Chapter 1 Resource Masters - Math Class - Home©Glencoe/McGraw-Hill v Glencoe Algebra 2 Assessment Options The assessment masters in the Chapter 1 Resource Masters offer a wide range

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

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

Glencoe/M cGraw-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 mg6 4 1 3

5. 370–,000 km 6. 370,00–0 km 7. 9.7 ( 104 g 8. 3.20 ( 10!2 g3 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 ( 2535 m2 23.1 63 cm

12. 11.01 mm ( 11 13. 908 yd ! 0.5 14. 38.6 m ( 4.0 m121.1 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 " 105c15. (7 ! 2.1x)3 " 2(3.5x ! 6) 16. (18 ! 6n " 12 " 3n)

0.7x " 9 20 ! 2n17. 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) " 8Comm. (") Assoc. (")

11. 5(3x " y) $ 5(3x " 1y) 12. 7n " 2n $ (7 " 2)nMult. Iden. Distributive

13. 3(2x)y $ (3 % 2)(xy) 14. 3x % 2y $ 3 % 2 % x % y 15. (6 " !6)y $ 0yAssoc. (') 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 _____

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Reading to Learn MathematicsProperties of Real Numbers

NAME ______________________________________________ DATE ____________ PERIOD _____

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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 _____

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Example 1Example 1

Example 2Example 2

Study Guide and InterventionSolving Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

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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

&ca&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 _____

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

4n " 7 5n ! 83. 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 $ 5

are 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 ! 2)r2%%2)r

x " 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 number

5(m " 1) 2y2 ! 1

Write a verbal expression to represent each equation. 5–8. Sample answers 5. 5 ! 2x $ 4 6. 3y $ 4y3

are 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 )%%w2

1&2

h " gt2%t

3c ! 1%2

2d " 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 _____

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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 _____

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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⏐ ! ⏐!2x⏐if x $ 6.

!4 ! !2x $ !4 ! !2 % 6$ !4 ! !12$ 4 ! 12$ !8

Evaluate ⏐2x ! 3y⏐if 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 ! 35yz !39

10. 5w " 2z ! 2y 34 11. z ! 42z " y !40 12. 10 ! xw 2

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

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

19. zz " xx !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 $ 172(10) ! 3 $ 17

20 ! 3 $ 1717 $ 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 $ 23x ! 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 _____

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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. 24w 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. 23w $ 12 {!2, 2}

21. 7x ! 3x " 2 $ 18 {!4, 4} 22. 47 ! y ! 1 $ 11 {4, 10}

23. 3n ! 2 $ % , & 24. 8d ! 4d " 5 $ 13 {!2, 2}

25. !56a " 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 " 4 123. !10d " a !15 4. 17c " 3b ! 5 114

5. !610a ! 12 !132 6. 2b ! 1 ! !8b " 5 !52

7. 5a ! 7 " 3c ! 4 23 8. 1 ! 7c ! a 33

9. !30.5c " 2 ! !0.5b !17.5 10. 4d " 5 ! 2a 12.6

11. a ! b " b ! a 14 12. 2 ! 2d ! 3b !19.2


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

15. 2y ! 3 $ 29 {!13, 16} 16. 7x " 3 $ 42 {!9, 3}

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

19. !34x ! 9 $ 24 , 20. !65 ! 2y $ !9 % , &21. 8 " p $ 2p ! 3 {11} 22. 4w ! 1 $ 5w " 37 {!4}

23. 42y ! 7 " 5 $ 9 {3, 4} 24. !27 ! 3y ! 6 $ !14 %1, &25. 24 ! s $ !3s {!8} 26. 5 ! 32 " 2w $ !7 {!3, 1}

27. 52r " 3 ! 5 $ 0 {!2, !1} 28. 3 ! 52d ! 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 _____

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Reading to Learn MathematicsSolving Absolute Value Equations

NAME ______________________________________________ DATE ____________ PERIOD _____

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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 ofthese 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 $ 5.5 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 there would 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 ! 3x " 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


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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 0 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 {xx ) 16}.

212019181716151413

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

17 ! 3w + 3517 ! 3w ! 17 + 35 ! 17

!3w + 18w 0 !6

The solution set is (!1, !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) 0 84 2. 3(9z " 4) ) 35z ! 4 3. 5(12 ! 3n) , 165

{a⏐a / 3} or (!∞, 3] {z⏐z 0 2} or (!∞, 2) {n⏐n - !7} or (!7, "∞)

4. 18 ! 4k , 2(k " 21) 5. 4(b ! 7) " 6 , 22 6. 2 " 3(m " 5) + 4(m" 3)

{k⏐k - !4} or (!4, "∞) {b⏐b 0 11} or (!∞, 11) {m⏐m / 5} or (!∞, 5]

7. 4x ! 2 ) !7(4x ! 2) 8. (2y ! 3) ) y " 2 9. 2.5d " 15 0 75

%x⏐x - & or # , "∞$ {y⏐y 0 !9} or (!∞, !9) {d⏐d / 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.

0 - / .

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) 0 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


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

1. + 2 {z⏐z / !8} or (!∞, !8] 2. 3a " 7 0 16 {a⏐a / 3} or (!∞, 3]

3. 16 , 3q " 4 {q⏐q - 4} or (4, ∞) 4. 20 ! 3s ) 7s {s⏐s 0 2} or (!∞, 2)

5. 3x + !9 {x⏐x . !3} or [!3, ∞) 6. 4b ! 9 0 7 {b⏐b / 4} or (!∞, 4]

7. 2z , !9 " 5z {z⏐z - 3} or (3, ∞) 8. 7f ! 9 ) 3f ! 1 {f⏐f - 2} or (2, ∞)

9. !3s ! 8 0 5s {s⏐s . !1} or [!1, ∞) 10. 7t ! (t ! 4) 0 25 %t⏐t / & or #!∞, '

11. 0.7m " 0.3m + 2m ! 4 {m⏐m / 4} 12. 4(5x " 7) 0 13 %x⏐x / ! & oror (!∞, 4]

#!∞, ! '13. 1.7y ! 0.78 ) 5 {y⏐y - 3.4} 14. 4x ! 9 ) 2x " 1 {x⏐x - 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 0 42; n 0 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 0 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 {x⏐x . 2} or [2, ∞) 2. 23 ! 4u , 11 {u⏐u - 3} or (3, ∞)

3. !16 ! 8r + 0 {r⏐r / !2} or (!∞, !2] 4. 14s , 9s " 5 {s⏐s 0 1} or (!∞, 1)

5. 9x ! 11 ) 6x ! 9 %x⏐x - & or # , ∞$ 6. !3(4w ! 1) ) 18 %w⏐w 0 ! &or #!∞, ! $

7. 1 ! 8u 0 3u ! 10 {u⏐u . 1} or [1, ∞) 8. 17.5 , 19 ! 2.5x {x⏐x 0 0.6} or (!∞, 0.6)

9. 9(2r ! 5) ! 3 , 7r ! 4 {r⏐r 0 4} 10. 1 " 5(x ! 8) 0 2 ! (x " 5) {x⏐x / 6} or (!∞, 4) or (!∞, 6]

11. + !3.5 {x⏐x . !1} or [!1, ∞) 12. q ! 2(2 ! q) 0 0 %q⏐q / & or #!∞, '

13. !36 ! 2(w " 77) ) !4(2w " 52) 14. 4n ! 5(n ! 3) ) 3(n " 1) ! 4 {w⏐w - !3} or (!3, ∞) {n⏐n 0 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 0 !20

16. Four times the sum of twice a number and !3 is less than 5.5 times that same number.4[2n " (!3)] 0 5.5n; n 0 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 0 600 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

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

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

4%3

4%3

4x ! 3&2

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

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


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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 0 400 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. {xx , !3} (!∞, !3)b. {xx + 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. 0 600b. A student may enroll in no more than six courses each semester. / 6c. To participate in a concert, you must be willing to attend at least ten rehearsals. . 10d. 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 0 !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


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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 0 !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 0 2x " 5 and 2x " 5 0 19!8 0 2x 2x 0 14!4 0 x x 0 7!4 0 x 0 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 0 !93y + 9 or 2y 0 !8y + 3 or y 0 !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 0 14 2. 3a " 8 , 23 or a ! 6 ) 7

{x⏐!4 0 x / 4} {a⏐a 0 5 or a - 52}

3. 18 , 4x ! 10 , 50 4. 5k " 2 , !13 or 8k ! 1 ) 19

{x⏐7 0 x 0 15} {k⏐k 0 !3 or k - 2.5}

5. 100 0 5y ! 45 0 225 6. b ! 2 ) 10 or b " 5 , !4

{y⏐29 / y / 54} {b⏐b 0 !12 or b - 18}

7. 22 , 6w !2 , 82 8. 4d ! 1 ) !9 or 2d " 5 , 11

{w⏐4 0 w 0 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 0 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⏐ 0 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 0 x 0 & 2. 4s " 1 ) 27 {s⏐s 0 !6.5 or s - 6.5}

3. ! 3 0 5 {c⏐!4 / c / 16} 4. a " 9 + 30 {a⏐a / !39 or a . 21}

5. 2f ! 11 ) 9 {f⏐f 0 1 or f - 10} 6. 5w " 2 , 28 {w⏐!6 0 w 0 5.2}

7. 10 ! 2k , 2 {k⏐4 0 k 0 6} 8. ! 5 " 2 ) 10 {x⏐x 0 !6 or x - 26}

9. 4b ! 11 , 17 %b⏐! 0 b 0 7& 10. 100 ! 3m ) 20 %m⏐m 0 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


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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⏐ 0 5

3. all numbers less than !1 or greater 4. all numbers between !6 and 6 ⏐n⏐ 0 6than 1 ⏐n⏐ - 1

Write an absolute value inequality for each graph.

