x

�2.

5

5.H

ow m

any

case

s w

ould

th

ere

be f

or a

n a

bsol

ute

val

ue

equ

atio

n c

onta

inin

g th

ree

sets

of

abso

lute

val

ue

sym

bols

?8

6.L

ist

each

cas

e an

d so

lve

|x�

2|�

|2x

�4

|�|x

�3

|.Ch

eck

you

r so

luti

on.

x�

2 �

2x�

4 �

x�

3 �

(x�

2) �

2x�

4 �

x�

3

x�

2 �

2x�

4 �

�(x

�3)

�

(x�

2) �

2x�

4 �

�(x

�3)

�(x

�2)

�(�

2x�

4) �

x�

3x

�2

�(�

2x�

4) �

x�

3

�(x

�2)

�(�

2x�

4) �

�(x

�3)

x�

2 �

(�2x

�4)

��

(x�

3)

No

so

luti

on

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-4

1-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-5

1-5

©G

lenc

oe/M

cGra

w-H

ill25

Gle

ncoe

Alg

ebra

2

Lesson 1-5

Solv

e In

equ

alit

ies

Th

e fo

llow

ing

prop

erti

es c

an b

e u

sed

to s

olve

in

equ

alit

ies.

Ad

dit

ion

an

d S

ub

trac

tio

n P

rop

erti

es f

or

Ineq

ual

itie

sM

ult

iplic

atio

n a

nd

Div

isio

n P

rop

erti

es f

or

Ineq

ual

itie

s

For

any

rea

l num

bers

a,

b, a

nd c

:F

or a

ny r

eal n

umbe

rs a

, b,

and

c,

with

c�

0:1.

If a

�b,

the

n a

�c

�b

�c

and

a�

c�

b�

c.1.

If c

is p

ositi

ve a

nd a

�b,

the

n ac

�bc

and

�.

2.If

a

b, t

hen

a�

c

b�

can

d a

�c

b

�c.

2.If

cis

pos

itive

and

a

b, t

hen

ac

bcan

d

.

3.If

cis

neg

ativ

e an

d a

�b,

the

n ac

bc

and

.

4.If

cis

neg

ativ

e an

d a

b,

the

n ac

�bc

and

�.

Th

ese

prop

erti

es a

re a

lso

tru

e fo

r �

and

�.

b � ca � c

b � ca � c

b � ca � c

b � ca � c

Sol

ve 2

x�

4

36.

Th

en g

rap

h t

he

solu

tion

set

on

an

um

ber

lin

e.

2x�

4 �

4

36 �

42x

32

x

16T

he

solu

tion

set

is

{xx

16

}.

2120

1918

1716

1514

13

Sol

ve 1

7 �

3w�

35.T

hen

grap

h t

he

solu

tion

set

on

a n

um

ber

lin

e.

17 �

3w�

3517

�3w

�17

�35

�17

�3w

�18

w�

�6

Th

e so

luti

on s

et i

s (�

�,�

6].

�9

�8

�7

�6

�5

�4

�3

�2

�1

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sol

ve e

ach

in

equ

alit

y.D

escr

ibe

the

solu

tion

set

usi

ng

set-

bu

ild

er o

r in

terv

aln

otat

ion

.Th

en g

rap

h t

he

solu

tion

set

on

a n

um

ber

lin

e.

1.7(

7a�

9) �

842.

3(9z

�4)

35

z�

43.

5(12

�3n

) �

165

{aa

�3}

or

(�∞

,3]

{zz

�2}

or

(�∞

,2)

{nn

�

7} o

r (�

7,�

∞)

4.18

�4k

�2(

k�

21)

5.4(

b�

7) �

6 �

226.

2 �

3(m

�5)

�4(

m�

3)

{kk

�

4} o

r (�

4,�

∞)

{bb

�11

} o

r (�

∞,1

1){m

m�

5} o

r (�

∞,5

]

7.4x

�2

�

7(4x

�2)

8.(2

y�

3)

y�

29.

2.5d

�15

�75

�xx

�o

r �

,�∞�

{yy

��

9} o

r (�

∞,�

9){d

d�

24}

or

(�∞

,24]

2122

1920

2324

2526

27�

12�

14�

10�

8�

6�

4�

3�

2�

10

12

34

1 � 21 � 2

1 � 3

23

01

45

67

88

96

710

1112

1314

�8

�7

�6

�5

�4

�3

�2

�1

0

�8

�7

�6

�5

�4

�3

�2

�1

0�

2�

1�

4�

30

12

34

�2

�1

�4

�3

01

23

4

©G

lenc

oe/M

cGra

w-H

ill26

Gle

ncoe

Alg

ebra

2

Rea

l-W

orl

d P

rob

lem

s w

ith

Ineq

ual

itie

sM

any

real

-wor

ld p

robl

ems

invo

lve

ineq

ual

itie

s.T

he

char

t be

low

sh

ows

som

e co

mm

on p

hra

ses

that

in

dica

te i

neq

ual

itie

s.

�

��

is le

ss t

han

is g

reat

er t

han

is a

t m

ost

is a

t le

ast

is f

ewer

tha

nis

mor

e th

anis

no

mor

e th

anis

no

less

tha

nis

less

tha

n or

equ

al t

ois

gre

ater

tha

n or

equ

al t

o

SPO

RTS

Th

e V

ikin

gs p

lay

36 g

ames

th

is y

ear.

At

mid

seas

on,t

hey

hav

e w

on 1

6 ga

mes

.How

man

y of

th

e re

mai

nin

g ga

mes

mu

st t

hey

win

in

ord

er t

ow

in a

t le

ast

80%

of

all

thei

r ga

mes

th

is s

easo

n?

Let

xbe

th

e n

um

ber

of r

emai

nin

g ga

mes

th

at t

he

Vik

ings

mu

st w

in.T

he

tota

l n

um

ber

ofga

mes

th

ey w

ill

hav

e w

on b

y th

e en

d of

th

e se

ason

is

16 �

x.T

hey

wan

t to

win

at

leas

t 80

%of

th

eir

gam

es.W

rite

an

in

equ

alit

y w

ith

�.

16 �

x�

0.8(

36)

x�

0.8(

36)

�16

x�

12.8

Sin

ce t

hey

can

not

win

a f

ract

ion

al p

art

of a

gam

e,th

e V

ikin

gs m

ust

win

at

leas

t 13

of

the

gam

es r

emai

nin

g.

1.PA

RK

ING

FEE

ST

he

city

par

kin

g lo

t ch

arge

s $2

.50

for

the

firs

t h

our

and

$0.2

5 fo

r ea

chad

diti

onal

hou

r.If

th

e m

ost

you

wan

t to

pay

for

par

kin

g is

$6.

