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C H A P T E R
12
Resource Manager Inequalities
Instructional Objectives Lesson (pages) ProblemSolving Workshop (503)
NCTM Standards 2000
Objectives Use the problemsolving strategy make a table to plan a menu.
1, 4, 5, 6, 8, 9, 10
12–1 (504–508)
Graph inequalities on a number line.
1, 6, 8, 9
12–2 (509–513)
Solve inequalities involving addition and subtraction.
1, 6, 8, 9
12–3 (514–518)
Solve inequalities involving multiplication and division.
1, 6, 8, 9
12–4 (519–523)
Solve inequalities involving more than one operation.
1, 6, 8, 9
12–5 (524–529)
Solve compound inequalities.
1, 6, 8, 9
12–6 (530–534)
Solve inequalities involving absolute value.
1, 6, 8, 9
12–7 (535–539)
Graph inequalities on the coordinate plane.
1, 3, 6, 8, 9
Solve and graph quadratic inequalities.
1, 3, 6, 8, 9, 10
Investigation (540–541)
State/Local Objectives
Key to NCTM Standards 2000 1 Number & Operations; 2 Algebra; 3 Geometry; 4 Measurement; 5 Data Analysis & Probability; 6 Problem Solving; 7 Reasoning & Proof; 8 Communications; 9 Connections; 10 Representation
Suggested Pacing
See page T13 for a complete courseplanning calendar.
Standard refers to schedules that provide 45 to 55minute periods that meet each day. Block refers to schedules that provide approximately 90minute periods which may meet every day for one semester or every other day over two semesters.
PACING
DAY 1
Standard Core (Chapters 1–13)
Lesson 12–1
Standard Enhanced (Chapters 1–15)
Lesson 12–1
Block Core (Chapters 1–13)
Chapter 11 Test & Lesson 12–1
Lesson 12–2
Block Enhanced (Chapters 1–15)
Chapter 11 Test & Lesson 12–1
Lessons 12–2 & 12–3
502a Chapter 12
DAY 2
DAY 3
DAY 4
DAY 5
DAY 6
Lesson 12–2
Lesson 12–3
Lesson 12–4
Lesson 12–5
Lesson 12–2
Lesson 12–3
Lesson 12–4
Lesson 12–5
Lessons 12–3 & 12–4
Lessons 12–5 & 12–6
Lesson 12–7 & INV
SG+A
Lessons 12–4 & 12–5
Lessons 12–6 & 12–7
INV & SG+A
Chapter Test & Lesson 13–1
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Instructional Resources
Transparencies A and B
Enrichment
Graphing Calculator Masters*
SchooltoWorkplace*
Practice
HandsOn Algebra*
Study Guide
Assessment and Evaluation
Blackline Masters (page numbers)
12–1
ruler 1, 2, 4 coordinate planes 4 masking tape
72
72
72
131
12–1
12–2
ruler 1, 2, 4 straws scissors 1, 2 pipe cleaners
73
73
73
132
12–2
12–3
ruler 1, 2, 4
74
74
74
231
133, 134
12–4
graphing calculator
75
75
75
230
135
12–5
ruler 1, 2, 4 coordinate planes 4 overhead transparency sheets permanent marker
76
76
76
136, 137
12–5
12–6
ruler 1, 2, 4 flowers or leaves
77
77
77
138
12–6
12–7
coordinate planes 4 ruler 1, 2, 4
78
78
78
Materials and Manipulatives Lesson
Investigation Study Guide & Assessment/ Chapter Test
(see below for Glencoe Manipulative Resources)
231
12
12–3 33
139
12–4
34, 35
12–7
grid paper 1, 4 ruler 1, 2, 4 yellow and blue colored paper ruler 1, 2, 4 coordinate planes 4
221–229, 232–234
*See page 502c for examples of these instructional materials. Key to Glencoe Manipulative Resources 1Classroom Manipulative Resources 2Student Manipulative Resources 3Overhead Manipulative Resources 4HandsOn Algebra Masters
INV = Investigation
SG+A = Study Guide and Assessment
DAY 7
DAY 8
Lesson 12–5
Lesson 12–6
Lesson 12–5
Lesson 12–6
DAY 9
DAY 10 Lesson 12–7
Lesson 12–7
INV
DAY 11
DAY 12
DAY 13
INV
SG+A
Chapter Test
SG+A
Chapter Test
Chapter Test & Lesson 13–1
Chapter 12
502b
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Resource Manager Applications
Manipulatives/Modeling
SchooltoWorkplace Masters, p. 12 NAME
12–3
DATE
HandsOn Algebra Masters, pp. 131–139 PERIOD
SchooltoWorkplace
The efficiency of an administrative assistant depends in part on the assistant’s keyboarding speed, measured in words per minute. The chart at the right shows the efficiency ratings for four administrative assistants. Lisa, for example, can keyboard between 45 words per minute and 55 words per minute.
55
Example 1: Consider the inequality d 16. Follow the steps below to graph this inequality on the number line provided below Step 2. Lisa
Step 1 Graph a bullet at 16, since d can equal 16.
Dan Ho Juanita Assistant
Step 2 Graph all numbers greater than 16 by drawing a line and an arrow to the right. Use a colored pencil to draw the line.
Let x represent the number of minutes it will take to do the keyboarding.
14
Since 45 words per minute is a lower estimate and 55 words per minute is an upper estimate, solve the inequalities 45x 1200 and 55x 1200. 55x 1200
45x 45
55x 55
1200 45
x 26.667
15
16
17
18
Your graph should show all values that are greater than or equal to 16. Example 2: Suppose we want to graph all numbers that are less than 16. We want to include all numbers up to 16, but not including 16. Follow the steps below to graph d 16 on the number line provided below Step 2.
1200 55
x 21.818
Step 1 Graph a circle at 16, since d does not equal 16.
It will take Lisa between 21 minutes and 27 minutes to keyboard the document.
This CDROM includes all of the blackline masters and transparencies available for this program.
Student Edition Page 505
Verbal phrases like greater than or less than are used to describe inequalities. Not only can inequalities be expressed through words and symbols, but they can also be graphed.
60 Keyboard 50
35
45x 1200
PERIOD
Materials colored pencils
Speed Words/Min45 40
Suppose a manager gives Lisa a document containing 1200 words. How long will it take Lisa to keyboard the document?
DATE
HandsOn Algebra Inequalities and Their Graphs
Keyboarding Times (Administrative Assistant)
The pages shown on this page are a small sample of the materials available on TeacherWorks: AllinOne Lesson Planner and Resource Center.
NAME
12–1
Student Edition Pages 514–518
Step 2 Graph all numbers less than 16 by drawing a line and an arrow to the left. Use a colored pencil to draw the line.
Solve.
14
1. Suppose a manager gives Ho a document containing 2800 words. How long will it take Ho to keyboard the document?
15
16
17
18
Your graph should show all values that are less than 16.
between 56 minutes and 62 minutes
Try These Graph each inequality on the number lines provided. 1. a 5
2. Suppose a manager gives Juanita a document containing 18,000 words. How many hours will it take her to enter the document on a computer?
3 2 1 0 1 2 3 4 5 6 7
between 5 hours and 5.45 hours 2. c 3 3. Which administrative assistant should be given a 12,000word document for keyboarding if the manager wants it entered in the least amount of time? Explain.
6 5 4 3 2 1 0 1 2 3 4
3. y 0
5 4 3 2 1 0 1 2 3 4 5
Juanita; she has the fastest keyboarding rates.
It also includes a lesson planner and interactive Teacher Edition, so you can customize lesson plans and reproduce classroom resources quickly and easily, from just about anywhere.