5. ⏐n⏐ 0 1 6. ⏐n⏐ . 4

7. ⏐n⏐ / 3 8. ⏐n⏐ - 2.5


9. 2c " 1 ) 5 or c , 0 {c⏐c - 2 10. !11 0 4y ! 3 0 1 {y⏐!2 / y / 1}or c 0 0}

11. 10 ) !5x ) 5 {x⏐!2 0 x 0 !1} 12. 4a + !8 or a , !3 {a⏐a . !2or a 0 !3}

13. 8 , 3x " 2 0 23 {x⏐2 0 x / 7} 14. w ! 4 0 10 or !2w 0 6 all realnumbers

15. t + 3 {t⏐t / !3 or t . 3} 16. 6x , 12 {x⏐!2 0 x 0 2}

17. !7r ) 14 {r⏐r 0 !2 or r - 2} 18. p " 2 0 !2 ,

19. n ! 5 , 7 {n⏐!2 0 n 0 12} 20. h " 1 + 5 {h⏐h / !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⏐ 0


5. !8 0 3y ! 20 , 52 {y⏐4 / y 0 24} 6. 3(5x ! 2) , 24 or 6x ! 4 ) 4 " 5x{x⏐x 0 2 or x - 8}

7. 2x ! 3 ) 15 or 3 ! 7x , 17 {x⏐x - !2} 8. 15 ! 5x 0 0 and 5x " 6 + !14 {x⏐x . 3}

9. 2w + 5 %w⏐w / ! or w . & 10. y " 5 , 2 {x⏐!7 0 x 0 !3}

11. x ! 8 + 3 {x⏐x / 5 or x . 11} 12. 2z ! 2 0 3 %z⏐! / z / &

13. 2x " 2 ! 7 0 !5 {x⏐!2 / x / 0} 14. x ) x ! 1 all real numbers

15. 3b " 5 0 !2 , 16. 3n ! 2 ! 2 , 1 %n⏐! 0 n 0 &

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; {r⏐r 0 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; {v⏐14.42 / v / 14.58}

4 3 2 1 0 1 2 3 40 1 2 3 4 1 2 3 4

5%3

1%3

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 40 2 4 6 8 10 12 14 16

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

4 3 2 1 0 1 2 3 4

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


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on

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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 0 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.{x⏐x / !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 0 !2

4. Write a statement equivalent to 3x " 7 , 8 that does not use the absolute valuesymbol. !8 0 3x " 7 0 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 0 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 ( 0 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 0 !11.

Solve each inequality. 3x + 18 or 3x 0 !4

x + 6 or x 0 !&43&

Every solution for the inequality is a replacement for x that makes either x + 6 or x 0 !&

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 , 0 8

x - 6 or x 0 !6; disjunction x / 15 and x . !1; conjunction

3. ,2x " 5 , , 1 4. ,x ! 1 , + 1

x 0 !2 and x - !3; conjunction x . 2 or x / 0; disjunction

5. ,3x ! 1 , 0 x 6. 7 ! ,2x , ) 5

x / %12% and x . %

14%; conjunction x 0 1 and x - !1; conjunction

7. , &2x& " 1 , + 7 8. ,&x !

34

& , , 4

x . 12 or x / !16; disjunction x 0 16 and x - !8; conjunction

9. ,8 ! x , ) 2 10. ,5 ! 2x , 0 3

x 0 6 or x - 10; disjunction x . 1 and x / 4; conjunction

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

1-61-6

Example 1Example 1

Example 2Example 2

© Glencoe/McGraw-Hill A2 Glencoe Algebra 2

Answers (Lesson 1-1)

Stu

dy

Gu

ide

and I

nte

rven

tion

Expr

essi

ons

and

Form

ulas

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

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____

____

__PE

RIO

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___

1-1

1-1

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Alg

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2

Lesson 1-1

Ord

er o

f O

per

atio

ns

1.Si

mpl

ify th

e ex

pres

sion

s in

side

gro

upin

g sy

mbo

ls.

Ord

er o

f2.

Eval

uate

all

pow

ers.

Ope

ratio

ns3.

Do

all m

ultip

licat

ions

and

div

isio

ns fr

om le

ft to

righ

t.4.

Do

all a

dditi

ons

and

subt

ract

ions

from

left

to ri

ght.

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)21

518

.!

6

19.a

c!

bd31

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.(b

!c)

2"

4a81

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."

6b!

5c!

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22.3

!"!

b!

2123

.cd

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24.d

(a"

c)!

6.1

25.a

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#c

7.45

26.b

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#d

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27.

!d

8.7

a& b

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b & dc & d

a & dc2!

1& b

!d

ab & d

1 % 26 "

9 #

3 "

15&

&8

!2

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4 #

2&

&4

#6

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16 "

23#

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1 & 4

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Form

ula

sA

for

mu

lais

a m

athe

mat

ical

sen

tenc

e th

at u

ses

vari

able

s to

exp

ress

the

rela

tion

ship

bet

wee

n ce

rtai

n qu

anti

ties

.If

you

know

the

val

ue o

f ev

ery

vari

able

exc

ept

one

in a

for

mul

a,yo

u ca

n us

e su

bsti

tuti

on a

nd t

he o

rder

of

oper

atio

ns t

o fi

nd t

he v

alue

of

the

unkn

own

vari

able

.

To c

alcu

late

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pap

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ded

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man

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of p

aper

mu

st y

ou b

uy

to p

rin

t 17

2 co

pie

s of

a 2

5-p

age

book

let?

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titu

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$17

2 an

d p

$25

into

the

for

mul

a r

$.

r$ $ $

8.6

You

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s of

pap

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ou w

ill n

eed

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rea

ms

to p

rint

172

cop

ies.

For

Exe

rcis

es 1

–3,u

se t

he

foll

owin

g in

form

atio

n.

For

a sc

ienc

e ex

peri

men

t,Sa

rah

coun

ts t

he n

umbe

r of

bre

aths

nee

ded

for

her

to b

low

up

abe

ach

ball.

She

will

the

n fi

nd t

he v

olum

e of

the

bea

ch b

all i

n cu

bic

cent

imet

ers

and

divi

deby

the

num

ber

of b

reat

hs t

o fi

nd t

he a

vera

ge v

olum

e of

air

per

bre

ath.

1.H

er b

each

bal

l has

a r

adiu

s of

9 in

ches

.Fir

st s

he c

onve

rts

the

radi

us t

o ce

ntim

eter

sus

ing

the

form

ula

C$

2.54

I,w

here

Cis

a le

ngth

in c

enti

met

ers

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Iis

the

sam

e le

ngth

in in

ches

.How

man

y ce

ntim

eter

s ar

e th

ere

in 9

inch

es?

22.8

6 cm

2.T

he v

olum

e of

a s

pher

e is

giv

en b

y th

e fo

rmul

a V

$'

r3,w

here

Vis

the

vol

ume

of t

he

sphe

re a

nd r

is it

s ra

dius

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t is

the

vol

ume

of t

he b

each

bal

l in

cubi

c ce

ntim

eter

s?(U

se 3

.14

for

'.)

50,0

15 c

m3

3.Sa

rah

take

s 40

bre

aths

to

blow

up

the

beac

h ba

ll.W

hat

is t

he a

vera

ge v

olum

e of

air

per

brea

th?

abou

t 125

0 cm

3

4.A

per

son’

s ba

sal m

etab

olic

rat

e (o

r B

MR

) is

the

num

ber

of c

alor

ies

need

ed t

o su

ppor

t hi

sor

her

bod

ily f

unct

ions

for

one

day

.The

BM

R o

f an

80-

year

-old

man

is g

iven

by

the

form

ula

BM

R $

12w

!(0

.02)

(6)1

2w,w

here

wis

the

man

’s w

eigh

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pou

nds.

Wha

t is

the

BM

R o

f an

80-

year

-old

man

who

wei

ghs

170

poun

ds?

1795

cal

orie

s

4 & 3

43,0

00&

500

(172

)(25

)&

&50

0

np & 500

np

% 500

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Expr

essi

ons

and

Form

ulas

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-1

1-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Expr

essi

ons

and

Form

ulas

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-1

1-1

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Fin

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1.18

#2

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272.

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rees

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sius

.Wha

t is

the

tem

pera

ture

in k

elvi

ns w

hen

the

tem

pera

ture

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5 de

gree

s C

elsi

us?

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TUR

ET

he f

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t w

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aith

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Pra

ctic

e (A

vera

ge)

Expr

essi

ons

and

Form

ulas

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-1

1-1



Rea

din

g t

o L

earn

Math

emati

csEx

pres

sion

s an

d Fo

rmul

as

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-1

1-1

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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

ucti

on t

o L

esso

n 1-

1 at

the

top

of

page

6 in

you

r te

xtbo

ok.

•N

urse

s us

e th

e fo

rmul

a F

$to

con

trol

the

flo

w r

ate

for

IVs.

Nam

e

the

quan

tity

tha

t ea

ch o

f th

e va

riab

les

in t

his

form

ula

repr

esen

ts a

nd t

heun

its

in w

hich

eac

h is

mea

sure

d.

Fre

pres

ents

the

an

d is

mea

sure

d in

pe

r m

inut

e.

Vre

pres

ents

the

of

sol

utio

n an

d is

mea

sure

d in

.

dre

pres

ents

the

an

d is

mea

sure

d in

pe

r m

illili

ter.

tre

pres

ents

an

d is

mea

sure

d in

.

•W

rite

the

exp

ress

ion

that

a n

urse

wou

ld u

se t

o ca

lcul

ate

the

flow

rat

e of

an

IV if

a d

octo

r or

ders

135

0 m

illili

ters

of

IV s

alin

e to

be

give

n ov

er

8 ho

urs,

wit

h a

drop

fac

tor

of 2

0 dr

ops

per

mill

ilite

r.D

o no

t fi

nd t

he v

alue

of t

his

expr

essi

on.

Rea

din

g t

he

Less

on

1.T

here

is a

cus

tom

ary

orde

r fo

r gr

oupi

ng s

ymbo

ls.B

rack

ets

are

used

out

side

of

pare

nthe

ses.

Bra

ces

are

used

out

side

of

brac

kets

.Ide

ntif

y th

e in

nerm

ost

expr

essi

on(s

) in

each

of

the

follo

win

g ex

pres

sion

s.

a.[(

3 !

22) "

8] #

4(3

!22

)b.

9 !

[5(8

!6)

"2(

10 "

7)]

(8 !

6) a

nd (1

0 "

7)c.

{14

![8

"(3

!12

)2]}

#(6

3!

100)

(3 !