50,s

olve

th

e in

equ

alit

y2.

50 �

0.25

(x�

1) �

6.50

to

dete

rmin

e fo

r h

ow m

any

hou

rs y

ou c

an p

ark

you

r ca

r.A

t m

ost

17

ho

urs

PLA

NN

ING

For

Exe

rcis

es 2

an

d 3

,use

th

e fo

llow

ing

info

rmat

ion

.

Eth

an i

s re

adin

g a

482-

page

boo

k fo

r a

book

rep

ort

due

on M

onda

y.H

e h

as a

lrea

dy r

ead

80 p

ages

.He

wan

ts t

o fi

gure

ou

t h

ow m

any

page

s pe

r h

our

he

nee

ds t

o re

ad i

n o

rder

to

fin

ish

th

e bo

ok i

n l

ess

than

6 h

ours

.

2.W

rite

an

in

equ

alit

y to

des

crib

e th

is s

itu

atio

n.

�6

or

6n�

482

�80

3.S

olve

the

ine

qual

ity

and

inte

rpre

t th

e so

luti

on.

Eth

an m

ust

rea

d a

t le

ast

67 p

ages

per

ho

ur

in o

rder

to

fin

ish

th

e b

oo

k in

less

th

an 6

ho

urs

.

BO

WLI

NG

For

Exe

rcis

es 4

an

d 5

,use

th

e fo

llow

ing

info

rmat

ion

.

Fou

r fr

ien

ds p

lan

to

spen

d F

rida

y ev

enin

g at

th

e bo

wli

ng

alle

y.T

hre

e of

th

e fr

ien

ds n

eed

tore

nt

shoe

s fo

r $3

.50

per

pers

on.A

str

ing

(gam

e) o

f bo

wli

ng

cost

s $1

.50

per

pers

on.I

f th

efr

ien

ds p

ool

thei

r $4

0,h

ow m

any

stri

ngs

can

th

ey a

ffor

d to

bow

l?

4.W

rite

an

equ

atio

n t

o de

scri

be t

his

sit

uat

ion

.3(

3.50

) �

4(1.

50)n

�40

5.S

olve

th

e in

equ

alit

y an

d in

terp

ret

the

solu

tion

.T

he

frie

nd

s ca

n b

ow

l at

mo

st

4 st

rin

gs.

482

�80

�� n

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

So

lvin

g In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-5

1-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

So

lvin

g In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-5

1-5

©G

lenc

oe/M

cGra

w-H

ill27

Gle

ncoe

Alg

ebra

2

Lesson 1-5

Sol

ve e

ach

in

equ

alit

y.D

escr

ibe

the

solu

tion

set

usi

ng

set-

bu

ild

er o

r in

terv

aln

otat

ion

.Th

en,g

rap

h t

he

solu

tion

set

on

a n

um

ber

lin

e.

1.�

2{z

z�

�8}

or

(�∞

,�8]

2.3a

�7

�16

{aa

�3}

or

(�∞

,3]

3.16

�3q

�4

{qq

4}

or

(4,∞

)4.

20 �

3s

7s{s

s�

2} o

r (�

∞,2

)

5.3x

��

9{x

x�

�3}

or

[�3,

∞)

6.4b

�9

�7

{bb

�4}

or

(�∞

,4]

7.2z

��

9 �

5z{z

z

3} o

r (3

,∞)

8.7f

�9

3f

�1

{ff

2}

or

(2,∞

)

9.�

3s�

8 �

5s{s

s�

�1}

or

[�1,

∞)

10.7

t�

(t�

4) �

25�t

t�

�or ��

∞,

�

11.0

.7m

�0.

3m�

2m�

4{m

m�

4}12

.4(5

x�

7) �

13�x

x�

��o

ro

r (�

∞,4

]��

∞,�

�13

.1.7

y�

0.78

5

{yy

3.

4}14

.4x

�9

2x

�1

{xx

5}

or

(5,∞

)o

r (3

.4,∞

)

Def

ine

a va

riab

le a

nd

wri

te a

n i

neq

ual

ity

for

each

pro

ble

m.T

hen

sol

ve.

15.N

inet

een

mor

e th

an a

nu

mbe

r is

les

s th

an 4

2.n

�19

�42

;n

�23

16.T

he

diff

eren

ce o

f th

ree

tim

es a

nu

mbe

r an

d 16

is

at l

east

8.

3n�

16 �

8;n

�8

17.O

ne h

alf

of a

num

ber

is m

ore

than

6 l

ess

than

the

sam

e nu

mbe

r.n

n

�6;

n�

12

18.F

ive

less

th

an t

he

prod

uct

of

6 an

d a

nu

mbe

r is

no

mor

e th

an t

wic

e th

at s

ame

nu

mbe

r.

6n�

5 �

2n;

n�

5 � 4

1 � 2

�1

01

23

45

67

�1

�2

01

23

45

6

3 � 4�

2�

1�

4�

30

12

34

�2

�1

01

23

45

6

3 � 4

�2

�1

�4

�3

01

23

4�

1�

2�

3�

40

12

34

7 � 27 � 2

�1

�2

�3

�4

01

23

4�

1�

20

12

34

56

�2

�1

01

23

45

6�

1�

2�

3�

40

12

34

�2

�1

�4

�3

01

23

4�

10

12

34

56

7

�2

�1

�4

�3

01

23

4�

7�

6�

9�

8�

5�

4�

3�

2�

1

z� �

4

©G

lenc

oe/M

cGra

w-H

ill28

Gle

ncoe

Alg

ebra

2

Sol

ve e

ach

in

equ

alit

y.D

escr

ibe

the

solu

tion

set

usi

ng

set-

bu

ild

er o

r in

terv

aln

otat

ion

.Th

en,g

rap

h t

he

solu

tion

set

on

a n

um

ber

lin

e.

1.8x

�6

�10

{x

x�

2} o

r [2

,∞)

2.23

�4u

�11

{u

u

3} o

r (3

,∞)

3.�

16 �

8r�

0 {r

r�

�2}

or

(�∞

,�2]

4.14

s�

9s�

5 {s

s�

1} o

r (�

∞,1

)

5.9x

�11

6x

�9 �x

x

�or �

,∞�

6.�

3(4w

�1)

18

�ww

��

�o

r ��

∞,�

�7.

1 �

8u�

3u�

10 {

uu

�1}

or

[1,∞

)8.

17.5

�19

�2.