©
T12
Glencoe/McGrawHill
Algebra: Concepts and Applications
©
131
Glencoe/McGrawHill
Algebra: Concepts and Applications
Technology/Multimedia Graphing Calculator Masters, pp. 33–35 124
NAME ______________________________________DATE __________PERIOD______
Casio Algebra FX 2.0
Student Edition Page 521
Solving Inequalities
127
NAME ______________________________________DATE __________PERIOD______
TI83 Plus
Programming and Inequalities
You can use a Casio Algebra FX 2.0 calculator to solve inequalities.
The calculator program, INEQUAL, can help you decide whether a coordinate pair is a solution of the inequality y x 3. For example, is the point at (2, 8) a solution?
Solve 5 – 8x 43 by using a graphing calculator. Step 1
Press
Step 2
Press F1 (TRNS) 5. After the parenthesis, enter the inequality. (The symbol is item 1, 3 in the EQUA folder.) Then press EXE .
ALPHA
[A].
Step 1
After entering the program, press Enter –x 3 as Y1.
Step 2
Press ZOOM 9 so that all coordinates on the coordinate plane will be integers.
Step 3
Use the arrow keys to move the cursor to a point in the upper right corner of the graphing screen. Press 2nd [QUIT] PRGM . Select the program INEQUAL.
Step 4
Press ENTER until you are prompted for the value of x. Then press X,T,,n ENTER . When you are prompted for the value of y, press ALPHA [Y] ENTER .
Step 5
To run the program again, press 2nd [QUIT] ENTER . If the coordinate pair that you enter is not a solution of the inequality, the point will not be graphed and the screen will simply read Done. Try the point at (0, 0). What happens?
GRPH
The solution, x 6, is shown on the screen. Try These Solve each inequality. Check your solution with a graphing calculator. 1. 9x 2 20
PROGRAM: INEQUAL : Disp "X Value?" : Input X : Disp "Y Value?" : Input Y : If Y>X+3↵ : PtOn (X,Y)
2. 5(x 4) 3(x – 4)
Y=
.
Try These Use the program to decide whether each point is a solution of y x 3. Write yes or no. 1. (1, 1)
2. (2, 4)
3. (3, 3)
4. (7, 2)
5. (3, 0)
6. (1, 2)
7. Edit the program to determine which of the points above are solutions of the inequality y 2x – 1. ©
502c Chapter 12
Glencoe/McGrawHill
33
Algebra: Concepts and Applications
©
Glencoe/McGrawHill
34
Algebra: Concepts and Applications
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D
V A N TA
A
G
E
GL
S
COE' EN
Assessment Resources Student Edition
Type Ongoing Assessment
Quizzes 1 and 2, pp. 518, 534
Mixed Review
Mixed Review, pp. 508, 513, 517, 523, 529, 534, 539 Standardized Test Practice, Chapters 1–12, pp. 546–547
Error Analysis
You Decide, pp. 507, 532
Standardized Test Prep
Standardized Test Practice, pp. 508, 513, 518, 523, 529, 534, 539 Standardized Test Practice, Chapters 1–12, pp. 546–547
OpenEnded Assessment
Math Journal, pp. 511, 538 ProblemSolving Workshop, p. 503 Investigation, pp. 540–541 Portfolio, pp. 503, 541
Chapter Assessment
Study Guide and Assessment, pp. 542–544 Chapter Test, p. 545
Teacher’s Wraparound Edition 5Minute Check, pp. 504, 509, 514, 519, 524, 530, 535
Assessment and Evaluation Masters MidChapter Test, p. 230 Quizzes A and B, p. 231 Cumulative Review, p. 232 Standardized Test Practice, pp. 233–234
Error Analysis, pp. 507, 512, 516, 522, 527, 533, 538 Standardized Test Practice, pp. 233–234
Act It Out: p. 529 Speaking: pp. 518, 539 Writing: pp. 513, 523, 534 Modeling: p. 508
Performance Assessment, p. 229
MultipleChoice Tests (Forms 1A, 1B), pp. 221–224 FreeResponse Tests (Forms 2A, 2B), pp. 225–228
Additional Chapter Resources Student Edition HandsOn Algebra, p. 511 Graphing Calculator Exploration, p. 521
Teacher’s Classroom Resources Manipulatives/Modeling Teacher’s Guide for Overhead Manipulative Resources
Meeting Individual Needs Prerequisite Skills Booklet Spanish Study Guide and Assessment, pp. 80–86, 127–128 Teaching Aids Answer Key Transparencies Block Schedule Planning Guide Lesson Planning Guide Solutions Manual
Glencoe Technology Instructional
Teaching Aids
AlgePASS CDROM, Lessons 29, 30 Interactive Chalkboard StudentWorks Multimedia Applications CDROM, Activity 7 Vocabulary PuzzleMaker
Answer Key Maker TeacherWorks Visit www.algconcepts.com for data updates, career information, games, and other interactive activities.
Assessment TestCheck and Worksheet Builder
Chapter 12
502d
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CHAPTER P R E V I E W
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12
Mathematics of the Chapter This chapter provides students with a thorough exploration of inequalities. Students will begin by graphing inequalities on a number line. Then they will learn to solve onestep and multistep linear inequalities. Next, students will solve compound inequalities and graph their solutions, and will extend these skills to solving inequalities involving absolute value. Finally, students will investigate inequalities in two variables and graph their solutions in the coordinate plane.
Prerequisite Skills
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C H A P T E R
12 >
Make this Foldable to help you organize information about the material in this chapter. Begin with four sheets of graph paper.
➊
Fold each sheet in half from top to bottom.
➋
Cut along fold. Staple the eight halfsheets together to form a booklet.
➌
Label each page with a lesson number and title.
Students will use the following concept in Chapter 12: • operations with decimals (Lesson 12–3).
▲
On each tab of this Foldable students should write rules for working with inequalities. Then students should solve an inequality of each type listed and graph the solution of the inequality.
Inequalities
12–1
Inequalitie
s
Reading and Writing As you read and study the chapter, use each page to write notes and to graph examples.
Math in the Workplace Students will learn how inequalities are used in sports, savings, and nutrition. Other realworld links and mathematics integration topics are listed in the chart below.
502 Chapter 12 Inequalities
CHAPTER 12 LINKS Lesson Math in the Workplace Applications and Connections
Math Integration
502 Chapter 12
12–1 12–2 Postal Service Sports Sports Budgeting Conservation Fundraising Entertainment Wrestling Safety Academics Finance Volunteerism
12–3 12–4 12–5 12–6 12–7 Savings School Nutrition Manufacturing Budgeting Safety Weather Pet Care Finance Sales Time Employment Construction Chemistry Animals Production Recreation Taxes Sales Airlines Budgeting Finance Cooking Travel Travel Contests Budgeting Chemistry Earthquakes Welding Geometry Geometry Measurement Probability
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ProblemSolving Workshop Objectives Students should: • plan a oneweek food menu that meets the given guidelines, • write inequalities to represent the guidelines, and • determine whether a sample menu meets the guidelines.
ProblemSolving Workshop Project Plan a oneweek menu of foods that require little or no preparation. 1. Total Calories must be between 1200 and 1800 per day. 2. Calories from fat must be at most 30% of the total Calories. Note that 1 gram of fat has 9 Calories. 3. Total sodium must be less than 2400 milligrams. 4. Total protein must be at least 46 grams.
Working on the Project
>
Work with a partner to solve the problem. sodium (mg), and protein (g) allowed per day. • Determine whether the sample menu in the table below satisfies the nutritional guidelines.
Technology Tools • Use a spreadsheet to help write a menu that meets all guidelines. • Use publishing software to illustrate your menu.
Food (one serving) canned chicken noodle soup granola cereal and skim milk canned tuna in water (2 ounces) frozen dinner white bread (3 slices)
Strategies Look for a pattern. Draw a diagram. Make a table. Work backward. Use an equation. Make a graph. Guess and check.