12)

2.R

ead

the

follo

win

g in

stru

ctio

ns.T

hen

use

grou

ping

sym

bols

to

show

how

the

inst

ruct

ions

can

be p

ut in

the

for

m o

f a

mat

hem

atic

al e

xpre

ssio

n.

Mul

tipl

y th

e di

ffer

ence

of

13 a

nd 5

by

the

sum

of

9 an

d 21

.Add

the

res

ult

to 1

0.T

hen

divi

de w

hat

you

get

by 2

.[(1

3 !

5)(9

"21

) "10

] #2

3.W

hy is

it im

port

ant

for

ever

yone

to

use

the

sam

e or

der

of o

pera

tion

s fo

r ev

alua

ting

expr

essi

ons?

Sam

ple

answ

er:I

f eve

ryon

e di

d no

t use

the

sam

e or

der o

fop

erat

ions

,diff

eren

t peo

ple

mig

ht g

et d

iffer

ent a

nsw

ers.

Hel

pin

g Y

ou

Rem

emb

er4.

Thi

nk o

f a

phra

se o

r se

nten

ce t

o he

lp y

ou r

emem

ber

the

orde

r of

ope

rati

ons.

Sam

ple

answ

er:P

leas

e ex

cuse

my

dear

Aun

t Sal

ly.(

pare

nthe

ses;

expo

nent

s;m

ultip

licat

ion

and

divi

sion

;add

ition

and

sub

trac

tion)

1350

'20

%%

8 '

60

min

utes

time

drop

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ctor

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Sign

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igits

All

mea

sure

men

ts a

re a

ppro

xim

atio

ns.T

he s

ign

ific

ant

dig

its

of a

n ap

prox

imat

enu

mbe

r ar

e th

ose

whi

ch in

dica

te t

he r

esul

ts o

f a

mea

sure

men

t.Fo

r ex

ampl

e,th

em

ass

of a

n ob

ject

,mea

sure

d to

the

nea

rest

gra

m,i

s 21

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is a

repe

atin

gde

cim

al b

ecau

se th

ere

is a

blo

ck o

f dig

its,5

7,th

at re

peat

s fo

reve

r,so

this

num

ber i

s ra

tiona

l.Th

e nu

mbe

r 0.0

1001

0001

… is

a n

on-r

epea

ting

deci

mal

bec

ause

,alth

ough

the

digi

ts fo

llow

a p

atte

rn,t

here

is n

o bl

ock

of d

igits

that

repe

ats.

So th

is n

umbe

r is

an ir

ratio

nal n

umbe

r.2.

Wri

te t

he A

ssoc

iati

ve P

rope

rty

of A

ddit

ion

in s

ymbo

ls.T

hen

illus

trat

e th

is p

rope

rty

byfi

ndin

g th

e su

m 1

2 "

18 "

45 in

tw

o di

ffer

ent

way

s.(a

"b)

"c

$a

"(b

"c)

;Sa

mpl

e an

swer

:(12

"18

) "45

$30

"45

$75

;12

"(1

8 "

45) $

12 "

63 $

753.

Con

side

r th

e eq

uati

ons

(a%

b) %

c$

a%

(b%

c) a

nd (

a%

b) %

c$

c%

(a%

b).O

ne o

f th

eeq

uati

ons

uses

the

Ass

ocia

tive

Pro

pert

y of

Mul

tipl

icat

ion

and

one

uses

the

Com

mut

ativ

eP

rope

rty

of M

ulti

plic

atio

n.H

ow c

an y

ou t

ell w

hich

pro

pert

y is

bei

ng u

sed

in e

ach

equa

tion

?Th

e fir

st e

quat

ion

uses

the

Ass

ocia

tive

Prop

erty

of

Mul

tiplic

atio

n.Th

e qu

antit

ies

a,b,

and

car

e us

ed in

the

sam

e or

der,

but

they

are

gro

uped

diff

eren

tly o

n th

e tw

o si

des

of th

e eq

uatio

n.Th

e se

cond

equa

tion

uses

the

quan

titie

s in

diff

eren

t ord

ers

on th

e tw

o si

des

of th

eeq

uatio

n.So

the

seco

nd e

quat

ion

uses

the

Com

mut

ativ

e Pr

oper

ty o

fM

ultip

licat

ion.

Hel

pin

g Y

ou

Rem

emb

er4.

How

can

the

mea

ning

s of

the

wor

ds c

omm

uter

and

asso

ciat

ion

help

you

to

rem

embe

r th

edi

ffer

ence

bet

wee

n th

e co

mm

utat

ive

and

asso

ciat

ive

prop

erti

es?

Sam

ple

answ

er:

A c

omm

uter

is s

omeo

ne w

ho tr

avel

s ba

ck a

nd fo

rth

to w

ork

or a

noth

erpl

ace,

and

the

com

mut

ativ

e pr

oper

ty s

ays

you

can

switc

h th

e or

derw

hen

two

num

bers

that

are

bei

ng a

dded

or m

ultip

lied.

An

asso

ciat

ion

is a

grou

p of

peo

ple

who

are

con

nect

ed o

r uni

ted,

and

the

asso

ciat

ive

prop

erty

say

s th

at y

ou c

an s

witc

h th

e gr

oupi

ngw

hen

thre

e nu

mbe

rs a

read

ded

or m

ultip

lied.

©G

lenc

oe/M

cGra

w-H

ill12

Gle

ncoe

Alg

ebra

2

Prop

ertie

s of

a G

roup

A s

et o

f nu

mbe

rs f

orm

s a

grou

p w

ith

resp

ect

to a

n op

erat

ion

if f

or t

hat

oper

atio

nth

e se

t ha

s (1

) th

e C

losu

re P

rope

rty,

(2)

the

Ass

ocia

tive

Pro

pert

y,(3

) a m

embe

rw

hich

is a

n id

enti

ty,a

nd (4

) an

inve

rse

for

each

mem

ber

of t

he s

et.

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 n

umbe

rs in

the

set

,is

a"

bin

the

set

? 0

"1

$1,

and

1 is

in t

he s

et;0

"2

$2,

and

2 is

in t

he s

et;a

nd s

o on

.The

set

has

clos

ure

for

addi

tion

.

Ass

ocia

tive

Pro

per

ty:

For

all n

umbe

rs in

the

set

,doe

s a

"(b

"c)

$(a

"b)

"c?

0

"(1

"2)

$(0

"1)

"2;

1 "

(2 "

3) $

(1 "

2) "

3;an

d so

on.

The

set

is a

ssoc

iati

ve f

or a

ddit

ion.

Iden

tity

:Is

the

re s

ome

num

ber,

i,in

the

set

suc

h th

at i

"a

$a

$a

"i

for

alla

? 0

"1

$1

$1

"0;

0 "

2 $

2 $

2 "

0;an

d so

on.

The

iden

tity

for

add

itio

n is

0.

Inve

rse:

Doe

s ea

ch n

umbe

r,a,

have

an

inve

rse,

a *,s

uch

that

a *

"a

$a

"a*

$i?

The

inte

ger

inve

rse

of 3

is !

3 si

nce

!3

"3

$0,

and

0 is

the

iden

tity

for

add

itio

n.B

ut t

he s

et d

oes

not

cont

ain

!3.

The

refo

re,t

here

is n

o in

vers

e fo

r 3.

The

set

is n

ot a

gro

up w

ith

resp

ect

to a

ddit

ion

beca

use

only

thr

ee o

f 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 s

et h

as c

losu

re f

or m

ulti

plic

atio

n.

Ass

ocia

tive

Pro

per

ty:

(!1)

[(!

1)(!

1)]

$(!

1)(1

) $

!1;

and

so o

nT

he s

et is

ass

ocia

tive

for

mul

tipl

icat

ion.

Iden

tity

:1(

!1)

$!

1;1(

1) $

1T

he id

enti

ty f

or m

ulti

plic

atio

n is

1.

Inve

rse:

!1

is t

he in

vers

e of

!1

sinc

e (!

1)(!

1) $

1,an

d 1

is t

he id

enti

ty.

1 is

the

inve

rse

of 1

sin

ce (

1)(1

) $1,

and

1 is

the

iden

tity

.E

ach

mem

ber

has

an in

vers

e.

The

set

{!1,

1} is

a g

roup

wit

h re

spec

t to

mul

tipl

icat

ion

beca

use

all f

our

prop

erti

es h

old.

Tell

wh

eth

er t

he

set

form

s a

grou

p w

ith

res

pec

t to

th

e gi

ven

op

erat

ion

.

1.{i

nteg

ers}

,add

itio

nye

s2.

{int

eger

s},m

ulti

plic

atio

nno

3.'&1 2& ,

&2 2& ,&3 2& ,

…(,a

ddit

ion

no4.

{mul

tipl

es o

f 5}

,mul

tipl

icat

ion

no

5.{x

,x2 ,

x3,x

4 ,…

} ad

diti

onno

6.{#

1$,#

2$,#

3$,…

},m

ulti

plic

atio

nno

7.{i

rrat

iona

l num

bers

},ad

diti

onno

8.{r

atio

nal n

umbe

rs},

addi

tion

yes

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-2

1-2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2



Stu

dy

Gu

ide

and I

nte

rven

tion

Solv

ing

Equa

tions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

The

cha

rt s

ugge

sts

som

e w

ays

tohe

lp y

ou t

rans

late

wor

d ex

pres

sion

s in

to a

lgeb

raic

exp

ress

ions

.Any

lett

er c

an b

e us

ed t

ore

pres

ent

a nu

mbe

r th

at is

not

kno

wn.

Wor

d Ex

pres

sion

Ope

ratio

n

and,

plu

s, s

um, i

ncre

ased

by,

mor

e th

anad

ditio

n

min

us, d

iffer

ence

, dec

reas

ed b

y, le

ss th

ansu

btra

ctio

n

times

, pro

duct

, of (

as in

of

a n

umbe

r)m

ultip

licat

ion

divi

ded

by, q

uotie

ntdi

visi

on

1 & 2

Wri

te a

n a

lgeb

raic

exp

ress

ion

to

rep

rese

nt

18 l

ess

than

the

quot

ien

t of

a n

um

ber

and

3.

!18

n & 3

Wri

te a

ver

bal s

ente

nce

to

rep

rese

nt

6(n

!2)

$14

.