5x{x

x�

0.6}

o

r (�

∞,0

.6)

9.9(

2r�

5) �

3 �

7r�

4 {r

r�

4}

10.1

�5(

x�

8) �

2 �

(x�

5) {

xx

�6}

o

r (�

∞,4

)o

r (�

∞,6

]

11.

��

3.5

{xx

��

1} o

r [�

1,∞

)12

.q�

2(2

�q)

�0 �q

q�

�or ��

∞,

�

13.�

36 �

2(w

�77

)

�4(

2w�

52)

14.4

n�

5(n

�3)

3(

n�

1) �

4 {w

w

�3}

or

(�3,

∞)

{nn

�4}

or

(�∞

,4)

Def

ine

a va

riab

le a

nd

wri

te a

n i

neq

ual

ity

for

each

pro

ble

m.T

hen

sol

ve.

15.T

wen

ty l

ess

than

a n

um

ber

is m

ore

than

tw

ice

the

sam

e n

um

ber.

n�

20

2n;

n�

�20

16.F

our

tim

es t

he

sum

of

twic

e a

nu

mbe

r an

d �

3 is

les

s th

an 5

.5 t

imes

th

at s

ame

nu

mbe

r.4[

2n�

(�3)

] �

5.5n

;n

�4.

8

17.H

OTE

LST

he

Lin

coln

’s h

otel

roo

m c

osts

$90

a n

igh

t.A

n a

ddit

ion

al 1

0% t

ax i

s ad

ded.

Hot

el p

arki

ng

is $

12 p

er d

ay.T

he

Lin

coln

’s e

xpec

t to

spe

nd

$30

in t

ips

duri

ng

thei

r st

ay.

Sol

ve t

he

ineq

ual

ity

90x

�90

(0.1

)x�

12x

�30

�60

0 to

fin

d h

ow m

any

nig

hts

th

eL

inco

ln’s

can

sta

y at

th

e h

otel

wit

hou

t ex

ceed

ing

tota

l h

otel

cos

ts o

f $6

00.

5 n

igh

ts

18.B

AN

KIN

GJa

n’s

acc

oun

t ba

lan

ce i

s $3

800.

Of

this

,$75

0 is

for

ren

t.Ja

n w

ants

to

keep

aba

lan

ce o

f at

lea

st $

500.

Wri

te a

nd

solv

e an

in

equ

alit

y de

scri

bin

g h

ow m

uch

sh

e ca

nw

ith

draw

an

d st

ill

mee

t th

ese

con

diti

ons.

3800

�75

0 �

w�

500;

w�

$255

0

0�

1�

21

23

45

6�

1�

2�

3�

40

12

34

0�

1�

2�

3�

41

23

4�

1�

2�

3�

40

12

34

4 � 34 � 3

4x�

3�

2

01

23

45

67

8�

2�

10

12

34

56

�2

�1

�4

�3

01

23

4�

1�

2�

3�

40

12

34

5 � 4�

2�

1�

4�

30

12

34

�1

�2

�3

�4

01

23

4

5 � 42 � 3

2 � 3

�2

�1

�4

�3

01

23

4�

2�

1�

4�

30

12

34

�1

�2

01

23

45

6�

1�

2�

3�

40

12

34

Pra

ctic

e (

Ave

rag

e)

So

lvin

g In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-5

1-5



Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-5

1-5

©G

lenc

oe/M

cGra

w-H

ill29

Gle

ncoe

Alg

ebra

2

Lesson 1-5

Pre-

Act

ivit

yH

ow c

an i

neq

ual

itie

s b

e u

sed

to

com

par

e p

hon

e p

lan

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

1-5

at

the

top

of p

age

33 i

n y

our

text

book

.

•W

rite

an

in

equ

alit

y co

mpa

rin

g th

e n

um

ber

of m

inu

tes

per

mon

thin

clu

ded

in t

he

two

phon

e pl

ans.

150

�40

0 o

r 40

0

150

•S

upp

ose

that

in

on

e m

onth

you

use

230

min

ute

s of

air

tim

e on

you

rw

irel

ess

phon

e.F

ind

you

r m

onth

ly c

ost

wit

h e

ach

pla

n.

Pla

n 1

:P

lan

2:

Wh

ich

pla

n s

hou

ld y

ou c

hoo

se?

Rea

din

g t

he

Less

on

1.T

her

e ar

e se

vera

l di

ffer

ent

way

s to

wri

te o

r sh

ow i

neq

ual

itie

s.W

rite

eac

h o

f th

efo

llow

ing

in i

nte

rval

not

atio

n.

a.{x

x�

�3}

(�∞

,�3)

b.

{xx

�5}

[5,�

∞)

c.(�

∞,2

]

d.

(�1,

�∞

)

2.S

how

how

you

can

wri

te a

n i

neq

ual

ity

sym

bol

foll

owed

by

a n

um

ber

to d

escr

ibe

each

of

the

foll

owin

g si

tuat

ion

s.

a.T

her

e ar

e fe

wer

th

an 6

00 s

tude

nts

in

th

e se

nio

r cl

ass.

�60

0

b.

A s

tude

nt

may

en

roll

in

no

mor

e th

an s

ix c

ours

es e

ach

sem

este

r.�

6

c.To

par

tici

pate

in

a co

ncer

t,yo

u m

ust

be w

illi

ng t

o at

tend

at

leas

t te

n re

hear

sals

.�

10

d.

Th

ere

is s

pace

for

at

mos

t 16

5 st

ude

nts

in

th

e h

igh

sch

ool

ban

d.�

165

Hel

pin

g Y

ou

Rem

emb

er

3.O

ne

way

to

rem

embe

r so

met

hin

g is

to

expl

ain

it

to a

not

her

per

son

.A c

omm

on s

tude

nt

erro

r in

sol

vin

g in

equ

alit

ies

is f

orge

ttin

g to

rev

erse

th

e in

equ

alit

y sy

mbo

l w

hen

mu

ltip

lyin

g or

div

idin

g bo

th s

ides

of

an i

neq

ual

ity

by a

neg

ativ

e n

um

ber.

Su

ppos

e th

atyo

ur

clas

smat

e is

hav

ing

trou

ble

rem

embe

rin

g th

is r

ule

.How

cou

ld y

ou e

xpla

in t

his

ru

leto

you

r cl

assm

ate?

Sam

ple

an

swer

:D

raw

a n

um

ber

lin

e.P

lot

two

po

siti

ven

um

ber

s,fo

r ex

amp

le,3

an

d 8

.Th

en p

lot

thei

r ad

dit

ive

inve

rses

,�3

and

�8.