Calories Fat (g) Sodium (mg) Protein (g) 90 270 60 450 240
2 9 0.5 20 3
You may want to introduce the workshop at the beginning of the chapter, with the intent that it be completed by the end of Chapter 12. This should motivate students to learn how inequalities apply to the fields of nutrition and diet.
940 20 250 1530 480
6 5 13 10 6
Problem Solving Pointer As students plan their menus, ask them to consider other criteria they might use in their selections, such as what foods they like, what foods go well together, and what kind of preparation the foods need.
▲
• Write inequalities to represent the Calories, fat (g),
How to Use the Workshop
You may want to have students record what they actually eat for one week and calculate the nutritional value for comparison to the menus they plan. The students should add their menus, nutritional information, inequality lists, and any tables or spreadsheets to their portfolios at this time.
Research For more information about nutrition, visit: www.algconcepts.com
Presenting the Project Compile a portfolio of your work for this project. Include the menu for each of seven days, the nutritional value of each food, and a list of inequalities to show that all guidelines were met.
Chapter 12 ProblemSolving Workshop 503
Quick Review Math Handbook Glencoe Algebra Lesson
hot words hottopics
Quick Review Math Handbook Hot Topic
121
64, 66
Internet Address Book
122
64, 66
123
64, 66
Record useful Internet addresses in the space at right for quick reference.
124
66
125
66
126
64
127
67
Chapter 12
503
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Lesson 12–1
12–1
Inequalities and Their Graphs
1 FOCUS
What You’ll Learn
You already know that equations are mathematical statements that describe two expressions with equal values. When the values of the two expressions are not equal, their relationship can be described in an inequality.
You’ll learn to graph inequalities on a number line.
Chapter 11
Why It’s Important Postal Service
1. Write the equation of the axis of symmetry and the coordinates of the vertex of the graph of the quadratic function y x2 2x 2. Then graph the function.
Inequalities can be used to describe postal regulations. See Exercise 12.
Verbal phrases like greater than or less than describe inequalities. For example, 6 is greater than 2. This is the same as saying 2 is less than 6. The chart below lists other phrases that indicate inequalities and their corresponding symbols.
Inequalities
y Inequalities: Lesson 3–1
3. 4.
5.
x 1; (1, 3) Describe how the graph of y (x 3)2 changes from the parent graph of y x2. Then name the vertex of the graph. same shape, but opens downward and shifted to the left 3 units; (3, 0) Solve s2 s 12 0 by factoring. 3, 4 Use the Quadratic Formula to solve z2 4z 2 0. 2 2 Find the amount of money in a bank account given an initial deposit of $2500, an annual rate of 6%, and a time of 12 years. $5030.49
Motivating the Lesson RealWorld Connection Point out to students that shortterm parking garages often charge in hours, with a portion of an hour charged as a full hour. For example, parking for 3 hours or 3 hours 59 minutes costs the same amount, but at 4 hours, the cost goes up. Have students describe this situation, 3 h 4, with as many verbal phrases as they can think of. Point out that these phrases are the language of inequalities, which they will begin to explore in this lesson.
504 Chapter 12
Example
1
• less than • fewer than
• less than or equal to • at most • no more than • a maximum of
• greater than • more than
• greater than or equal to • at least • no less than • a minimum of
Suppose the minimum driving age in your state is 16. Write an inequality to describe people who are not of legal driving age in your state. Let d represent the ages of people who are not of legal driving age. The ages of all drivers
d
are greater than or equal to
16 years.
2.
x
O
d 16 is the same as 16 d.
5Minute Check
16
Then d is less than 16, or d 16.
Your Turn
a. Let n number of papers; n 50.
a. Lisa can carry no more than 50 newspapers on her paper route. Express the number of papers that Lisa can carry as an inequality.
Not only can inequalities be expressed through words and symbols, but they can also be graphed.
504 Chapter 12 Inequalities
Resource Manager Reproducible Masters • Study Guide, p. 72 • Practice, p. 72 • Enrichment, p. 72 • HandsOn Algebra, p. 131
Transparencies • 5Minute Check, 12–1 • Teaching, 12–1 • Answer Key, 12–1
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Consider the inequality d 16.
2 TEACH
Step 1 Graph a bullet at 16 to show that d can equal 16. Step 2 Graph all numbers greater than 16 by drawing a line and an arrow to the right.
On a graph, a circle means that the number is not included. A bullet means that the number is included.
14 15
16 17 18
d 16 The graph shows all values that are greater than or equal to 16.
Teaching Tip Before discussing Example 1, point out to students that the and symbols each contain the bottom half of an equal sign. This might help students distinguish the meanings of these symbols from those of or symbols.
Now suppose we want to graph all numbers that are less than 16. We want to include all numbers up to 16, but not including 16. Step 1 Graph a circle at 16 to show that d does not equal 16.
InClass Example
Step 2 Graph all numbers less than 16 by drawing a line and an arrow to the left.
Example 1
14 15
16 17 18
d 16 The graph shows all values that are less than 16.
Examples
Many movie theaters give a seniorcitizen discount to people who are 65 or over. Write an inequality that describes those who are eligible to receive the discount. a 65
Graph each inequality on a number line.
2
x 4
3
Since x cannot equal 4, graph a circle at 4 and shade to the right.
n 1.5 Since n can equal 1.5, graph a bullet at 1.5 and shade to the right.
6 5 4 3 2
1
0
1
2
3
InClass Examples
Your Turn
Examples 2–3
c. p 1
b. y 3 1
2
3
4
2
5
1
0
1
You can also write inequalities given their graphs.
Examples
Write an inequality for each graph.
4
1
0
1
2
Graph each inequality on a number line. 2 x 1 –3 –2 –1
3 k
0
1
3 4
3
Locate where the graph begins. This graph begins at 1, but 1 is not included. Also note that the arrow is to the left. The graph describes values that are less than 1. So, x 1.
www.algconcepts.com/extra_examples
Teaching Tip While discussing Examples 2–4, point out to students that to check their graphs, they can pick a point in the shaded region. This point should satisfy the inequality.
Lesson 12–1 Inequalities and Their Graphs 505
0
1– 4
1– 2
3– 4
1
Example 4 Write an inequality for the graph. x 2 0
1
2
3
4
Lesson 12–1
505
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In Class Examples
5
0
Example 5
2 4
3 4
1
1 4
Note that the arrow is to the right. The graph describes values 1 greater than or equal to .
0
4
1 4
So, x .
2
y 3
Your Turn
2 3 e.
d. x 2 e. x
d.
Example 6
al Wo
Example
6
Sports Link
10 11 12 13 14
0
0
1 3
Answers
To play junior league soccer, you must be at least 14 years of age.
a 14 B. Graph the inequality on a number line. To graph the inequality, first graph a bullet at 14. Then include all ages greater than 14 by drawing a line and an arrow to the right. 12
Page 507
13 14
15 16
C. The Valdez children are 10, 13, 14, and 16 years old. Which of the children can play junior league soccer?
7. 4
5
6
7
8
0
1
2
3
4
3
4
5
6
7
5
6
7
8
9
The set of the children’s ages, {10, 13, 14, and 16}, can be called a replacement set. It includes possible values of the variable a. In this case, only 14 and 16 satisfy the inequality a ≥ 14 and are members of the solution set. So, the two Valdez children that are 14 and 16 years old can play junior league soccer.
8. 9. 19.