Six

tim

es t

he d

iffe

renc

e of

a n

umbe

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

umbe

r an

d 25

6n"

25

2.fo

ur t

imes

the

sum

of

a nu

mbe

r an

d 3

4(n

"3)

3.7

less

tha

n fi

ftee

n ti

mes

a n

umbe

r15

n !

7

4.th

e di

ffer

ence

of

nine

tim

es a

num

ber

and

the

quot

ient

of

6 an

d th

e sa

me

num

ber9n

!

5.th

e su

m o

f 10

0 an

d fo

ur t

imes

a n

umbe

r10

0 "

4n

6.th

e pr

oduc

t of

3 a

nd t

he s

um o

f 11

and

a n

umbe

r3(

11 "

n)

7.fo

ur t

imes

the

squ

are

of a

num

ber

incr

ease

d by

fiv

e ti

mes

the

sam

e nu

mbe

r4n

2"

5n

8.23

mor

e th

an t

he p

rodu

ct o

f 7

and

a nu

mbe

r7n

"23

Wri

te a

ver

bal

sen

ten

ce t

o re

pre

sen

t ea

ch e

quat

ion

.Sa

mpl

e an

swer

s ar

e gi

ven.

9.3n

!35

$79

The

diffe

renc

e of

thre

e tim

es a

num

ber a

nd 3

5 is

equ

al to

79.

10.2

(n3

"3n

2 ) $

4nTw

ice

the

sum

of t

he c

ube

of a

num

ber a

nd th

ree

times

the

squa

re o

f the

num

ber i

s eq

ual t

o fo

ur ti

mes

the

num

ber.

11.&

n5 "n

3&

$n

!8

The

quot

ient

of f

ive

times

a n

umbe

r and

the

sum

of t

henu

mbe

rand

3 is

equ

al to

the

diffe

renc

e of

the

num

ber 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 ca

n so

lve

equa

tion

s by

usi

ng a

ddit

ion,

subt

ract

ion,

mul

tipl

icat

ion,

or d

ivis

ion.

Add

ition

and

Sub

trac

tion

For a

ny re

al n

umbe

rs a

, b, a

nd c

,if a

$b,

Prop

ertie

s of

Equ

ality

then

a"

c$

b"

can

d a

!c

$b

!c.

Mul

tiplic

atio

n an

d D

ivis

ion

For a

ny re

al n

umbe

rs a

, b, a

nd c

,if a

$b,

Prop

ertie

s of

Equ

ality

then

a%

c$

b%

can

d, if

cis

not

zer

o,

$.

b & ca & c

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Solv

ing

Equa

tions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

equ

atio

n.C

hec

k y

our

solu

tion

.

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

equ

atio

n o

r fo

rmu

la f

or t

he

spec

ifie

d v

aria

ble.

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) $

108,

for

jj$

18 "

23.3

.5s

!42

$14

t,fo

r s

s$

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

Solv

ing

Equa

tions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

num

ber,

incr

ease

d by

72.

8 le

ss t

han

5 ti

mes

a n

umbe

r

4n"

75n

!8

3.6

tim

es t

he s

um o

f a

num

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 d

iffe

renc

e of

4 a

nd a

num

ber

3(4

!n)

6.th

e pr

oduc

t of

!11

and

the

squ

are

of a

num

ber

!11

n2

Wri

te a

ver

bal e

xpre

ssio

n t

o re

pres

ent

each

equ

atio

n.

7–10

.Sam

ple

answ

ers

7.

n!

8 $

168.

8 "

3x$

5ar

e gi

ven.

The

diffe

renc

e of

a n

umbe

r Th

e su

m o

f 8 a

nd 3

tim

es a

an

d 8

is 1

6.nu

mbe

r is

5.9.

b2"

3 $

b10

.$

2 !

2y

Thre

e ad

ded

to th

e sq

uare

of

A n

umbe

r div

ided

by

3 is

the

a nu

mbe

r is

the

num

ber.

diffe

renc

e of

2 a

nd tw

ice

the

num

ber.

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

,the

n a

$10

.12

.If

d"

1 $

f,th

en d

$f

!1.

Tran

sitiv

e ($

)Su

btra

ctio

n ($

)13

.If

!7x

$14

,the

n 14

$!

7x.

14.I

f (8

"7)

r$

30,t

hen

15r

$30

.Sy

mm

etric

($)

Subs

titut

ion

($)

Sol

ve e

ach

equ

atio

n.C

hec

k y

our

solu

tion

.

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

equ

atio

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or y

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nt

each

ver

bal

exp

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ion

.

1.2

mor

e th

an t

he q

uoti

ent

of a

num

ber

and

52.

the

sum

of

two

cons

ecut

ive

inte

gers

"2

n"

(n"

1)3.

5 ti

mes

the

sum

of

a nu

mbe

r an

d 1

4.1

less

tha

n tw

ice

the

squa

re o

f a

num

ber

5(m

"1)

2y2

!1

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te a

ver

bal e

xpre

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n t

o re

pres

ent

each

equ

atio

n.

5–8.

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ple

answ

ers

5.

5 !

2x$

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3y$

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are

give

n.

The

diffe

renc

e of

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nd tw

ice

a Th

ree

times

a n

umbe

r is

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es

num

ber i

s 4.

the

cube

of t

he n

umbe

r.7.

3c$

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3(2m

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quot

ient

Th

ree

times

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r is

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e th

eof

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r and

5 is

3 ti

mes

the

diffe

renc

e of

the

num

ber a

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.su

m o

f tw

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.

Nam

e th

e p

rop

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ust

rate

d b

y ea

ch s

tate

men

t.

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ach

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k y

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tion

.

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v$

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ine

a va

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rite

an

equ

atio

n,a

nd

sol

ve t

he

pro

blem

.

25.G

EOM

ETRY

The

leng

th o

f a

rect

angl

e is

tw

ice

the

wid

th.F

ind

the

wid

th if

the

peri

met

er is

60

cent

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ers.

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d th

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t go

lfing

.Tw

o of

the

frie

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rent

ed c

lubs

for

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ach.

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l cos

t of

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ren

ted

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d th

e gr

een

fees

for

each

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was

$76

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t w

as t

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ost

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reen

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son?

g$

gree

n fe

es p

er p

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____

____

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1-3

1-3



Rea

din

g t

o L

earn

Math

emati

csSo

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g Eq

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ns

NAM

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____

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____

____

____

____

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1-3

1-3

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lenc

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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

ucti

on t

o L

esso

n 1-

3 at

the

top

of

page

20

in y

our

text

book

.

•To

fin

d yo

ur t

arge

t he

art

rate

,wha

t tw

o pi

eces

of

info

rmat

ion

mus

t yo

usu

pply

?ag

e (A

) and

des

ired

inte

nsity

leve

l (I)

•W

rite

an

equa

tion

tha

t sh

ows

how

to

calc

ulat

e yo

ur t

arge

t he

art

rate

.

P$

or P

$(2

20 !

A) *

I#6

Rea

din

g t

he

Less

on

1.a.

How

are

alg

ebra

ic e

xpre

ssio

ns a

nd e

quat

ions

alik

e?Sa

mpl

e an

swer

:Bot

h co

ntai

n va

riabl

es,c

onst

ants

,and

ope

ratio

nsi

gns.

b.H

ow a

re a

lgeb

raic

exp

ress

ions

and

equ

atio

ns d

iffe

rent

?Sa

mpl

e an

swer

:Equ

atio

ns c

onta

in e

qual

sig

ns;e

xpre

ssio

ns d

o no

t.

c.H

ow a

re a

lgeb

raic

exp

ress

ions

and

equ

atio

ns r

elat

ed?

Sam

ple

answ

er:A

n eq

uatio

n is

a s

tate

men

t tha

t say

s th

at tw

oal

gebr

aic

expr

essi

ons

are

equa

l.

Rea

d t

he

foll

owin

g p

robl

em a

nd

th

en w

rite

an

equ

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 t

he

nu

mbe

rof

mil

es d

rive

n.

2.W

hen

Lou

isa

rent

ed a

mov

ing

truc

k,sh

e ag

reed

to

pay

$28

per

day

plus

$0.

42 p

er m

ile.

If s

he k

ept

the

truc

k fo

r 3

days

and

the

ren

tal c

harg

es (

wit

hout

tax

) w

ere

$153

.72,

how

man

y m

iles

did

Lou

isa

driv

e th

e tr

uck?

3(28

) "0.

42m

$15

3.72

Hel

pin

g Y

ou

Rem

emb

er

3.H

ow c

an t

he w

ords

ref

lect

ion

and

sym

met

ryhe

lp y

ou r

emem

ber

and

dist

ingu

ish

betw

een

the

refl

exiv

e an

d sy

mm

etri

c pr

oper

ties

of

equa

lity?

Thi

nk a

bout

how

the

se w

ords

are

used

in e

very

day

life

or in

geo

met

ry.

Sam

ple

answ

er:W

hen

you

look

at y

our r

efle

ctio

n,yo

u ar

e lo

okin

g at

your

self.

The

refle

xive

pro

pert

y sa

ys th

at e

very

num

ber i

s eq

ual t

o its

elf.

In g

eom

etry

,sym

met

ry w

ith re

spec

t to

a lin

e m

eans

that

the

part

s of

afig

ure

on th

e tw

o si

des

of a

line

are

iden

tical

.The

sym

met

ric p

rope

rty

ofeq

ualit

y al

low

s yo

u to

inte

rcha

nge

the

two

side

s of

an

equa

tion.

The

equa

l sig

n is

like

the

line

of s

ymm

etry

.

(220

!A

) *I

%% 6

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ill18

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Alg

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2

Venn

Dia

gram

sR

elat

ions

hips

am

ong

sets

can

be

show

n us

ing

Ven

n di

agra

ms.

Stud

y th

edi

agra

ms

belo

w.T

he c

ircl

es r

epre

sent

set

s A

and

B,w

hich

are

sub

sets

of

set

S.

The

uni

on o

f Aan

d B

cons

ists

of

all e

lem

ents

in e

ithe

r A

or B

.T

he in

ters

ecti

on o

f Aan

d B

cons

ists

of

all e

lem

ents

in b

oth

Aan

d B

.T

he c

ompl

emen

t of

Aco

nsis

ts o

f al

l ele

men

ts n

otin

A.