Wri

te a

n in

equ

alit

y th

at c

om

par

es t

he

po

siti

ve n

um

ber

s an

d o

ne

that

com

par

es t

he

neg

ativ

e n

um

ber

s.N

oti

ce t

hat

8

3,bu

t �

8 �

�3.

Th

eo

rder

ch

ang

es w

hen

yo

u m

ult

iply

by

�1.

32

54

10

�1

�2

�3

�4

�5

�5

�4

�3

�2

�1

01

23

45

Pla

n 2

$55

$67

©G

lenc

oe/M

cGra

w-H

ill30

Gle

ncoe

Alg

ebra

2

Eq

uiv

alen

ce R

elat

ion

sA

rel

atio

n R

on

a s

et A

is a

n e

quiv

alen

ce r

elat

ion

if i

t h

as t

he

foll

owin

g pr

oper

ties

.

Ref

lexi

ve P

rop

erty

For

an

y el

emen

t a

of s

et A

,aR

a.

Sym

met

ric

Pro

per

tyF

or a

ll e

lem

ents

aan

d b

of s

et A

,if

aR

b,t

hen

bR

a.

Tra

nsi

tive

Pro

per

tyF

or a

ll e

lem

ents

a,b

,an

d c

of s

et A

,if

aR

ban

d b

R c

,th

en a

R c

.

Equ

alit

y on

th

e se

t of

all

rea

l n

um

bers

is

refl

exiv

e,sy

mm

etri

c,an

d tr

ansi

tive

.T

her

efor

e,it

is

an e

quiv

alen

ce r

elat

ion

.

In e

ach

of

the

foll

owin

g,a

rela

tion

an

d a

set

are

giv

en.W

rite

yes

if t

he

rela

tion

is

an e

qu

ival

ence

rel

atio

n o

n t

he

give

n s

et.I

f it

is

not

,tel

l w

hic

h o

f th

e p

rop

erti

es i

t fa

ils

to e

xhib

it.

1.�

,{al

l n

um

bers

}n

o;

refl

exiv

e,sy

mm

etri

c

2.

,{al

l tr

ian

gles

in

a p

lan

e}ye

s

3.is

th

e si

ster

of,

{all

wom

en i

n T

enn

esse

e}n

o;

refl

exiv

e

4.�

,{al

l n

um

bers

}n

o;

sym

met

ric

5.is

a f

acto

r of

,{al

l n

onze

ro i

nte

gers

}n

o;

sym

met

ric

6.

,{al

l po

lygo

ns

in a

pla

ne}

yes

7.is

th

e sp

ouse

of,

{all

peo

ple

in R

oan

oke,

Vir

gin

ia}

no

;re

flex

ive,

tran

siti

ve

8.⊥

,{al

l li

nes

in

a p

lan

e}n

o;

refl

exiv

e,tr

ansi

tive

9.is

a m

ult

iple

of,

{all

in

tege

rs}

no

;sy

mm

etri

c

10.

is t

he

squ

are

of,{

all

nu

mbe

rs}

no

;re

flex

ive,

sym

met

ric,

tran

siti

ve

11.

��,{a

ll l

ines

in

a p

lan

e}n

o;

refl

exiv

e

12.

has

th

e sa

me

colo

r ey

es a

s,{a

ll m

embe

rs o

f th

e C

leve

lan

d S

ymph

ony

Orc

hes

tra}

yes

13.

is t

he

grea

test

in

tege

r n

ot g

reat

er t

han

,{al

l n

um

bers

}n

o;

refl

exiv

e,sy

mm

etri

c,tr

ansi

tive

14.

is t

he

grea

test

in

tege

r n

ot g

reat

er t

han

,{al

l in

tege

rs}

yes

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-5

1-5


An

swer

s


Stu

dy G

uid

e a

nd I

nte

rven

tion

So

lvin

g C

om

po

un

d a

nd

Ab

solu

te V

alu

e In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-6

1-6

©G

lenc

oe/M

cGra

w-H

ill31

Gle

ncoe

Alg

ebra

2

Lesson 1-6

Co

mp

ou

nd

Ineq

ual

itie

sA

com

pou

nd

ineq

ual

ity

con

sist

s of

tw

o in

equ

alit

ies

join

ed b

yth

e w

ord

and

or t

he

wor

d or

.To

solv

e a

com

pou

nd

ineq

ual

ity,

you

mu

st s

olve

eac

h p

art

sepa

rate

ly.

Exa

mpl

e: x

�

4 an

d x

�3

The

gra

ph is

the

inte

rsec

tion

of s

olut

ion

sets

of

two

ineq

ualit

ies.

Exa

mpl

e:x

��

3 or

x

1T

he g

raph

is t

he u

nion

of

solu

tion

sets

of

two

ineq

ualit

ies.

�5

�4

�3

�2

�1

01

23

45

Or

Co

mp

ou

nd

Ineq

ual

itie

s

�3

�2

�5

�4

�1

01

23

45

An

dC

om

po

un

dIn

equ

alit

ies

Sol

ve �

3 �

2x�

5 �

19.

Gra

ph

th

e so

luti

on s

et o

n a

nu

mb

er l

ine.

�3

�2x

�5

and

2x�

5 �

19�

8 �

2x2x

�14

�4

�x

x�

7

�4

�x

�7

�4

�2

�8

�6

02

46

8

Sol

ve 3

y�

2 �

7 or

2y

�1

��

9.G

rap

h t

he

solu

tion

set

on a

nu

mb

er l

ine.

3y�

2 �

7or

2y�

1 �

�9

3y�

9or

2y�

�8

y�

3or

y�

�4

�8

�6

�4

�2

02

46

8

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sol

ve e

ach

in

equ

alit

y.G

rap

h t

he

solu

tion

set

on

a n

um

ber

lin

e.

1.�

10 �

3x�

2 �

142.

3a�

8 �

23 o

r a

�6

7

{x�

4 �

x�

4}{a

a�

5 o

r a

52

}

3.18

�4x

�10

�50

4.5k

�2

��

13 o

r 8k

�1

19

{x7

�x

�15

}{k

k�

�3

or

k

2.5}

5.10

0 �

5y�

45 �

225

6.b

�2

10

or

b�

5 �

�4

{y2

9 �

y�

54}

{bb

��

12 o

r b

18

}

7.22

�6w

�2

�82

8.4d

�1

�

9 or

2d

�5

�11

{w4

�w

�14

}{a

ll re

al n

um

ber

s}

0�

1�

2�

3�

41

23

40

24

68

1012

1416

�24

�12

012

240

1020

3040

5060

7080

3 � 42 � 3

�4

�3

�2

�1

01

23

43

57

911

1315

1719

�10

010

2030

4050

6070

�8

�6

�4

�2

02

46

8

1 � 4

©G

lenc

oe/M

cGra

w-H

ill32

Gle

ncoe

Alg

ebra

2

Ab

solu

te V

alu

e In

equ

alit

ies

Use

th

e de

fin

itio

n o

f ab

solu

te v

alu

e to

rew

rite

an

abso

lute

val

ue

ineq

ual

ity

as a

com

pou

nd

ineq

ual

ity.