Check for Understanding
Study Guide Masters, p. 72 NAME
DATE
PERIOD
Study Guide
Student Edition Pages 504–508
Inequalities and Their Graphs Many mathematical relationships can be expressed with inequalities. For example, the President of the United States must be at least 35 years old. If a represents his or her age, this can be expressed with the inequality a 35. Some verbal phrases that can be used for inequalities are listed in the chart below. less than or equal to at most no more than a maximum of
• greater than • greater than or equal to • more than • at least • no less than • a minimum of
Communicating Mathematics
1. Match each inequality symbol with its description. a. i. less than or equal to b b. ii. at least a c. iii. fewer than d d. iv. greater than c
506 Chapter 12 Inequalities
Example 1: Write and graph an inequality to describe the age of people who cannot be President of the United States. Let a represent the age of a person who is less than 35 years old. Then a 35. Since a cannot equal 35, graph a circle at 35. Then graph all numbers less than 35 by drawing a line and an arrow to the left. 32 33 34 35 36 37
Example 2: Graph n 2.5 on a number line. Since n can equal 2.5, graph a bullet at 2.5. Then graph all numbers greater than 2.5 by drawing a line and an arrow to the right. 1
2
3
4
5
6
Write an inequality to describe each number. Then graph the inequality on a number line. 1. a number less than 3 x 3 –1 0
1
2
3
4
2. a number more than 2 x 2
5
–4 –3 –2 –1 0
3. a maximum number of 8 x 8 4
5
6
7
8
9 10
–3 –2 –1 0
1
1
2
3
4
5
x
6
7
8
2
506 Chapter 12
2
3
1
1
3
9 10 11 12
Glencoe/McGrawHill
1
6. a minimum number of 6 5
1 1 2
7. a number that is at most 10.2 x 10.2
©
1
4. a number that is at least 1 x 1
5. a number greater than or equal to 1 2 –1 0
1
13
Let a represent the ages of people who can play junior league soccer. Then write an inequality using since the soccer players have to be greater than or equal to 14 years of age.
2. There is a bullet at 12 and an arrow pointing to the right. 3. The graph of x 4 has a bullet at 4. The graph of x 4 has a circle at 4.
• • • •
1
A. Write an inequality to represent this situation.
s 12
• less than • fewer than
2 3
Inequalities are commonly used in the real world.
rld
Re
4 3 2 1
No more than 12 students can fit in a large school van. Write an inequality to express this information. Then graph the inequality on a number line.
12–1
1 4
Locate where the graph begins. This graph begins at and includes .
Write an inequality for the graph. – 4–3 –1 – 2–3 – 1–3
1 4
4
6
7
8
9
x 6 5
8. a number less than 2.4 x 2.4 –1 0
T72
5
1
2
3
4
5
Algebra: Concepts and Applications
Reteaching Activity Kinesthetic Learners Make a large number line on the floor using masking tape. Mark off the integers from 5 to 5 and ask 11 students to stand on the integers. Write both verbal and mathematical inequalities on the board. For each inequality, have the 11 students raise their hands if their point satisfies the inequality.
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2. Describe the graph of x 12. 2–3. See margin. 3. Explain the difference between the graphs of x 4 and x 4. 4. Soto says that x 7 is the same as 7 x. Darrell says that it is not. Who is correct? Explain. Soto; in both,
3 PRACTICE/APPLY Error Analysis
the inequality symbol is opening toward the x. (Example 1)
Write an inequality to describe each number.
Guided Practice
x 14
5. a number less than 14
x0
6. a number no less than 0
Graph each inequality on a number line. (Examples 2 & 3) 7. x 6
7–9. See margin.
8. x 2
9. 5 x
Write an inequality for each graph. (Examples 4 & 5)
x 8 11.
10. 6
7
8
9
3 2 1
10
0
x 1
1
12. Mail All letters mailed with one firstclass stamp can weigh no more than 1 ounce. Write an inequality to represent this situation. Then graph the inequality. (Example 6) 1 1
Exercises
•
•
•
•
•
•
See Examples 1 2, 3
31–38
4, 5
39–42
6
•
3
•
•
4
•
•
•
14. x 13
13. a number greater than 4 x 4 14. a number less than or equal to 13 16. a number no more than 7 x 7 15. a number less than 9 x 9 17. a number that is at least 8 18. a number more than 6
Homework Help 13–18 19–30
•
Write an inequality to describe each number.
Practice
For Exercises
• • • • •
2
x 8
x 6
Graph each inequality on a number line. 19–30. See margin. 19. x 7 22. a 5 25. x 2.5
Extra Practice See page 715.
20. y 3 23. x 4 26. d 1.9
3 4
21. b 0 24. 1 z 27. x 3.4
1 2
Teaching Tip You may want to discuss the domain of the variable in Exercise 12. Can the weight be a negative number or zero? This realworld example actually requires the compound inequality greater than 0 and less than or equal to 1. Students will explore compound inequalities in Lesson 12–5. Assignment Guide Basic: 13–41 odd, 42–50 Average: 14–38 even, 39–50
1 3
29. 1 x
28. g
Watch for students who reverse the direction of the inequality sign when translating verbal inequalities that involve a negation, such as no less than in Exercise 6 or not greater than. Prevent by encouraging students to translate the original verbal statement to a statement they understand better before writing the mathematical inequality. For example, students may recognize that no less than means the same thing as at least, which they may feel more comfortable translating into mathematical symbols.
30. h
Write an inequality for each graph. 31.
32.
8 7 6 5 4
x 6
35.
34.
37.
2
3
4
5
x3 3
36. 4
5
6
7
x5 2
0
1
20.
2
1
x0
33. 1
Answer 2 1
38. 3
11 10 9 8 7
0
4
DATE
PERIOD
Practice
Student Edition Pages 504–508
Inequalities and Their Graphs Write an inequality to describe each number. 1. a number less than or equal to 11
1.5
5
NAME
12–1
1
x 1
4
2
2. a number greater than 3
x 11
2.5
x3 4. a number that is no less than 7
3. a number that is at least 6
x 1.75 Lesson 12–1 Inequalities and Their Graphs 507
x 2.5
3
Practice Masters, p. 72
x 9 3 2 1
2
x6
x 7
6. a number that is less than 2
5. a maximum number of 9
x9
x 2
Graph each inequality on a number line. 7. x 4 1
Answers 21. 0
1
3
4
5
6
1.5
1
2
16. x
3
6 5 4 3 2
24. 1
1
2
3
25.
1 4
2 4
3 4
1
3
4
2
1
2
3
13
1 3
1
0
8
1
5
2
0
1
6
–4
2
1
–2
8
9 10 11 12 13
12. 7 g 7
8
9 10 11
15. 2.5 h
–1
1
7
9 10 11
–3
–2
–1
1 18. 2 x
17. m 2
20. 2
3
4
5
6
7
4
–1
0
2
1
23. –3 –2 –1 0
3
–8 –7 –6 –5 –4 –3 –2
21.
x 6 1
2
3
x 1
3
4
–2 –1 0
5
Glencoe/McGrawHill
6
26.
5
6
7
8
9 10 11
x8 1
2
3
4
24.
x0
x 4.5
©
23
7
–4 –3 –2 –1 0
3
22.
25.
30. 2
1
1 3
19. 1
0
9. y 9
6
14. x 0.5
x4
29. 0
5
7
Write an inequality for each graph.
28.
23.
6
11. p 2
0
3.5
4
7
5
–7 –6 –5 –4 –3 –2 –1
27.
22.
4
13. y 1.5 0
2
3
10. 5 x
26. 2 1
8. x 8 2
–3
–2
–1
0
x 1.5
1
2
3
27.
–1
0
3 x
2 x 1 3
4
T72
Algebra: Concepts and Applications
Lesson 12–1
507
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Modeling Draw a large number line on a poster board or on the chalkboard. You may want to include decimal or fractional values. Make two large arrows out of construction paper, one with an open circle at one end and the other with a closed circle at one end. Have students take turns writing mathematical inequalities on the board or overhead or announcing verbal inequalities. Have other students place the appropriate arrow in the appropriate position on the number line to graph each inequality.