You

can

com

bine

the

ope

rati

ons

of u

nion

,int

erse

ctio

n,an

d fi

ndin

g th

e co

mpl

emen

t.

Sh

ade

the

regi

on (

A∩

B)+

.

(A "

B)*

mea

ns t

he c

ompl

emen

t of

the

inte

rsec

tion

of A

and

B.

Fir

st f

ind

the

inte

rsec

tion

of A

and

B.T

hen

find

its

com

plem

ent.

Dra

w a

Ven

n d

iagr

am a

nd

sh

ade

the

regi

on i

nd

icat

ed.

See

stud

ents

’dia

gram

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

stud

ents

’dia

gram

s.

7.(A

# B

) #C

*8.

(A #

B)*

"C

*

9.A

# (B

#C

)10

.(A

# B

) #C

11.I

s th

e un

ion

oper

atio

n as

soci

ativ

e?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

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

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RIO

D__

___

1-3

1-3

Exam

ple

Exam

ple


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Solv

ing

Abs

olut

e Va

lue

Equa

tions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

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____

____

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RIO

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___

1-4

1-4

©G

lenc

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Gle

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Alg

ebra

2

Lesson 1-4

Ab

solu

te V

alu

e Ex

pre

ssio

ns

The

abs

olu

te v

alu

eof

a n

umbe

r is

the

num

ber

ofun

its

it is

fro

m 0

on

a nu

mbe

r lin

e.T

he s

ymbo

l x

is u

sed

to r

epre

sent

the

abs

olut

e va

lue

of a

num

ber

x.

•W

ords

For a

ny re

al n

umbe

r a, i

f ais

pos

itive

or z

ero,

the

abso

lute

val

ue o

f ais

a.

Abs

olut

e Va

lue

If a

is n

egat

ive,

the

abso

lute

val

ue o

f ais

the

oppo

site

of a

.•

Sym

bols

For a

ny re

al n

umbe

r a,

a $

a, if

a+

0, a

nd

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

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y$

,an

d z

$!

6.

1.2

x!

84

2.6

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!!

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73.

5 "

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z15

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y

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46.

7 !

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11

7.w

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12

8.w

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239.

z

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2

13.

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1 & 4

1 % 2

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Ab

solu

te V

alu

e Eq

uat

ion

sU

se t

he d

efin

itio

n of

abs

olut

e va

lue

to s

olve

equ

atio

nsco

ntai

ning

abs

olut

e va

lue

expr

essi

ons.

For a

ny re

al n

umbe

rs a

and

b, w

here

b+

0, if

a

$b

then

a$

bor

a$

!b.

Alw

ays

chec

k yo

ur a

nsw

ers

by s

ubst

itut

ing

them

into

the

ori

gina

l equ

atio

n.So

met

imes

com

pute

d so

luti

ons

are

not

actu

al s

olut

ions

.

Sol

ve ⏐

2x!

3⏐$

17.C

hec

k y

our

solu

tion

s.

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Solv

ing

Abs

olut

e Va

lue

Equa

tions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-4

1-4

Exam

ple

Exam

ple

Cas

e 1

a$

b2x

!3

$17

2x!

3 "

3 $

17 "

32x

$20

x$

10

CH

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2(1

0) !

3$

172

0 !

3$

171

7$

1717

$17

✓

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e 2

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2x!

3 $

!17

2x!

3 "

3 $

!17

"3

2x$

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x$

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CH

ECK

2(!

7) !

3$

17!

14 !

3$

17!

17

$17

17 $

17 ✓

The

re a

re t

wo

solu

tion

s,10

and

!7.

Sol

ve e

ach

equ

atio

n.C

hec

k y

our

solu

tion

s.

1.x

"15

$

37

{!52

,22}

2.t

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9}

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0}4.

m"

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5.5

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1 % 21 & 2

1 & 3

1 % 71 & 3

11 % 2

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Solv

ing

Abs

olut

e Va

lue

Equa

tions

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

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____

____

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RIO

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___

1-4

1-4

©G

lenc

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Gle

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2

Lesson 1-4

Eva

luat

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xpre

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f w

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4,x

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y$

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and

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.

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y!

z17

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0

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31

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x!

3y

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y"

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z"

1!

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44 "

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9.2

4w

3.2

10.3

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1.

6

11.

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!

4!

412

.6.4

"w

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7

Sol

ve e

ach

equ

atio

n.C

hec

k y

our

solu

tion

s.

13.

y"

3$

2{!

5,!

1}14

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$

10{!

2,2}

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3k!

6$

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,&

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3}

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0 $

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3}

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p!

7$

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w

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4}22

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!4d

"

5 $

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2,2}

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,&

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k"

10 $

9,

1 % 65 % 6

5 % 61 % 2

1 & 2

8 % 34 % 3

Skil

ls P

ract

ice

Solv

ing

Abs

olut

e Va

lue

Equa

tions

NAM

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1-4

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f w

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.

1.5

w

22.

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0

5.!1

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y"

5x

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7.25

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z"

1!

248.

44 "

!2x

!y

45

9.2

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3.2

10.3

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1.

6

11.

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!2y

!

4!

412

.6.4

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ve e

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atio

n.C

hec

k y

our

solu

tion

s.

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

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"x

$9

{!3,

3}

19.

p!

7$

!14

,20

.23

w

$12

{!2,

2}

21.

7x!

3x

"2

$18

{!4,

4}22

.47

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!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


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csSo

lvin

g A

bsol

ute

Valu

e Eq

uatio

ns

NAM

E__

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____

____

____

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____

____

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1-4

1-4

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Alg

ebra

2

Lesson 1-4

Pre-

Act

ivit

yH

ow c

an a

n a

bsol

ute

val

ue

equ

atio

n d

escr

ibe

the

mag

nit

ud

e of

an

eart

hqu

ake?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 1-

4 at

the

top

of

page

28

in y

our

text

book

.

•W

hat

is a

sei

smol

ogis

t an

d w

hat

does

mag

nitu

de o

f an

ear

thqu

ake

mea

n?a

scie

ntis

t who

stu

dies

ear

thqu

akes

;a n

umbe

r fro

m 1

to 1

0th

at te

lls h

ow s

trong

an

eart

hqua

ke is

•W

hy is

an

abso

lute

val

ue e

quat

ion

rath

er t

han

an e

quat

ion

wit

hout

abso

lute

val

ue u

sed

to f

ind

the

extr

emes

in t

he a

ctua

l mag

nitu

de o

f an

eart

hqua

ke in

rel

atio

n to

its

mea

sure

d va

lue

on t

he R

icht

er s

cale

?Sa

mpl

e an

swer

:The

act

ual m

agni

tude

can

var

y fro

m th

em

easu

red

mag

nitu

de b

y up

to 0

.3 u

nit i

n ei

ther

dire

ctio

n,so

an a

bsol

ute

valu

e eq

uatio

n is

nee

ded.

•If

the

mag

nitu

de o

f an

ear

thqu

ake

is e

stim

ated

to

be 6

.9 o

n th

e R

icht

er

scal

e,it

mig

ht a

ctua

lly h

ave

a m

agni

tude

as

low

as

or a

s hi

gh

as

.

Rea

din

g t

he

Less

on

1.E

xpla

in h

ow !

aco

uld

repr

esen

t a

posi

tive

num

ber.

Giv

e an

exa

mpl

e.Sa

mpl

ean

swer

:If a

is n

egat

ive,

then

!a

is p

ositi

ve.E

xam

ple:

If a

$!

25,t

hen

!a

$!

(!25

) $25

.

2.E

xpla

in w

hy t

he a

bsol

ute

valu

e of

a n

umbe

r ca

n ne

ver

be n

egat

ive.

Sam

ple

answ

er:

The

abso

lute

val

ue is

the

num

ber o

f uni

ts it

is fr

om 0

on

the

num

ber l

ine.

The

num

ber o

f uni

ts is

nev

er n

egat

ive.

3.W

hat

does

the

sen

tenc

e b

+0

mea

n?Sa

mpl

e an

swer

:The

num

ber b

is 0

or

grea

ter t

han

0.

4.W

hat

does

the

sym

bol -

mea

n as

a s

olut

ion

set?

Sam

ple

answ

er:I

f a s

olut

ion

set

is ,

,the

n th

ere

are

no s

olut

ions

.

Hel

pin

g Y

ou

Rem

emb

er

5.H

ow c

an t

he n

umbe

r lin

e m

odel

for

abs

olut

e va

lue

that

is s

how

n on

pag

e 28

of

your

text

book

hel

p yo

u re

mem

ber

that

man

y ab

solu

te v

alue

equ

atio

ns h

ave

two

solu

tion

s?Sa

mpl

e an

swer

:The

num

ber l

ine

show

s th

at fo

r eve

ry p

ositi

ve n

umbe

r,th

ere

are

two

num

bers

that

hav

e th

at n

umbe

r as

thei

r abs

olut

e va

lue.

7.2

6.6

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Con

side

ring

All

Cas

es in

Abs

olut

e Va

lue

Equa

tions

Yo

u ha

ve le

arne

d th

at a

bsol

ute

valu

e eq

uati

ons

wit

h on

e se

t of

abs

olut

e va

lue

sym

bols

hav

e tw

o ca

ses

that

mus

t be

con

side

red.

For

exam

ple,

|x"

3|$

5 m

ust

be b

roke

n in

tox

"3

$5

or !

(x"

3) $

5.Fo

r an

equ

atio

n w

ith

two

sets

of

abso

lute

val

ue s

ymbo

ls,f

our

case

s m

ust

be c

onsi

dere

d.

Con

side

r th

e pr

oble

m |x

"2|"

3 $

|x"

6|.F

irst

we

mus

t w

rite

the

equ

atio

nsfo

r th

e ca

se w

here

x"

6 +

0 an

d w

here

x"

6 ,

0.H

ere

are

the

equa

tion

s fo

rth

ese

two

case

s:

|x"

2|"

3 $

x"

6

|x"

2|"

3 $

!(x

"6)

Eac

h of

the

se e

quat

ions

als

o ha

s tw

o ca

ses.