For

all

real

num

bers

aan

d b,

b

0, t

he f

ollo

win

g st

atem

ents

are

tru

e.

1.If

a

�b,

the

n �

b�

a�

b.2.

If a

b, t

hen

a

bor

a�

�b.

The

se s

tate

men

ts a

re a

lso

true

for

�an

d �

.

Stu

dy G

uid

e a

nd I

nte

rven

tion

(c

onti

nued

)

So

lvin

g C

om

po

un

d a

nd

Ab

solu

te V

alu

e In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-6

1-6

Sol

ve

x�

2

4.G

rap

hth

e so

luti

on s

et o

n a

nu

mb

er l

ine.

By

stat

emen

t 2

abov

e,if

x

�2

4,

then

x

�2

4

or x

�2

��

4.S

ubt

ract

ing

2fr

om b

oth

sid

es o

f ea

ch i

neq

ual

ity

give

s x

2

or x

��

6.

�8

�6

�4

�2

02

46

8

Sol

ve

2x�

1�

5.G

rap

h t

he

solu

tion

set

on

a n

um

ber

lin

e.

By

stat

emen

t 1

abov

e,if

2x

�1

�5,

then

�5

�2x

�1

�5.

Add

ing

1 to

all

th

ree

part

sof

th

e in

equ

alit

y gi

ves

�4

�2x

�6.

Div

idin

g by

2 g

ives

�2

�x

�3.

�4

�2

�8

�6

02

46

8

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Sol

ve e

ach

in

equ

alit

y.G

rap

h t

he

solu

tion

set

on

a n

um

ber

lin

e.

1.3

x�

4�

8�x

�4

�x

��

2.4

s�

1

27{s

s�

�6.

5 o

r s

6.

5}

3.

�3 �

5{c

�4

�c

�16

}4.

a�

9�

30{a

a�

�39

or

a�

21}

5.2

f�

11

9

{ff

�1

or

f

10}

6.5

w�

2�

28{w

�6

�w

�5.

2}

7.1

0 �

2k

�2

{k4

�k

�6}

8.

�5 �

2

10{x

x�

�6

or

x

26}

9.4

b�

11

�17

�b�

�b

�7 �

10.

100

�3m

20�m

m�

26o

r m

40

�0

1020

305

1525

3540

�4

04

8�

22

610

12

2 � 33 � 2

�10

010

20�

55

1525

300

24

61

35

78

x � 2�8

�4

04

�6

�2

26

8�

40

48

�2

26

1012

�40

�20

020

40�

80

816

�4

412

2024

c � 2

�8

�4

04

�6

�2

26

8�

5�

4�

3�

2�

10

12

3

4 � 3



Skil

ls P

ract

ice

So

lvin

g C

om

po

un

d a

nd

Ab

solu

te V

alu

e In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-6

1-6

©G

lenc

oe/M

cGra

w-H

ill33

Gle

ncoe

Alg

ebra

2

Lesson 1-6

Wri

te a

n a

bso

lute

val

ue

ineq

ual

ity

for

each

of

the

foll

owin

g.T

hen

gra

ph

th

eso

luti

on s

et o

n a

nu

mb

er l

ine.

1.al

l n

um

bers

gre

ater

th

an o

r eq

ual

to

22.

all

nu

mbe

rs l

ess

than

5 a

nd

grea

ter

or l

ess

than

or

equ

al t

o �

2n

�

2th

an �

5n

�

5

3.al

l n

um

bers

les

s th

an �

1 or

gre

ater

4.

all

nu

mbe

rs b

etw

een

�6

and

6n

�

6th

an 1

n

1

Wri

te a

n a

bso

lute

val

ue

ineq

ual

ity

for

each

gra

ph

.

5.n

�

16.

n

�4

7.n

�

38.

n

2.

5

Sol

ve e

ach

in

equ

alit

y.G

rap

h t

he

solu

tion

set

on

a n

um

ber

lin

e.

9.2c

�1

5

or c

�0

{cc

2

10.�

11 �

4y�

3 �

1{y

�2

�y

�1}

or

c�

0}

11.1

0

�5x

5

{x�

2 �

x�

�1}

12.4

a�

�8

or a

��

3{a

a�

�2

or

a�

�3}

13.8

�3x

�2

�23

{x2

�x

�7}

14.w

�4

�10

or

�2w

�6

all r

eal

nu

mb

ers

15.

t�

3{t

t�

�3

or

t�

3}16

.6x

�

12{x

�2

�x

�2}

17.

�7r

14{r

r�

�2

or

r

2}18

.p

�2

��

2�

19.

n�

5�

7{n

�2

�n

�12

}20

.h

�1

�5

{hh

��

6 o

r h

�4}

�8

�6

�4

�2

02

46

8�

4�

20

24

68

1012

0�

1�

2�

3�

41

23

4�

4�

3�

2�

10

12

34

�4

�3

�2

�1

01

23

4�

4�

3�

2�

10

12

34

0�

1�

2�

3�

41

23

40

12

34

56

78

�4

�3

�2

�1

01

23

4�

4�

3�

2�

10

12

34

�4

�3

�2

�1

01

23

4�

4�

3�

2�

10

12

34

�4

�3

�2

�1

01

23

4�

2�

1�

4�

30

12

34

�4

�3

�2

�1

01

23

4�

2�

1�

4�

30

12

34

�8

�6

�4

�2

02

46

8�

4�

3�

2�

10

12

34

�8

�6

�4

�2

02

46

8�

4�

3�

2�

10

12

34

©G

lenc

oe/M

cGra

w-H

ill34

Gle

ncoe

Alg

ebra

2

Wri

te a

n a

bso

lute

val

ue

ineq

ual

ity

for

each

of

the

foll

owin

g.T

hen

gra

ph

th

eso

luti

on s

et o

n a

nu

mb

er l

ine.

1.al

l n

um

bers

gre

ater

th

an 4

or

less

th

an �

4n

4

2.al

l n

um

bers

bet

wee

n �

1.5

and

1.5,

incl

udi

ng

�1.

5 an

d 1.

5n

�

1.5

Wri

te a

n a

bso

lute

val

ue

ineq

ual

ity

for

each

gra

ph

.