For Exercises 39–41, write an inequality to represent each situation. Then graph the inequality on a number line.
al Wo
39. Conservation At a pond, you must return any fish that you catch if it weighs less than 3 pounds. f 3
rld
OpenEnded Assessment
Re
Applications and Problem Solving
4 ASSESS
1
2
3
4
5
40. Entertainment A roller coaster requires a person to be at least 42 inches tall to ride it. h 42 40
41 42 43
44
41. Safety The elevators in an office building have been approved to hold a maximum of 3600 pounds. w 3600 3400
Internet Use Surfing Internet
84% 84%
Going online more than once a day
71% 71%
Spending 1– 4 hours a day online
61% 61%
Regularly visiting shopping sites
3600
3800
42. Critical Thinking The graph shows survey results on the Internet use of 100 college students. Write an inequality to represent the number of students that may spend at least 5 hours online each day.
n 39
32% 32% Source: USA TODAY
Mixed Review
43. Finance Mr. Johnson invested $10,000 at an annual interest rate of 6%. Graph the function B(t) 10,000(1.06)t, where t is the time in years. How many years will it take before the balance is over $15,000? (Lesson 11–7) 7 yr Solve each equation by using the Quadratic Formula. (Lesson 11–6) 44. t2 3t 40 0
45. 3a2 a 1 0
5, 8
46. x2 6x 7 0
no real solutions
3 兹2 苶
47. Find the prime factorization of 24. How many distinct prime factors are there? (Lesson 10–1) 23 3; two Simplify each expression. 48. 兹16 苶 (Lesson 8–5) 4
Standardized Test Practice
Enrichment Masters, p. 72 NAME
12–1
DATE
PERIOD
Enrichment
Student Edition Pages 504–508
Intersection and Union The intersection of two sets is the set of elements that are in both of the sets. The intersection of sets A and B is written A B. The union of two sets is the set of elements in either A, or in B, or in both. The union is written A B. In the drawings below, suppose A is the set of points inside the circle and B is the set of points inside the square. Then, the shaded areas show the intersection and union.
A
B
A
AB
Intersection
AB
Union
Write A B and A B for each of the following. A B {q, r, s} A B {p, q, r, s, t} 1. A {p, q, r, s, t} B {q, r, s} A B {3} 2. A {the integers between 2 and 7} B {0, 3, 8} A B {0, 3, 4, 5, 6, 8} A B {Kansas} A B {Hawaii, Kansas, 3. A {the states whose names start with K} Kentucky} B {the states whose capital is Honolulu or Topeka} A B {1, 2, 3, 4, 6, 8} 4. A {the positive integer factors of 24} A B {1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24} B {the counting numbers less than 10} Suppose A {numbers x such that x 3}, B {numbers x such that x 1}, and C {numbers x such that x 1.5}. Graph each of the following. 5. A B –4
–3
6. A B –2
–1
0
1
2
3
4
–4
7. B C –4
–3
–2
–1
0
1
2
3
–4
–3
–2
–1
–4
4
9. (A C) B
©
–3
–2
–1
0
1
2
3
4
–3
–2
–1
0
1
2
3
4
–1
0
1
2
3
4
8. B C
10. A (B C) 0
1
2
3
4
Glencoe/McGrawHill
508 Chapter 12
–4
T72
–3
–2
Algebra: Concepts and Applications
6x 2y
50. Multiple Choice If two lines are perpendicular to each other, which of the following statements could be true? (Lesson 7–7) I. The slopes of both lines are equal. B II. One slope is positive, and the other slope is negative. III. The slope of one line is the reciprocal of the slope of the other line. A I only B II only C III only D I and II E II and III
508 Chapter 12 Inequalities
B
18x3y 3x
49. (Lesson 8–2)
Extra Credit Three players on a basketball team have each scored at least 16 points per game in the last three games. Write an inequality for the total number of points the three players have scored in the three games. p 144
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12–2 What You’ll Learn You’ll learn to solve inequalities involving addition and subtraction.
Why It’s Important Sports You can use inequalities to set goals in competitions. See Example 2.
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Solving Addition and Subtraction Inequalities
Lesson 12–2
The Iditarod, nicknamed “The Last Great Race on Earth,” is a dogsled race in Alaska. The 12 to 16dog teams cover over 1150 miles in subzero weather. The record time for finishing the race is 218 hours (9 days, 2 hours).
1 FOCUS 5Minute Check Lesson 12–1 Write an inequality for each graph. 1. –3 –2 –1
A dogsled race
Suppose a team takes 73 hours to reach the first checkpoint of the Iditarod and 98 hours to reach the second. How much time can be spent on the last leg of the race to beat the record time? Let t represent the time for the last leg of the race. Write an inequality.
73 98 t
218
The sum of the times must be less than the record time. 171 t 218
x 1 2. – 1–3
x
Words:
Example
1
For any inequality, if the same quantity is added or subtracted to each side, the resulting inequality is true. Symbols: For all numbers a, b, and c, 1. if a b, then a c b c and a c b c. 2. if a b, then a c b c and a c b c. Numbers: 5 1 2 4 5 2 1 2 2 3 4 3 3 3 1 7
Solve x 14 5. Check your solution. x 14 5 x 14 14 5 14 Subtract 14 from each side. x 9 (continued on the next page)
Lesson 12–2 Solving Addition and Subtraction Inequalities
0
1– 3
2– 3
1
1 3
–1.4
509
–1
–0.6
4. w 11 9
Addition and Subtraction Properties for Inequalities
1
Graph each inequality on a number line. 3. b 1.2
If this were an equation, we would subtract 171 from each side to solve for t. The same procedure can be used with inequalities, as explained by the properties below. This problem will be solved in Example 2.
These properties are also true for and .
0
10 11 12 13
5. A small theater can seat no more than 230 people. Write an inequality to represent this situation. p 230
Motivating the Lesson RealWorld Connection Pose the following application to students. Suppose that you plan to run at least 7.5 miles per week during the summer to train for the crosscountry season in the fall. During one week, you run 2 miles on Sunday and 2.5 miles on Wednesday. How many more miles do you need to run before the week is over? Have students explain why an inequality is involved, and have them write an inequality to describe the situation.
Resource Manager Reproducible Masters • Study Guide, p. 73 • Practice, p. 73 • Enrichment, p. 73 • HandsOn Algebra, p. 132
Transparencies • 5Minute Check, 12–2 • Teaching, 12–2 • Answer Key, 12–2 Technology/Multimedia • AlgePASS, Lesson 29 Lesson 12–2
509
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Check: Substitute a number less than 9, the number 9, and a number greater than 9 into the inequality. Let x 10. Let x 9. Let x 0. x45 x 14 5 x 14 5 ? ? ? 9 14 5 0 14 5 10 14 5 4 5 false 5 5 true 14 5 true The solution is {all numbers greater than or equal to 9}.
2 TEACH Teaching Tip Students may better understand the check for Example 1 if they first sketch the solution on a number line. Point out that they can then pick the endpoint of the ray and one number on either side of the endpoint to test in the original inequality. Those numbers that make the inequality true should be in the shaded area of the graph.
Your Turn
a. {all numbers less than 5} b. {all numbers greater than or equal to 18}
InClass Examples
Solve each inequality. Check your solution.
a. x 2 7
b. x 6 12
A more concise way to express the solution to an inequality is to use setbuilder notation. The solution in Example 1 in setbuilder notation is {xx 9}.
In Lesson 12–1, you learned that you can show the solution to an inequality on a line graph. The solution, {xx 9}, is shown below.
Re
11 10 9 8 7
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rld
Teaching Tip Draw students’ attention to the equivalent statements 16 y and y 16 in Example 3. Explain that any inequality can be reversed in this manner. Stress that the symbol is still in the same relationship with the variable. In this case, the small end of the inequality points toward the variable y.
The set of all numbers x such that x is greater than or equal to 9.