By

wri

ting

the

equ

atio

ns f

or b

oth

case

s of

eac

h eq

uati

on a

bove

,you

end

up

wit

h th

e fo

llow

ing

four

equ

atio

ns:

x"

2 "

3 $

x"

6x

"2

"3

$!

(x"

6)

!(x

"2)

"3

$x

"6

!x

!2

"3

$!

(x"

6)

Solv

e ea

ch o

fthe

se e

quat

ions

and

che

ck y

our

solu

tion

s in

the

ori

gina

l equ

atio

n,

|x"

2|"

3 $

|x"

6|.T

he o

nly

solu

tion

to

this

equ

atio

n is

!&5 2& .

Sol

ve e

ach

abs

olu

te v

alu

e eq

uat

ion

.Ch

eck

you

r so

luti

on.

1.|x

!4|$

|x"

7|x

$5.

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 th

ere

wou

ld b

e fo

r an

abs

olut

e va

lue

equa

tion

con

tain

ing

thre

e se

ts o

f ab

solu

te v

alue

sym

bols

?8

6.L

ist

each

cas

e an

d so

lve |x

"2|"

|2x

!4|$

|x!

3|.C

heck

you

r so

luti

on.

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

x"

2 "

(!2x

!4)

$x

!3

!(x

"2)

"(!

2x!

4) $

!(x

!3)

x"

2 "

(!2x

!4)

$!

(x!

3)

No

solu

tion

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-4

1-4



Stu

dy

Gu

ide

and I

nte

rven

tion

Solv

ing

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-5

1-5

©G

lenc

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ncoe

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2

Lesson 1-5

Solv

e In

equ

alit

ies

The

fol

low

ing

prop

erti

es c

an b

e us

ed t

o so

lve

ineq

ualit

ies.

Add

ition

and

Sub

trac

tion

Prop

ertie

s fo

r Ine

qual

ities

Mul

tiplic

atio

n an

d D

ivis

ion

Prop

ertie

s fo

r Ine

qual

ities

For a

ny re

al n

umbe

rs a

, b, a

nd c

:Fo

r any

real

num

bers

a, b

, and

c, w

ith c

.0:

1.If

a,

b, th

en a

"c

,b

"c

and

a!

c,

b!

c.1.

If c

is p

ositi

ve a

nd a

,b,

then

ac

,bc

and

,.

2.If

a)

b, th

en a

"c

)b

"c

and

a!

c)

b!

c.2.

If c

is p

ositi

ve a

nd a

)b,

then

ac

)bc

and

).

3.If

cis

neg

ativ

e an

d a

,b,

then

ac

)bc

and

).

4.If

cis

neg

ativ

e an

d a

)b,

then

ac

,bc

and

,.

The

se p

rope

rtie

s ar

e al

so t

rue

for

/an

d +

.

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 s

olut

ion

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

The

sol

utio

n se

t is

(!0

,!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-

buil

der

or

inte

rval

not

atio

n.T

hen

gra

ph

th

e so

luti

on s

et o

n a

nu

mbe

r li

ne.

1.7(

7a!

9) /

842.

3(9z

"4)

)35

z!

43.

5(12

!3n

) ,16

5

{a⏐a

/3}

or (

!∞

,3]

{z⏐z

02}

or (

!∞

,2)

{n⏐n

-!

7} o

r (!

7,"

∞)

4.18

!4k

,2(

k"

21)

5.4(

b!

7) "

6 ,

226.

2 "

3(m

"5)

+4(

m"

3)

{k⏐k

-!

4} o

r (!

4,"

∞)

{b⏐b

011

} or (

!∞

,11)

{m⏐m

/5}

or (

!∞

,5]

7.4x

!2

)!

7(4x

!2)

8.(2

y!

3) )

y"

29.

2.5d

"15

/75

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-&o

r #,"

∞$

{y⏐y

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r (!

∞,!

9){d

⏐d/

24} o

r (!

∞,2

4]

2122

1920

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27!

12!

14!

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10

12

34

1 % 21 % 2

1 & 3

23

01

45

67

88

96

710

1112

1314

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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

ualit

ies.

The

cha

rt b

elow

sho

ws

som

e co

mm

on p

hras

es t

hat

indi

cate

ineq

ualit

ies.

0-

/.

is le

ss th

anis

gre

ater

than

is a

t mos

tis

at l

east

is fe

wer

than

is m

ore

than

is n

o m

ore

than

is n

o le

ss th

anis

less

than

or e

qual

tois

gre

ater

than

or e

qual

to

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

the

num

ber

of r

emai

ning

gam

es t

hat

the

Vik

ings

mus

t w

in.T

he t

otal

num

ber

ofga

mes

the

y w

ill h

ave

won

by

the

end

of t

he s

easo

n is

16

"x.

The

y w

ant

to w

in a

t le

ast

80%

of t

heir

gam

es.W

rite

an

ineq

ualit

y w

ith

+.

16 "

x+

0.8(

36)

x+

0.8(

36)

!16

x+

12.8

Sinc

e th

ey c

anno

t w

in a

fra

ctio

nal p

art

of a

gam

e,th

e V

ikin

gs m

ust

win

at

leas

t 13

of

the

gam

es r

emai

ning

.

1.PA

RK

ING

FEE

ST

he c

ity

park

ing

lot

char

ges

$2.5

0 fo

r th

e fi

rst

hour

and

$0.

25 f

or e

ach

addi

tion

al h

our.

If t

he m

ost

you

wan

t to

pay

for

par

king

is $

6.50

,sol

ve t

he in

equa

lity

2.50

"0.

25(x

!1)

/6.

50 t

o de

term

ine

for

how

man

y ho

urs

you

can

park

you

r ca

r.A

t mos

t 17

hour

s

PLA

NN

ING

For

Exe

rcis

es 2

an

d 3

,use

th

e fo

llow

ing

info

rmat

ion

.

Eth

an is

rea

ding

a 4

82-p

age

book

for

a b

ook

repo

rt d

ue o

n M

onda

y.H

e ha

s al

read

y re

ad

80 p

ages

.He

wan

ts t

o fi

gure

out

how

man

y pa

ges

per

hour

he

need

s to

rea

d in

ord

er t

ofi

nish

the

boo

k in

less

tha

n 6

hour

s.

2.W

rite

an

ineq

ualit

y to

des

crib

e th

is s

itua

tion

./

6 or

6n

.48

2 !

80

3.So

lve

the

ineq

ualit

y an

d in

terp

ret

the

solu

tion

.Et

han

mus

t rea

d at

leas

t 67

page

spe

r hou

r in

orde

r to

finis

h th

e bo

ok in

less

than

6 h

ours

.

BO

WLI

NG

For

Exe

rcis

es 4

an

d 5

,use

th

e fo

llow

ing

info

rmat

ion

.

Four

fri

ends

pla

n to

spe

nd F

rida

y ev

enin

g at

the

bow

ling

alle

y.T

hree

of

the

frie

nds

need

to

rent

sho

es f

or $

3.50

per

per

son.

A s

trin

g (g

ame)

of

bow

ling

cost

s $1

.50

per

pers

on.I

f th

efr

iend

s po

ol t

heir

$40

,how

man

y st

ring

s ca

n th

ey a

ffor

d to

bow

l?

4.W

rite

an

equa

tion

to

desc

ribe

thi

s si

tuat

ion.

3(3.

50) "

4(1.

50)n

/40

5.So

lve

the

ineq

ualit

y an

d in

terp

ret

the

solu

tion

.Th

e fr

iend

s ca

n bo

wl a

t mos

t 4

strin

gs.

482

!80

%% n

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Solv

ing

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-5

1-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Solv

ing

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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-

buil

der

or

inte

rval

not

atio

n.T

hen

,gra

ph

th

e so

luti

on s

et o

n a

nu

mbe

r li

ne.

1.+

2{z

⏐z/

!8}

or (

!∞

,!8]

2.3a

"7

/16

{a⏐a

/3}

or (

!∞

,3]

3.16

,3q

"4

{q⏐q

-4}

or (

4,∞

)4.

20 !

3s)

7s{s

⏐s0

2} o

r (!

∞,2

)

5.3x

+!

9{x

⏐x.

!3}

or [

!3,

∞)

6.4b

!9

/7

{b⏐b

/4}

or (

!∞

,4]

7.2z

,!

9 "

5z{z

⏐z-

3} o

r (3,

∞)

8.7f

!9

)3f

!1

{f⏐f

-2}

or (

2,∞

)

9.!

3s!

8 /

5s{s

⏐s.

!1}

or [

!1,

∞)

10.7

t!

(t!

4) /

25%t⏐

t/&o

r #!∞

,'

11.0

.7m

"0.

3m+

2m!

4{m

⏐m/

4}12

.4(5

x"

7) /

13%x⏐

x/

!&o

ror

(!∞

,4]

#!∞

,!'

13.1

.7y

!0.

78 )

5{y

⏐y-

3.4}

14.4

x!

9 )

2x"

1{x

⏐x-

5} o

r (5,

∞)

or (3

.4,∞

)

Def

ine

a va

riab

le a

nd

wri

te a

n i

neq

ual

ity

for

each

pro

blem

.Th

en s

olve

.

15.N

inet

een

mor

e th

an a

num

ber

is le

ss t

han

42.

n"

19 0

42;n

023

16.T

he d

iffe

renc

e of

thr

ee t

imes

a n

umbe

r an

d 16

is a

t le

ast

8.3n

!16

.8;

n.

8

17.O

ne h

alf o

f a n

umbe

r is

mor

e th

an 6

less

tha

n th

e sa

me

num

ber.

n-

n!

6;n

012

18.F

ive

less

tha

n th

e pr

oduc

t of

6 a

nd a

num

ber

is n

o m

ore

than

tw

ice

that

sam

e 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-

buil

der

or

inte

rval

not

atio

n.T

hen

,gra

ph

th

e so

luti

on s

et o

n a

nu

mbe

r li

ne.

1.8x

!6

+10

{x⏐x

.2}

or [

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

⏐s0

1} o

r (!

∞,1

)

5.9x

!11

)6x

!9 %x⏐

x-

&or #

,∞$

6.!

3(4w

!1)

)18

%w⏐w

0!

&or

#!∞

,!$

7.1

!8u

/3u

!10

{u⏐u

.1}

or [

1,∞

)8.

17.5

,19

!2.