3.n

�

104.

n

�

Sol

ve e

ach

in

equ

alit

y.G

rap

h t

he

solu

tion

set

on

a n

um

ber

lin

e.

5.�

8 �

3y�

20 �

52{y

4 �

y�

24}

6.3(

5x�

2) �

24 o

r 6x

�4

4

�5x

{xx

�2

or

x

8}

7.2x

�3

15

or

3 �

7x�

17{x

x

�2}

8.15

�5x

�0

and

5x�

6 �

�14

{xx

�3}

9.2

w

�5

�ww

��

or

w�

�10

.y

�5

�2

{x�

7 �

x�

�3}

11.

x�

8�

3{x

x�

5 o

r x

�11

}12

.2z

�2

�3

�z�

�z

��

13.

2x�

2�

7 �

�5

{x�

2 �

x�

0}14

.x

x

�1

all r

eal n

um

ber

s

15.

3b�

5�

�2

�16

.3n

�2

�2

�1

�n�

�n

��

17.R

AIN

FALL

In 9

0% o

f th

e la

st 3

0 ye

ars,

the

rain

fall

at

Sh

ell

Bea

ch h

as v

arie

d n

o m

ore

than

6.5

in

ches

fro

m i

ts m

ean

val

ue

of 2

4 in

ches

.Wri

te a

nd

solv

e an

abs

olu

te v

alu

ein

equ

alit

y to

des

crib

e th

e ra

infa

ll i

n t

he

oth

er 1

0% o

f th

e la

st 3

0 ye

ars.

r�

24

6.

5;{r

r�

17.5

or

r

30.5

}

18.M

AN

UFA

CTU

RIN

GA

com

pany

’s g

uide

line

s ca

ll f

or e

ach

can

of s

oup

prod

uced

not

to

vary

from

its

sta

ted

volu

me

of 1

4.5

flu

id o

un

ces

by m

ore

than

0.0

8 ou

nce

s.W

rite

an

d so

lve

anab

solu

te v

alu

e in

equ

alit

y to

des

crib

e ac

cept

able

can

vol

um

es.

v�

14.5

�

0.08

;{v

14.

42 �

v�

14.5

8}�4

�3

�2

�1

01

23

40

�1

�2

�3

�4

12

34

5 � 31 � 3

0�

1�

2�

3�

41

23

4�

4�

3�

2�

10

12

34

�4

�3

�2

�1

01

23

40

24

68

1012

1416

5 � 21 � 2

�8

�7

�6

�5

�4

�3

�2

�1

0�

4�

3�

2�

10

12

34

5 � 25 � 2

�1

�2

�3

�4

01

23

4�

1�

2�

3�

40

12

34

�2

02

46

810

1214

04

812

1620

2428

32

4 � 3�

4�

3�

2�

10

12

34

�20

�10

010

20

�4

�3

�2

�1

01

23

4

�8

�6

�4

�2

02

46

8

Pra

ctic

e (

Ave

rag

e)

So

lvin

g C

om

po

un

d a

nd

Ab

solu

te V

alu

e In

equ

alit

ies

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-6

1-6


An

swer

s


Readin

g t

o L

earn

Math

em

ati

csS

olv

ing

Co

mp

ou

nd

an

d A

bso

lute

Val

ue

Ineq

ual

itie

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-6

1-6

©G

lenc

oe/M

cGra

w-H

ill35

Gle

ncoe

Alg

ebra

2

Lesson 1-6

Pre-

Act

ivit

yH

ow a

re c

omp

oun

d i

neq

ual

itie

s u

sed

in

med

icin

e?

Rea

d th

e in

trod

uct

ion

to

Les

son

1-6

at

the

top

of p

age

40 i

n y

our

text

book

.

•F

ive

pati

ents

arr

ive

at a

med

ical

lab

orat

ory

at 1

1:30

A.M

.for

a g

luco

seto

lera

nce

tes

t.E

ach

of

them

is

aske

d w

hen

th

ey l

ast

had

som

eth

ing

toea

t or

dri

nk.

Som

e of

th

e pa

tien

ts a

re g

iven

th

e te

st a

nd

oth

ers

are

told

that

th

ey m

ust

com

e ba

ck a

not

her

day

.Eac

h o

f th

e pa

tien

ts i

s li

sted

belo

w w

ith

th

e ti

mes

wh

en t

hey

sta

rted

to

fast

.(T

he

P.M

.tim

es r

efer

to

the

nig

ht

befo

re.)

Wh

ich

of

the

pati

ents

wer

e ac

cept

ed f

or t

he

test

?

Ora

5:00

A.M

.Ju

anit

a11

:30

P.M

.Ja

son

an

d J

uan

ita

Jaso

n1:

30 A

.M.

Sam

ir5:

00 P

.M.

Rea

din

g t

he

Less

on

1.a.

Wri

te a

com

pou

nd

ineq

ual

ity

that

say

s,“x

is g

reat

er t

han

�3

and

xis

les

s th

an o

req

ual

to

4.”

�3

�x

�4

b.

Gra

ph t

he

ineq

ual

ity

that

you

wro

te i

n p

art

a on

a n

um

ber

lin

e.

2.U

se a

com

pou

nd

ineq

ual

ity

and

set-

buil

der

not

atio

n t

o de

scri

be t

he

foll

owin

g gr

aph

.{x

x�

�1

or

x

3}

3.W

rite

a s

tate

men

t eq

uiv

alen

t to

4x

�5

2

that

doe

s n

ot u

se t

he

abso

lute

val

ue

sym

bol.

4x�

5

2 o

r 4x

�5

��

2

4.W

rite

a s

tate

men

t eq

uiv

alen

t to

3x

�7

�8

that

doe

s n

ot u

se t

he

abso

lute

val

ue

sym

bol.

�8

�3x

�7

�8

Hel

pin

g Y

ou

Rem

emb

er

5.M

any

stu

den

ts h

ave

trou

ble

know

ing

wh

eth

er a

n a

bsol

ute

val

ue

ineq

ual

ity

shou

ld b

etr

ansl

ated

in

to a

n a

nd

or a

n o

rco

mpo

un

d in

equ

alit

y.D

escr

ibe

a w

ay t

o re

mem

ber

wh

ich

of t

hes

e ap

plie

s to

an

abs

olu

te v

alu

e in

equ

alit

y.A

lso

desc

ribe

how

to

reco

gniz

e th

edi

ffer

ence

fro

m a

nu

mbe

r li

ne

grap

h.