Example 2 Refer to the application in the Motivating the Lesson note in the side column of page 509. Solve 2 2.5 m 7.5 to find how many more miles must be run. m 3
Solve y 5 2. Check your solution. y 3
x 9}
{x
Example 1
Examples
2
Sports Link
Refer to the application at the beginning of the lesson. Solve 171 t 218 to find the time needed to finish the last leg of the Iditarod and beat the record. 171 t 218 171 t 171 218 171 Subtract 171 from each side. t 47 This means all numbers less than 47. The solution can be written as {tt 47}. So any time less than 47 hours will beat the record.
3
Solve 7y 4 8y 12. Graph the solution.
InClass Example
7y 4 8y 12 7y 4 7y 8y 12 7y Subtract 7y from each side. 4 y 12 4 12 y 12 12 Add 12 to each side. 16 y
Example 3
Since 16 y is the same as y 16, the solution is {yy 16}.
Solve 4w 8 3w 10. Graph the solution. {w  w 2} –1
0
1
2
The graph of the solution has a circle at 16, since 16 is not included. The arrow points to the left.
15 16
17 18
3
510 Chapter 12 Inequalities
Reteaching Activity Intrapersonal Learners Have students make a twocolumn table. In the first column, have them list examples of mistakes to avoid when solving and graphing inequalities. In the second column, opposite each possible mistake, have students write a suggestion to themselves about how to avoid making the mistake.
510 Chapter 12
14
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Your Turn
c. {yy 6} 4
5
6
c. 5y 3 6y 9 7
Answers
Solve each inequality. Graph the solution.
1. They are the same. 2. Sample answer: y 4 and y40
d. 3r 7 2r 4
8
d. {rr 3}
You can also use inequalities to describe some geometry concepts.
5 4 3 2 1
Materials:
straws
scissors
pipe cleaners Step 1
Step 2
1a. Yes, the sum of any two sides is greater than the third. 2. side 1 side 2 side 3
Cut one straw so that it is 3 inches long. Cut a second straw 4 inches long. Cut a third straw 5 inches long. Insert a pipe cleaner into each straw. Then form a triangle. Label each side like the one shown at the right.
Math th Jo Journal
5 in. 4 in. y
z x 3 in.
Try These 1. Repeat Steps 1 and 2 for each of the side lengths listed below. In each case, can a triangle be formed? Explain. a. x 4 in., y 8 in., z 6 in. 5. b. x 3 in., y 5 in., z 2 in. No, 2 3 c. x 1 in., y 6 in., z 3 in. No, 1 3 6. 2. Study the triangles in Exercise 1. Use an inequality to describe what must be true about the side lengths to form a triangle. 3. Can a triangle be formed with side lengths 9 cm, 12 cm, and 15 cm? Explain. Yes, the sum of any two sides is greater than the third.
Check for Understanding Communicating Mathematics
ruler
1–2. See margin.
1. Compare and contrast solving inequalities by using addition and subtraction with solving setbuilder notation equations by using addition and subtraction. 2. Write two inequalities that each have {yy 4} as their solution. 3. List three situations in which solving an inequality may be helpful.
See students’ work. Guided Practice
Getting Ready
Study Guide Masters, p. 73
Write an inequality for each statement.
Sample: Five more than a number is greater than sixteen. Solution: n 5 16
PERIOD Student Edition Pages 509–513
Suppose you already have $50 and want to earn at least enough money to buy a DVD player for $325. Let m the amount of money you earn. You can represent this situation with the inequality m 50 325. Then the solution to m 50 325 is the amount of money you must earn. You can use the Addition and Subtraction Properties for Inequalities to solve inequalities involving addition or subtraction. The properties are summarized below. Addition and Subtraction Properties for Inequalities
511
For all numbers a, b, and c, 1. if a b, then a c b 2. if a b, then a c b 3. if a b, then a c b 4. if a b, then a c b Example:
HandsOn Algebra Cooperative Learning In this activity students discover the Triangle Inequality, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Point out that this requirement must be true of any two sides when compared to the third side. In practice, however, students need test only that the sum of the lengths of the two shortest sides is greater than the length of the third side. HandsOn Algebra Masters, p. 132
DATE
Study Guide
Solving Addition and Subtraction Inequalities
4. A number minus three is greater than or equal to ten. n 3 10 5. The sum of 8 and a number is at most 12. 8 n 12
Lesson 12–2 Solving Addition and Subtraction Inequalities
NAME
12–2
c, c, c, c,
and and and and
a a a a
c c c c
b b b b
c. c. c. c.
Solve m 50 325. Check your solution. m 50 325 m 50 50 325 50 Subtract 50 from each side. m 275 Check: Substitute a number less than 275, the number 275, and a number greater than 275 into the inequality. Let m 200. Let m 275. Let m 300. m 50 325 m 50 325 m 50 325 ? ? ? 275 50 325 300 50 325 200 50 325 250 325; false 325 325; true 350 325; true In setbuilder notation the solution is {all numbers greater than or equal to 275}, or {mm 275}.
Solve each inequality. Check your solution. 1. n 3 6 {n n 3}
2. x 6 2 {x  x 4}
3. 2 y 8 { y  y 10}
4. x 4 12 {x  x 16}
5. 3 t 2 {t  t 5}
6. 1 p 1 { p  p 2}
7. y 1.2 3.4 { y  y 2.2} 9. 1.8 y 0 { y  y 1.8} 11. 1 y ©
2 3
{ y y 1 23 }
Glencoe/McGrawHill
8. 2.6 x 1.9 {x  x 4.5} 1 2
12. p
1 8
1 2
T73
3 4
{ x x 1 14 } { p p 1 58 }
10. x
1
Algebra: Concepts and Applications
Lesson 12–2
511
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Solve each inequality. Check your solution. (Examples 1 & 2)
3 PRACTICE/APPLY
6. x 9 12 {xx 3} 8. a 6 14 {aa 8} 10. 4.6 x 2.1 {xx 2.5}
Error Analysis Watch for students who come up with coefficients of 1 while solving Exercises 12 and 13. Prevent by encouraging students always to subtract the smaller variable term from each side. Remind students that they may have to rewrite the resulting solution to have the variable on the left in setbuilder notation.
Basic: 15–43 odd, 45–54 Average: 16–40 even, 41–54
Data Update For the latest prices on computer hardware, visit: www.algconcepts.com
Practice
18 17 16 15 14
13. 1
33. 11
12
13
14
15
8
9
10
11
12
33–38. See margin for graphs. 35. {xx 16}
36. 7 6 5 4 3
37. 10
11
9
10
11
12
13
DATE
PERIOD
Re
Solve each inequality. Check your solution. 2. b 4 3
{w  w 4}
8. 10 c 13
{n n 8}
9. q 9 4
{c  c 3}
10. 5 d 7
{q  q 13}
11. 17 v 11
{d  d 2}
12. 14 h 9
{v  v 6}
13. x 1.7 5.8
{h  h 23}
14. 2.9 s 5.7
{x x 4.1}
6. 3 w 1
{a  a 11}
7. n 1 7
16. y
{ y  y 6}
5. a 2 9
{f  f 15}
1 2
3. y 6 12
{b  b 7}
15. 0.3 g 4.4
{s s 2.8}
3 2 4
17.