5x{x

⏐x0

0.6}

or

(!∞

,0.6

)

9.9(

2r!

5) !

3 ,

7r!

4 {r

⏐r0

4}

10.1

"5(

x!

8) /

2 !

(x"

5) {x

⏐x/

6}

or (!

∞,4

)or

(!∞

,6]

11.

+!

3.5

{x⏐x

.!

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,

∞)

{n⏐n

04}

or (

!∞

,4)

Def

ine

a va

riab

le a

nd

wri

te a

n i

neq

ual

ity

for

each

pro

blem

.Th

en s

olve

.

15.T

wen

ty le

ss t

han

a nu

mbe

r is

mor

e th

an t

wic

e th

e sa

me

num

ber.

n!

20 -

2n;n

0!

20

16.F

our

tim

es t

he s

um o

f tw

ice

a nu

mbe

r an

d !

3 is

less

tha

n 5.

5 ti

mes

tha

t sa

me

num

ber.

4[2n

"(!

3)] 0

5.5n

;n0

4.8

17.H

OTE

LST

he L

inco

ln’s

hot

el r

oom

cos

ts $

90 a

nig

ht.A

n ad

diti

onal

10%

tax

is a

dded

.H

otel

par

king

is $

12 p

er d

ay.T

he L

inco

ln’s

exp

ect

to s

pend

$30

in t

ips

duri

ng t

heir

sta

y.So

lve

the

ineq

ualit

y 90

x"

90(0

.1)x

"12

x"

30 /

600

to f

ind

how

man

y ni

ghts

the

Lin

coln

’s c

an s

tay

at t

he h

otel

wit

hout

exc

eedi

ng t

otal

hot

el c

osts

of

$600

.5

nigh

ts

18.B

AN

KIN

GJa

n’s

acco

unt

bala

nce

is $

3800

.Of

this

,$75

0 is

for

ren

t.Ja

n w

ants

to

keep

aba

lanc

e of

at

leas

t $5

00.W

rite

and

sol

ve a

n in

equa

lity

desc

ribi

ng h

ow m

uch

she

can

wit

hdra

w a

nd s

till

mee

t th

ese

cond

itio

ns.

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 (A

vera

ge)

Solv

ing

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-5

1-5



Rea

din

g t

o L

earn

Math

emati

csSo

lvin

g In

equa

litie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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 be

use

d t

o co

mp

are

ph

one

pla

ns?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 1-

5 at

the

top

of

page

33

in y

our

text

book

.

•W

rite

an

ineq

ualit

y co

mpa

ring

the

num

ber

of m

inut

es p

er m

onth

incl

uded

in t

he t

wo

phon

e pl

ans.

150

040

0 or

400

-15

0•

Supp

ose

that

in o

ne m

onth

you

use

230

min

utes

of

airt

ime

on y

our

wir

eles

s ph

one.

Fin

d yo

ur m

onth

ly c

ost

wit

h ea

ch p

lan.

Pla

n 1:

Pla

n 2:

Whi

ch p

lan

shou

ld y

ou c

hoos

e?

Rea

din

g t

he

Less

on

1.T

here

are

sev

eral

dif

fere

nt w

ays

to w

rite

or

show

ineq

ualit

ies.

Wri

te e

ach

of t

hefo

llow

ing

in in

terv

al n

otat

ion.

a.{xx

,!

3}(!

∞,!

3)b.

{xx

+5}

[5,"

∞)

c.(!

∞,2

]

d.(!

1,"

∞)

2.Sh

ow h

ow y

ou c

an w

rite

an

ineq

ualit

y sy

mbo

l fol

low

ed b

y a

num

ber

to d

escr

ibe

each

of

the

follo

win

g si

tuat

ions

.

a.T

here

are

few

er t

han

600

stud

ents

in t

he s

enio

r cl

ass.

060

0b.

A s

tude

nt m

ay e

nrol

l in

no m

ore

than

six

cou

rses

eac

h se

mes

ter.

/6

c.To

par

tici

pate

in a

con

cert

,you

mus

t be

will

ing

to a

tten

d at

leas

t te

n re

hear

sals

..

10d.

The

re is

spa

ce f

or a

t m

ost

165

stud

ents

in t

he h

igh

scho

ol b

and.

/16

5

Hel

pin

g Y

ou

Rem

emb

er

3.O

ne w

ay t

o re

mem

ber

som

ethi

ng is

to

expl

ain

it t

o an

othe

r pe

rson

.A c

omm

on s

tude

nter

ror

in s

olvi

ng in

equa

litie

s is

for

gett

ing

to r

ever

se t

he in

equa

lity

sym

bol w

hen

mul

tipl

ying

or

divi

ding

bot

h si

des

of a

n in

equa

lity

by a

neg

ativ

e nu

mbe

r.Su

ppos

e th

atyo

ur c

lass

mat

e is

hav

ing

trou

ble

rem

embe

ring

thi

s ru

le.H

ow c

ould

you

exp

lain

thi

s ru

leto

you

r cl

assm

ate?

Sam

ple

answ

er:D

raw

a n

umbe

r lin

e.Pl

ot tw

o po

sitiv

enu

mbe

rs,f

or e

xam

ple,

3 an

d 8.

Then

plo

t the

ir ad

ditiv

e in

vers

es,!

3 an

d!

8.W

rite

an in

equa

lity

that

com

pare

s th

e po

sitiv

e nu

mbe

rs a

nd o

ne th

atco

mpa

res

the

nega

tive

num

bers

.Not

ice

that

8 -

3,bu

t !8

0!

3.Th

eor

der c

hang

es w

hen

you

mul

tiply

by

!1.

32

54

10

!1

!2

!3

!4

!5

!5

!4

!3

!2

!1

01

23

45

Plan

2$5

5$6

7

©G

lenc

oe/M

cGra

w-H

ill30

Gle

ncoe

Alg

ebra

2

Equ

ival

ence

Rel

atio

nsA

rel

atio

n R

on

a se

t A

is a

n eq

uiva

lenc

e re

lati

onif

it h

as t

he f

ollo

win

g pr

oper

ties

.

Ref

lexi

ve P

rope

rty

For

any

elem

ent

aof

set

A,a

R a

.Sy

mm

etri

c P

rope

rty

For

all e

lem

ents

aan

d b

of s

et A

,if

aR

b,t

hen

bR

a.

Tra

nsi

tive

Pro

pert

yFo

r al

l ele

men

ts a

,b,a

nd c

of s

et A

,if

aR

ban

d b

R c

,the

n a

R c

.

Equ

alit

y on

the

set

of

all r

eal n

umbe

rs is

ref

lexi

ve,s

ymm

etri

c,an

d tr

ansi

tive

.T

here

fore

,it

is a

n eq

uiva

lenc

e re

lati

on.

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

quiv

alen

ce r

elat

ion

on

th

e gi

ven

set

.If

it i

s n

ot,t

ell

wh

ich

of

the

pro

per

ties

it

fail

s to

exh

ibit

.

1.,

,{al

l num

bers

}no

;ref

lexi

ve,s

ymm

etric

2.)

,{al

l tri

angl

es in

a p

lane

}ye

s

3.is

the

sis

ter

of,{

all w

omen

in T

enne

ssee

}no

;ref

lexi

ve

4.+

,{al

l num

bers

}no

;sym

met

ric

5.is

a f

acto

r of

,{al

l non

zero

inte

gers

}no

;sym

met

ric

6.*

,{al

l pol

ygon

s in

a p

lane

}ye

s

7.is

the

spo

use

of,{

all p

eopl

e in

Roa

noke

,Vir

gini

a}no

;ref

lexi

ve,t

rans

itive

8.⊥

,{al

l lin

es in

a p

lane

}no

;ref

lexi

ve,t

rans

itive

9.is

a m

ulti

ple

of,{

all i

nteg

ers}

no;s

ymm

etric

10.

is t

he s

quar

e of

,{al

l num

bers

}no

;ref

lexi

ve,s

ymm

etric

,tra

nsiti

ve

11.

++,{a

ll lin

es in

a p

lane

}no

;ref

lexi

ve

12.

has

the

sam

e co

lor

eyes

as,

{all

mem

bers

of

the

Cle

vela

nd S

ymph

ony

Orc

hest

ra}

yes

13.

is t

he g

reat

est

inte

ger

not

grea

ter

than

,{al

l num

bers

}no

;ref

lexi

ve,s

ymm

etric

,tra

nsiti

ve14

.is

the

gre

ates

t in

tege

r no

t gr

eate

r th

an,{

all i

nteg

ers}

yes

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-5

1-5


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Solv

ing

Com

poun

d an

d A

bsol

ute

Valu

e In

equa

litie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

poun

d in

equa

lity

cons

ists

of

two

ineq

ualit

ies

join

ed b

yth

e w

ord

and

or t

he w

ord

or.T

o so

lve

a co

mpo

und

ineq

ualit

y,yo

u m

ust

solv

e ea

ch p

art

sepa

rate

ly.

Exam

ple:

x)

!4

and

x,

3Th

e gr

aph

is th

e in

ters

ectio

n of

sol

utio

n se

ts o

f tw

o in

equa

litie

s.

Exam

ple:

x/

!3

or x

)1

The

grap

h is

the

unio

n of

sol

utio

n se

ts o

f tw

o in

equa

litie

s.!

5!

4!

3!

2!

10

12

34

5

Or

Com

poun

dIn

equa

litie

s

!3

!2

!5

!4

!1

01

23

45

And

Com

poun

dIn

equa

litie

s

Sol

ve !

3 /

2x"

5 /

19.

Gra

ph

th

e so

luti

on s

et o

n a

nu

mbe

r li

ne.

!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

mbe

r li

ne.

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 0

x/

4}{a

⏐a0

5 or

a-

52}

3.18

,4x

!10

,50

4.5k

"2

,!

13 o

r 8k

!1

)19

{x⏐7

0x

015

}{k

⏐k0

!3

or k

-2.

5}

5.10

0 /

5y!

45 /

225

6.b

!2

)10

or

b"

5 ,

!4

{y⏐2

9 /

y/

54}

{b⏐b

0!

12 o

r b-

18}

7.22

,6w

!2

,82

8.4d

!1

)!