Sam

ple

an

swer

:If

th

e ab

solu

te v

alu

eq

uan

tity

is f

ollo

wed

by

a �

or

�sy

mb

ol,

the

exp

ress

ion

insi

de

the

abso

lute

val

ue

bar

s m

ust

be

bet

wee

ntw

o n

um

ber

s,so

th

is b

eco

mes

an

and

ineq

ual

ity.

Th

e n

um

ber

lin

e g

rap

h w

ill s

ho

w a

sin

gle

inte

rval

bet

wee

ntw

o n

um

ber

s.If

th

e ab

solu

te v

alu

e q

uan

tity

is f

ollo

wed

by

a

or

�sy

mb

ol,

it b

eco

mes

an

or

ineq

ual

ity,

and

th

e g

rap

h w

ill s

ho

w t

wo

dis

con

nec

ted

inte

rval

s w

ith

arr

ow

s g

oin

g in

op

po

site

dir

ecti

on

s.

�4

�5

�3

�2

�1

01

23

45

�4

�3

�2

�1

01

23

5�

54

©G

lenc

oe/M

cGra

w-H

ill36

Gle

ncoe

Alg

ebra

2

Co

nju

nct

ion

s an

d D

isju

nct

ion

sA

n a

bsol

ute

val

ue

ineq

ual

ity

may

be

solv

ed a

s a

com

pou

nd

sen

ten

ce.

Sol

ve �2

x��

10.

�2x�

�10

mea

ns

2x�

10 a

nd

2x

�10

.

Sol

ve e

ach

in

equ

alit

y.x

�5

and

x

�5.

Eve

ry s

olu

tion

for

�2x�

�10

is

a re

plac

emen

t fo

r x

that

mak

es b

oth

x�

5 an

d x

�

5 tr

ue.

A c

ompo

un

d se

nte

nce

th

at c

ombi

nes

tw

o st

atem

ents

by

the

wor

d an

dis

a

con

jun

ctio

n.

Sol

ve �3

x�

7��

11.

�3x

�7

��11

mea

ns

3x�

7 �

11 o

r 3x

�7

��

11.

Sol

ve e

ach

in

equ

alit

y.3x

�18

or

3x�

�4

x�

6 or

x

��

�4 3�

Eve

ry s

olu

tion

for

th

e in

equ

alit

y is

a r

epla

cem

ent

for

xth

at m

akes

eit

her

x�

6 or

x�

��4 3�

tru

e.

A c

ompo

un

d se

nte

nce

th

at c

ombi

nes

tw

o st

atem

ents

by

the

wor

d or

is

a d

isju

nct

ion

.

Sol

ve e

ach

in

equ

alit

y.T

hen

wri

te w

het

her

th

e so

luti

on i

s a

con

jun

ctio

n o

rd

isju

nct

ion

.

1.�4

x�

242.

�x�

7��

8

x

6 o

r x

��

6;d

isju

nct

ion

x�

15 a

nd

x�

�1;

con

jun

ctio

n

3.�2

x�

5��

14.

�x�

1��

1

x�

�2

and

x

�3;

con

jun

ctio

nx

�2

or

x�

0;d

isju

nct

ion

5.�3

x�

1��

x6.

7 �

�2x�

5

x�

�1 2�an

d x

��1 4� ;

con

jun

ctio

nx

�1

and

x

�1;

con

jun

ctio

n

7.�� 2x �

�1

��7

8.��x

� 34

��

4

x�

12 o

r x

��

16;

dis

jun

ctio

nx

�16

an

d x

�

8;co

nju

nct

ion

9.�8

�x�

2

10.�

5�

2x�

�3

x�

6 o

r x

10

;d

isju

nct

ion

x�

1 an

d x

�4;

con

jun

ctio

n

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

1-6

1-6

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2


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.

11. D

B

B

C

C

A

B

C

D

A

C

�

C

A

B

C

D

B

A

B

D

C

C

D

A

B

A

C

A

D

B

B

Chapter 1 Assessment Answer KeyForm 1 Form 2APage 37 Page 38 Page 39

An

swer

s

(continued on the next page)


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 � x � 0}

B

D

A

B

C

C

A

B

D

C

A

C

A

B

A

D

B

C

A

D

|g � 80| � 5

A

A

D

C

C

B

A

D

A

Chapter 1 Assessment Answer KeyForm 2A (continued) Form 2BPage 40 Page 41 Page 42


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:�75

�

g � number of additionalgames to be won;

�g

1�62

56� � 0.60;

at least 42 games

�1�2�3�4�5�6 0 1 2

{x � �4 x 1} or [�4, 1]

�1�2 0 1 2 4 5 63

{x � x � �2 or x � 6} or(�, �2) � (6, �)

�1�2�3�4 0 1 2 43

all real numbers or (�, �)

1 2 4 5 6 7 83

{n �2 � n 5} or (2, 5]

222120 23 24 26 27 2825

{x � x 24} or (�, 24]

642 8 10 14 16 1812

{t � t � 12} or (12, �)

{�4, 8}

{�5, 2}

11

�92

�

n3 � 10

15v

Additive Identity

Multiplicative Inverse

Q, R

N, W, Z, Q, R

Q, R

$531.25

3.5

�13

�79

�

20

An

swer

s

Chapter 1 Assessment Answer KeyForm 2CPage 43 Page 44

a � number of adult tickets;12.00a � 7.50(a � 8) � 138;4 adults’ tickets and 12 children’s tickets

n � the number;2n � 6 � 28; 11


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B: �5

d � the number of dimes; 0.10d �

0.05(25 � d) � 1.44;at least 4 dimes

�4�8 0 4 8

{x ��5 x 4} or [�5, 4]

6420 8 10

{x �x � 2 or x � 8} or (�, 2) � (8, �)

�1�2 0 1 2 4 5 63

{n�n � �1 or n � 3} or (�, 1) � [3, �)

�1�2�3 0 1 2 43

{x ��1 � x � 1} or (�1, 1)

�1�2 0 1 2 4 5 63

{x �x � 5} or [5, �)

�1�2 0 1 2 4 5 63

{t �t � 2} or (�, 2)

w � width;2[(w � 7) � w] � 38;

width: 6 ft, length: 13 ft

n � number;3n � 1 � 25; 8

{1, 7}

{�2, 1}

7

3

5(7 � n)

10x � 23

Multiplicative Identity

Additive Inverse

Q, R

Q, R

N, W, Z, Q, R

$180

1.5

�3

�161�

5

Chapter 1 Assessment Answer KeyForm 2DPage 45 Page 46


1.

2.

3.

4.