{ y y 2 14 }
1 1 4
{g g 4.7}
5 4 8
m
1
18. 2 r 6
{m m 3 83 }
2 3
{r  r 2 56 }
Solve each inequality. Graph the solution. 19. 5x 2 6x
{x  x 2} –4 –3 –2 –1 0
1
2
22. 7p 3(2p 1)
©
1
2
21. 2y 6 3y 9
{ n n 8}
{ y  y 3}
6
7
8
9 10 11 12
23. 9m 6 8m 5
{p  p 3} –1 0
20. n 7 2n 1
{m  m 1} 3
4
5
–2 –1 0
Glencoe/McGrawHill
512 Chapter 12
1
T73
–6 –5 –4 –3 –2 –1 0
24. 2h 11 3h 7
{ h h 4} 2
3
4
–6 –5 –4 –3 –2 –1 0
Algebra: Concepts and Applications
al Wo
rld
Solving Addition and Subtraction Inequalities
4. f 9 24
Applications and Problem Solving
Student Edition Pages 509–513
{x  x 9}
•
•
•
•
•
•
•
•
•
•
•
n 6 9 {nn 3} 3 b 8 {bb 5} r 8 11 {rr 19} w 9 13 {ww 22} t 5 5 {tt 0} 2 c 2 {cc 4} d 1.4 6.8 {dd 5.4}
}
x 7 3 {xx 4} g 12 5 {gg 7} x 6 14 {xx 8} 4 p 1 {pp 3} x 3 19 {xx 16} 11 4 m {mm 15} 3 x 11.9 {xx 14.9} 1 3 30. s (2) ss 1 4 4 1 2 32. x xx 7 2 3 6 16. 18. 20. 22. 24. 26. 28.
{
{
}
}
33. 7 8g 7g 6 {gg 13} 35. 3x 12 2x 4 37. (t 9) 0 {tt 9}
34. 5n 4n 10 {nn 10} 36. 6a 5(a 1) {aa 5} 38. 3(v 5) 2(v 2) {vv 11}
5 10 x 25, x 10
Practice
1. x 7 16
•
39. Eight less than a number is not greater than 12. x 8 12, x 20 40. The sum of 5, 10, and a number is more than 25.
Practice Masters, p. 73 NAME
•
Write an inequality for each statement. Then solve.
38.
12–2
• • • • •
Solve each inequality. Graph the solution.
18 17 16 15 14
9
14. Budgeting Antonio can spend no more than $1000 on a new computer system. The hard drive he wants costs $220. The monitor costs $300. How much money does Antonio have to spend on other components? (Example 2) m $480
{
35.
8
13. 4x 1 5x {xx 1}
12–13. See margin for graphs.
29. 0.2 0.3 z {zz 0.5} 1 3 7 31. 3 v 5 vv 2 8 8 2
34.
7
12. z 12 2 z 4 {zz 16}
15. 17. 19. 21. 23. 25. 27.
Answers
0
}
Solve each inequality. Check your solution.
12.
3 2 1
{
Solve each inequality. Graph the solution. (Example 3)
Exercises
Assignment Guide
7. v 12 5 {vv 17} 9. h 7 14 {hh 21} 2 1 11. y 1 yy 15 3 6 6
41. Fundraising Westfield High School is having a raffle to raise money for Habitat for Humanity. Any homeroom that sells at least 150 tickets will get to help build a home. Ms. Martinez’ homeroom is keeping a table of the number of tickets sold each day. How many more tickets do they need to sell to help build a home? t 99
512 Chapter 12 Inequalities
Day Tickets 1 2 3 4 5
32 19 ? ? ?
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42. Sports Carlos weighs 175 pounds. At least how much weight must he lose to wrestle in the 171pound weight class? w 4 43. Academics Alissa must earn 475 out of 550 points to receive a grade of B. So far, she has earned 244 test points, 82 quiz points, and 50 homework points. How many points must she score on her final exam to earn at least a B in the class? p 99 44. Volunteerism Each summer, 70 men bicycle in the Journey of Hope to raise money for people with disabilities. They begin their journey in San Francisco, California, and end in Washington, D.C. The map shows the miles they cycle in California. It takes them at most 285 miles to cycle to Nevada. How many miles is it from Jackson, California, to their first stop in Nevada? m 115
Carson City
Yuba City
Auburn
4 ASSESS OpenEnded Assessment Writing Have students write a brief summary describing the properties they can use to solve an addition or subtraction inequality. Have them include examples to illustrate the properties.
Sacramento
65 mi
48 mi
Napa Fairfield
Jackson
Lodi
57 mi
Stockton
Berkeley
San Francisco
Modesto
45. Critical Thinking Choose the correct term and justify your answer. The solution set of an inequality is (sometimes, always, never) the empty set ( ). Sometimes; examples are x x and 2x 1 2x.
Mixed Review
Write an inequality to describe each number.
(Lesson 12–1)
46. a number no more than 1 x 1 47. a number less than 8 x 8 48. a number greater than 3 x 3 49. a minimum number of 5 x 5 y
50. The graph of which of the following equations is shown at the right: y 2x, y 2x 1, y 2x 1, or y 2x? (Lesson 11–7) y 2x 1 O
51. Factor the trinomial 3x2 13x 10.
x
(Lesson 10–4) (3x 2)(x 5)
52. Write 3x y 8 in slopeintercept form. (Lesson 7–3) y 3x 8
54. Multiple Choice Suppose **x** 4x 2. If **x** 10, then what is the value of x? (Lesson 4–4) B A 2 B 3 C 38 D 40
Lesson 12–2 Solving Addition and Subtraction Inequalities
513
12–2
NAME
DATE
PERIOD
Enrichment
Student Edition Pages 509–513
Consecutive Integers and Inequalities Consecutive integers follow one after another. For example, 4, 5, 6, and 7 are consecutive integers, as are 8, 7, 6. Each number to the right in the series is one greater than the one that comes before it. If x the first consecutive integer, then x 1 the second consecutive integer, x 2 the third consecutive integer, x 3 the fourth consecutive integer, and so on. Example: Find three consecutive positive integers whose sum is less than 12. first integer
www.algconcepts.com/self_check_quiz
Enrichment Masters, p. 73
x
second integer
third integer
53. Grid In If y varies inversely as x and y 8 when x 24, find y when x 6. (Lesson 6–6) 32
Standardized Test Practice
x 1 x 2 12
Simplify the expression by combining like terms.
3x 3 12 3x 3 3 12 3
Subtract 3 from each side.
3x 9 3x 9 3 3
Divide each side by 3.
x3
So x could equal 1 or 2.
If x 1, then x 1 2, x 2 3, and {1, 2, 3} is one solution.
Extra Credit
If x 2, then x 1 3, x 2 4, and {2, 3, 4} is another solution. Each of the two solutions must be considered in the answer. The solution set is {1, 2, 3; 2, 3, 4}.
Write a realworld problem that involves an addition or subtraction inequality. Define a variable and write the inequality. Then solve the inequality and graph its solution. See students’ work.
Solve. Show all possible solutions. 1. Find three consecutive positive integers whose sum is less than 15.
2. Find two consecutive positive even integers whose sum is less than 10.
{1, 2, 3; 2, 3, 4; 3, 4, 5}
{2, 4}
3. Find three consecutive positive integers such that the second plus four times the first is less than 21.
4. Find three consecutive positive even integers such that the third plus twice the second is less than 26.
{1, 2, 3; 2, 3, 4; 3, 4, 5} ©
Glencoe/McGrawHill
{2, 4, 6; 4, 6, 8} T73
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Lesson 12–2
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Lesson 12–3
12–3
1 FOCUS
What You’ll Learn
5Minute Check
You’ll learn to solve inequalities involving multiplication and division.
Lesson 12–2
Why It’s Important
Solve each inequality. Check your solution. 1. r 10 2 {r  r 8} 2. 3 x 2.1 {x  x 0.9} 3. 4 3t 2t 7 {t  t 3}
Savings You can use
Solving Multiplication and Division Inequalities In the previous lesson, you learned that addition and subtraction inequalities are solved with the same procedures as addition and subtraction equations. Can you use the procedures for solving multiplication and division equations to solve inequalities as well? For example, to solve 4x 36, we would divide each side by 4. Will this work when solving inequalities? Consider the inequality 4 36.
inequalities when you are trying to budget your money. See Exercise 37.
4 36 4 ? 36 4 4
1 9 true
4. The chart shows the time Rachel has jogged so far this week. How many more minutes m must she jog this week for her weekly total to be at least 2 hours?