9 or

2d

"5

,11

{w⏐4

0w

014

}{a

ll re

al n

umbe

rs}

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

the

def

init

ion

of a

bsol

ute

valu

e to

rew

rite

an

abso

lute

val

ue in

equa

lity

as a

com

poun

d in

equa

lity.

For a

ll re

al n

umbe

rs a

and

b, b

)0,

the

follo

win

g st

atem

ents

are

true

.1.

If a

,

b, th

en !

b,

a,

b.2.

If a

)

b, th

en a

)b

or a

,!

b.Th

ese

stat

emen

ts a

re a

lso

true

for /

and

+.

Stu

dy

Gu

ide

and I

nte

rven

tion

(c

onti

nued

)

Solv

ing

Com

poun

d an

d A

bsol

ute

Valu

e In

equa

litie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-6

1-6

Sol

ve ⏐

x"

2⏐-

4.G

rap

hth

e so

luti

on s

et o

n a

nu

mbe

r li

ne.

By

stat

emen

t 2

abov

e,if

x

"2

)4,

then

x

"2

)4

or x

"2

,!

4.Su

btra

ctin

g 2

from

bot

h si

des

of e

ach

ineq

ualit

y gi

ves

x)

2 or

x,

!6.

!8

!6

!4

!2

02

46

8

Sol

ve ⏐

2x!

1⏐0

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

thre

e pa

rts

of t

he in

equa

lity

give

s !

4 ,

2x,

6.D

ivid

ing

by 2

giv

es !

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

0x

0&

2.4

s"

1 )

27{s

⏐s0

!6.

5 or

s-

6.5}

3.

!3

/5

{c⏐!

4 /

c/

16}

4.a

"9

+30

{a⏐a

/!

39 o

r a.

21}

5.2

f!11

)

9{f⏐

f01

or f

-10

}6.

5w

"2

,28

{w⏐!

6 0

w0

5.2}

7.1

0 !

2k

,2

{k⏐4

0k

06}

8.

!5

"2

)10

{x⏐x

0!

6 or

x-

26}

9.4

b!

11

,17

%b⏐!

0b

07 &

10.

100

!3m

)

20%m

⏐m0

26or

m-

40&

010

2030

515

2535

40!

40

48

!2

26

1012

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

Solv

ing

Com

poun

d an

d A

bsol

ute

Valu

e In

equa

litie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

bsol

ute

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

mbe

r li

ne.

1.al

l num

bers

gre

ater

tha

n or

equ

al t

o 2

2.al

l num

bers

less

tha

n 5

and

grea

ter

or le

ss t

han

or e

qual

to

!2

⏐n⏐

.2

than

!5

⏐n⏐

05

3.al

l num

bers

less

tha

n !

1 or

gre

ater

4.

all n

umbe

rs b

etw

een

!6

and

6⏐n

⏐0

6th

an 1

⏐n⏐

-1

Wri

te a

n a

bsol

ute

val

ue

ineq

ual

ity

for

each

gra

ph

.

5.⏐n

⏐0

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

{c⏐c

-2

10.!

11 /

4y!

3 /

1{y

⏐!2

/y

/1}

or c

00}

11.1

0 )

!5x

)5

{x⏐!

2 0

x0

!1}

12.4

a+

!8

or a

,!

3{a

⏐a.

!2

or a

0!

3}

13.8

,3x

"2

/23

{x⏐2

0x

/7}

14.w

!4

/10

or

!2w

/6

all r

eal

num

bers

15.

t+

3{t

⏐t/

!3

or t

.3}

16.

6x

,12

{x⏐!

2 0

x0

2}

17.

!7r

)14

{r⏐r

0!

2 or

r-

2}18

.p

"2

/!

2,

19.

n!

5,

7{n

⏐!2

0n

012

}20

.h

"1

+5

{h⏐h

/!

6 or

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

bsol

ute

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

mbe

r li

ne.

1.al

l num

bers

gre

ater

tha

n 4

or le

ss t

han

!4

⏐n⏐

-4

2.al

l num

bers

bet

wee

n !

1.5

and

1.5,

incl

udin

g !

1.5

and

1.5

⏐n⏐

/1.

5

Wri

te a

n a

bsol

ute

val

ue

ineq

ual

ity

for

each

gra

ph

.

3.⏐n

⏐.

104.

⏐n⏐

0

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 /

y0

24}

6.3(

5x!

2) ,

24 o

r 6x

!4

)4

"5x

{x⏐x

02

or x

-8}

7.2x

!3

)15

or

3 !

7x,

17{x

⏐x-

!2}

8.15

!5x

/0

and

5x"

6 +

!14

{x⏐x

.3}

9.2

w

+5

%w⏐w

/!

or w

.&

10.

y"

5,

2{x

⏐!7

0x

0!

3}

11.

x!

8+

3{x

⏐x/

5 or

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

umbe

rs

15.

3b"

5/

!2

,16

.3n

!2

!2

,1

%n⏐!

0n

0&

17.R

AIN

FALL

In 9

0% o

f th

e la

st 3

0 ye

ars,

the

rain

fall

at S

hell

Bea

ch h

as v

arie

d no

mor

eth

an 6

.5 in

ches

fro

m it

s m

ean

valu

e of

24

inch

es.W

rite

and

sol

ve a

n ab

solu

te v

alue

ineq

ualit

y to

des

crib

e th

e ra

infa

ll in

the

oth

er 1

0% o

f th

e la

st 3

0 ye

ars.

⏐r!

24⏐

-6.

5;{r

⏐r0

17.5

or r

-30

.5}

18.M

AN

UFA

CTU

RIN

GA

com

pany

’s gu

idel

ines

cal

l for

eac

h ca

n of

sou

p pr

oduc

ed n

ot t

o va

ryfr

om it

s st

ated

vol

ume

of 1

4.5

flui

d ou

nces

by

mor

e th

an 0

.08

ounc

es.W

rite

and

sol

ve a

nab

solu

te v

alue

ineq

ualit

y to

des

crib

e ac

cept

able

can

vol

umes

.⏐v

!14

.5⏐

/0.

08;{

v⏐14

.42

/v

/14

.58}!

4!

3!

2!

10

12

34

0!

1!

2!

3!

41

23

4

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 (A

vera

ge)

Solv

ing

Com

poun

d an

d A

bsol

ute

Valu

e In

equa

litie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

1-6

1-6


An

swer

s


Rea

din

g t

o L

earn

Math

emati

csSo

lvin

g C

ompo

und

and

Abs

olut

e Va

lue

Ineq

ualit

ies

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DAT

E__

____

____

__PE

RIO

D__

___

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

ucti

on t

o L

esso

n 1-

6 at

the

top

of

page

40

in y

our

text

book

.

•F

ive

pati

ents

arr

ive

at a

med

ical

labo

rato

ry a

t 11

:30

A.M

.for

a g

luco

seto

lera

nce

test

.Eac

h of

the

m is

ask

ed w

hen

they

last

had

som

ethi

ng t

oea

t or

dri

nk.S

ome

of t

he p

atie

nts

are

give

n th

e te

st a

nd o

ther

s ar

e to

ldth

at t

hey

mus

t co

me

back

ano

ther

day

.Eac

h of

the

pat

ient

s is

list

edbe

low

wit

h th

e ti

mes

whe

n th

ey s

tart

ed t

o fa

st.(

The

P.M

.tim

es r

efer

to

the

nigh

t be

fore

.) W

hich

of

the

pati

ents

wer

e ac

cept

ed f

or t

he t

est?

Ora

5:00

A.M

.Ju

anit

a11

:30

P.M

.Ja

son

and

Juan

itaJa

son

1:30

A.M

.Sa

mir

5:00

P.M

.

Rea

din

g t

he

Less

on

1.a.

Wri

te a

com

poun

d in

equa

lity

that

say

s,“x

is g

reat

er t

han

!3

and

xis

less

tha

n or

equa

l to

4.”

!3

0x

/4

b.G

raph

the

ineq

ualit

y th

at y

ou w

rote

in p

art

a on

a n

umbe

r lin

e.

2.U

se a

com

poun

d in

equa

lity

and

set-

build

er n

otat

ion

to d

escr

ibe

the

follo

win

g gr

aph.

{x⏐x

/!

1 or

x-

3}

3.W

rite

a s

tate

men

t eq

uiva

lent

to 4

x!

5)

2 th

at d

oes

not

use

the

abso

lute

val

uesy

mbo

l.4x

!5

-2

or 4

x!

5 0

!2

4.W

rite

a s

tate

men

t eq

uiva

lent

to 3

x"

7,

8 th

at d

oes

not

use

the

abso

lute

val

uesy

mbo

l.!

8 0

3x"

7 0

8

Hel

pin

g Y

ou

Rem

emb

er

5.M

any

stud

ents

hav

e tr

oubl

e kn

owin

g w

heth

er a

n ab

solu

te v

alue

ineq

ualit

y sh

ould

be

tran

slat

ed in

to a

n an

dor

an

orco

mpo

und

ineq

ualit

y.D

escr

ibe

a w

ay t

o re

mem

ber

whi

chof

the

se a

pplie

s to

an

abso

lute

val

ue in

equa

lity.

Als

o de

scri

be h

ow t

o re

cogn

ize

the

diff

eren

ce f

rom

a n

umbe

r lin

e gr

aph.

Sam

ple

answ

er:I

f the

abs

olut

e va

lue

quan

tity

is fo

llow

ed b

y a

0or

/sy

mbo

l,th

e ex

pres

sion

insi

de th

eab

solu

te v

alue

bar

s m

ust b

e be

twee

ntw

o nu

mbe

rs,s

o th

is b

ecom

es a

nan

din

equa

lity.

The

num

ber l

ine

grap

h w

ill s

how

a s

ingl

e in

terv

al b

etw

een

two

num

bers

.If t

he a

bsol

ute

valu

e qu

antit

y is

follo

wed

by

a -

or .

sym

bol,

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En

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NAM

E__

____

____

____

____

____

____

____

____

____

____

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DAT

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____

____

__PE

RIO

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___

1-6

1-6

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

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Chapter 1 Resource Masters - Math Class - Home©Glencoe/McGraw-Hill v Glencoe Algebra 2 Assessment Options The assessment masters in the Chapter 1 Resource Masters offer a wide range