5. a.

b.

c.

d.

e.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

B: more than 1.75 h

a � amount invested instock;

0.08a � 0.06(5000 � a) �3; at least $2500

2 353

43

73

113

103

83

�w ��53

� w 3� or ��53

�, 3��2�4�6 0 2 4 8 106

{x �x � �2 or x � 8} or (�, �2] � (8, �)

0 1� 14� 2

414

34

54

24

�w ��14

� w 1�or ��

14

�, 1�

�2�4 0 2 4 8 10 146

{x �x �4 or x � 10} or (�, �4] � (10, �)

0 1�1 � 13� 2

313

43

23

�y �y ��13

�� or ��, ��13

��43 17

4154

134

194

92

72

�x �x � �147�� or ��

147�, ��

A � �12

�b(18 � b)

� � length;

2�� 14

�� 3�� 2� � 10;

length: 8 meters width: 5 meters

�

a � �2hA� � b

��23

�, 4�

all real numbers

four times the sum of thecube of a number and

twice the same number

6x � 10y � 5

N, W, Z, Q, RQ, R

W, Z, Q, RI, R

Z, Q, R

13.5648 in3

25

�3

An

swer

s

Chapter 1 Assessment Answer KeyForm 3Page 47 Page 48

Sometimes, since when a � �2b, thevalue of theexpression is zero.


Chapter 1 Assessment Answer KeyPage 49, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts of order ofoperations, properties of real numbers, simplifying andevaluating expressions, solving equations and inequalitiesincluding those with absolute value, and graphinginequalities.

• 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 order ofoperations, properties of real numbers, simplifying andevaluating expressions, solving equations and inequalitiesincluding those with absolute value, and graphinginequalities.

• 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 of orderof operations, properties of real numbers, simplifying andevaluating expressions, solving equations and inequalitiesincluding those with absolute value, and graphinginequalities.

• 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 conceptsof order of operations, properties of real numbers,simplifying and evaluating expressions, solving equationsand inequalities including those with absolute value, andgraphing 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

Chapter 1 Assessment Answer KeyPage 49, Open-Ended Assessment

Sample Answers


1a. GIVENAddition Property of Equality Commutative Property of Addition Associative Property of Addition Inverse Property of Addition Identity Property of Addition Distributive Property SUBSTITUTIONMultiplication Property of Equality Associative Property of

Multiplication Inverse Property of Multiplication Identity Property of Multiplication Symmetric Property of Equality

1b. Sample student solution:6(7 � x) � 3 � 9x42 � 6x � 3 � 9x

45 � 6x � 6x � 9x � 6x

�4155�

� �1155x

�

3 � xStudents should note that theirsolutions are considerably brieferthough the answers are the same. Theyshould understand that they did, in fact,use all of the same properties but thatthey applied many of them mentally.

2a. Students may select any negativenumber for k. Their explanations shouldinclude the fact that an absolute valuemay never be less than zero.

2b. The only possible value of k is zero.Students should indicate that the onlynumber that is zero units away from 3on the number line is 3 itself.

2c. Students may select any value for kbetween 0 and 2. They should indicatethat the solution of this inequality willnot contain 5 if the distance from 3 onthe number line is less than 2 units.

3a. Sample word problem:Anoki is packing a box to ship to ascience fair. The box must weigh nomore than 10 pounds. He will put in anexhibit frame that weighs two pounds.How many rocks can he include if eachrock weighs one-fourth of a pound?

3b. {x � x � 32} and x is a whole number; Forthe sample problem, this would meanthat no more than 32 rocks can bepacked.

3c. Students should graph {x � x � 32} andindicate that the graph includesnegative numbers and numbers that are not integers. These numbers have nomeaning in this context. Only 0, 1, 2, …, 32 are possible for the number of rocks.

In addition to the scoring rubric found on page A26, the following sample answers may be used as guidance in evaluating open-ended assessment items.

An

swer

s


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

Page 50 Page 51 Page 52

1. Identity Property

2. rational numbers

3. Symmetric Property

4. Reflexive Property

5. intersection

6. set-builder notation

7. CommutativeProperty

8. Transitive Property

9. compoundinequality

10. absolute value

11. Sample answer: Anirrational number isa real number thatis not rational. Thismeans that anirrational numbercannot be written asa ratio of twointegers.

12. Sample answer: TheTrichotomy Propertysays that if youcompare two realnumbers you willfind that either thefirst one is smallerthan the second,they are equal, orthe first one islarger than thesecond.

1.

2.

3.

4.

5.

Quiz (Lesson 1–3)

Page 51

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

Quiz (Lesson 1–6)

Page 52

1.

2.

3.

4.

5.

�1�2�3�4 0 1 2 43

all real numbers or (�, �)

0 1 2 4 5 6 73

{x �1 x 6} or [1, 6]

�3�11

{x �x � �11 or x � �3}or (�, �11) U (�3, �)

�1 8

{m��1 � m � 8} or (�1, 8)

�1 0 1 2 4 5 63

{x �x � 2 or x � 3} or (�, 2) � (3, �)


�41

8�2

g� � 0.70;

at least 17 games

0 1� 15

15

25

35

45

65

�x �x � �15

��

�

��43

�, 2�

5

d � number of daysrunning 7 miles;

8 � 7d � 99; 13 days

x � �y

m� b�

��110�

�16

�

B

8v � �76

�

I, R

47.8 m

72

13


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

16.

17.


�57

16�2

g� � 0.65; at least

49 games

n � the number;48 � 3n � 36; 4

�1�2�3�4�5�6�7 0 1

{x �x � �7 or x � 1} or (�, �7) � (1, �)

�1�2�3�4 0 1 2 43

all real numbers (�, �)

0 1 2 4 5 6 7 83

{t �t � 5} or [5, �)

�

{�4, 5}

8

n2 � n3

7x � 2

N, W, Z, Q, R

77

17

20

0.49

�36

14

b � �2ah

�11.5

t � the number ofstudents’ tickets sold;5(295 � t) � 2t � 950;175 students’ tickets

20

3x � 7

A

C

A

B

D

Chapter 1 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 53 Page 54

An

swer

s


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12. 13.

14. 15.

16.

17.

18.

19. DCBA

DCBA

DCBA

DCBA

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 9 / 3

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

. 0 9

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

1 0

0 0 0

.. ./ /

.

99 9 987654321

87654321

87654321

87654321

3 5

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

HGFE

DCBA

Chapter 1 Assessment Answer KeyStandardized Test Practice

Page 55 Page 56

Documents

Chapter 1 Resource Masters - ktlmathclass.weebly.comktlmathclass.weebly.com/.../5/7/6/25760552/alg_2_resource_ws_ch_1.pdf©Glencoe/McGraw-Hill v Glencoe Algebra 2 Assessment Options