So, it is possible to solve an inequality when dividing by a positive number. What happens when you divide by a negative number? Consider the inequality –4 36. –4 36 – 4 ? 36 –4 –4
Day Mon Tue Wed Minutes 25 30 28 Jogged m 37 5. The sum of the lengths of any two sides of a triangle is greater than the length of the third side. What length must x be greater than for the figure below to be a triangle? What length must x be less than? 4 in.; 14 in. 5 in.
Divide each side by 4.
Divide each side by –4.
1 –9 false When dividing by a negative number, you must reverse the symbol for the inequality to remain true. This example leads us to the Division Property for Inequalities. Words:
This property is also true for and .
Division Property for Inequalities
x in. 9 in.
If you divide each side of an inequality by a positive number, the inequality remains true. If you divide each side of an inequality by a negative number, the inequality symbol must be reversed for the inequality to remain true. Symbols: For all numbers a, b, and c, a b 1. if c is positive and a b, then , and
c c a b c c a b 2. if c is negative and a b, then , and c c a b if c is negative and a b, then . c c 9 6 Numbers: If 9 6, then or 3 2. 3 3 9 6 If 9 6, then or –3 2. 3 3
if c is positive and a b, then .
514 Chapter 12 Inequalities
Resource Manager
MODELING Alternative handson options are available for teaching this lesson.
514 Chapter 12
Reproducible Masters • Study Guide, p. 74 • Practice, p. 74 • Enrichment, p. 74 • HandsOn Algebra, pp. 133–134 • Assessment and Evaluation, p. 231 • SchooltoWorkplace, p. 12
Transparencies • 5Minute Check, 12–3 • Teaching, 12–3 • Answer Key, 12–3 Technology/Multimedia • AlgePASS, Lesson 29
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Examples
1
Sports Link
Carmen runs at least 15 miles in the park every day. If she runs 5 miles per hour, how long does she run every day? Recall that rate times time equals distance, or rt = d. 5t 15 5t 15 5 5
Divide each side by 5.
t3
Keep the symbol facing the same direction.
Carmen runs at least 3 hours every day.
Prerequisite Skills Review Operations with Decimals, p. 684
2
Solve 10x 25.6. Check your solution. 10x 25.6 10x 25.6 10 10
Divide each side by –10 and reverse the symbol.
x 2.56
Check: Substitute 2.56 and a number greater than 2.56, such as 0, into the inequality. Let x 2.56. 10x 25.6 ?
10(2.56) 25.6
Motivating the Lesson HandsOn Activity Have students write the integers from 5 to 5 on small squares of paper and order the integers from least to greatest. Have them multiply each integer by 2, write the product in pencil on the back of each square, and order the products. Point out that the order is the same. Next have students erase the original products, write the product of the original integers and 1, and order the products. Seeing that the order reverses should help students understand why an inequality symbol reverses when an inequality is multiplied or divided by a negative number.
Let x 0. 10x 25.6 ?
10(0) 25.6
25.6 25.6 true
2 TEACH
0 25.6 true
InClass Example
The solution set is {xx 2.56}.
Example 1 Your Turn Solve each inequality. Check your solution. a. 8x 40 {xx 5}
b. x 4.7 {xx 4.7}
We can also solve inequalities by multiplying. Use the Multiplication Property for Inequalities. Words: This property is also true for and .
Multiplication Property for Inequalities
If you multiply each side of an inequality by a positive number, the inequality remains true. If you multiply each side of an inequality by a negative number, the inequality symbol must be reversed for the inequality to remain true. Symbols: For all numbers a, b, and c, 1. if c is positive and a b, then ac bc, and if c is positive and a b, then ac bc. 2. if c is negative and a b, then ac bc, and if c is negative and a b, then ac bc. Numbers: If 3 7, then 3(2) 7(2) or 6 14. If 3 7, then 3(2) 7(2) or 6 14.
www.algconcepts.com/extra_examples
Lesson 12–3 Solving Multiplication and Division Inequalities 515
An electric car that needs to be recharged every 260 miles should travel no more than 130 miles from the charger. If you drive at an average speed of 50 miles per hour, what are the lengths of time t you can drive away from the charger and then still make it back without running out of energy? t 2.6 h, or 2 h 36 min Teaching Tip As you discuss Example 2, remind students to check their answers in the original inequality. Point out that the endpoint and one different number in the solution set should be checked. Also point out that if the two sides of the original inequality do not have the same value for the endpoint students have tested, then they have not found the correct endpoint.
InClass Example Example 2 Solve 2.5x 10. Check your solution. {x  x 4}
Lesson 12–3
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InClass Example
Example
3
Example 3
x 3
Solve 6. Check your solution. x 3 x 3 3(6) 3
6
1
Solve x 5. Check your 2
solution. {x  x 10}
Multiply each side by –3 and reverse the symbol.
x 18 Check: Substitute 18 and a number greater than 18, such as 21, into the inequality. Let x 18.
3 PRACTICE/APPLY Error Analysis Watch for students who make mistakes of sign or inequality direction when dividing each side of an inequality by a negative number as in Exercises 3 5, 6, and 9. Prevent by suggesting that students who are having trouble solve these inequalities in smaller steps. Have these students take a separate step to multiply each side by 1 and reverse the inequality before dividing each side by the coefficient of the variable. This way, they may be less likely to make a mistake.
1. For both, reverse the symbol when dividing or multiplying by a negative number. 2. Five times four equals twenty. It is not greater than twenty. 3. 3
4
5
NAME
6 6 false
DATE
PERIOD
Study Guide
Student Edition Pages 514–518
7 6
Your Turn Solve each inequality. Check your solution. x c. 8 {xx 32}
2 d. x 22 {xx 33}
4
Communicating Mathematics
3
1–3. See margin.
1. Compare and contrast the Division Property for Inequalities and the Multiplication Property for Inequalities. 2. Explain why 5 is not a solution of the inequality 4x 20. 3. Graph the solution of 2x 8 on a number line.
4. {zz 6} 5. {xx 9} 6. {bb 2} 9. {aa 30}
Solve each inequality. Check your solution. 4. 2z 12 5. 3x 27 6. 7b 14 x 7. 5 4
r 6
8. 6
516 Chapter 12 Inequalities
Multiplication and Division Properties for Inequalities For all numbers a, b, and c, a b . c c a b positive and a b, then ac bc and c . c a b . negative and a b, then ac bc and c c a b negative and a b, then ac bc and c . c
1. if c is positive and a b, then ac bc and
4. if c is
Example 1: Solve 25g 300.
Example 2: Solve
25g 300 25g 25
300 25
Divide by 25.
2
g 12 {gg 12}
y 2
y 2 y 2
8.
2(8) Multiply by 2 and reverse the symbol.
x 4
2.
18 {x  x 72}
4. 3x 9 {x  x 3}
5. 8
t 2
{t  t 16}
7. 2.4y 4.8 { y  y 2} 8. 1.5x 7.2
{x  x 4.8}
10.
x 12
3 {x  x 36} 11.
n –3
1.4
6
1
3. 2y 8 { y  y 4}
{
1 2
{
1 2
6. 2p 1 p  p 9. 6.2y 3.1 y  y
}
}
12. 7p 7 { p p 1}
{n n 4.2} ©
Glencoe/McGrawHill
516 Chapter 12
T74
Logical Learners Have students work with a partner. One student writes a multiplication or division inequality for the other to solve. As the second student solves the inequality, he or she explains in detail to the first student the steps taken to solve the inequality, including why the direction of the inequality symbol is or is not reversed. After completion, the students can reverse roles. 10 6
1006
Solve each inequality. Check your solution. 1. 3n 6 {n n 2}
Reteaching Activity
8
y 16 {yy 16}
(Example 3)
routes, but the trip is at most 10 miles. Considering the stops at traffic lights, she thinks her average driving speed is about 40 miles per hour. How much time does it take Cherise to get to work? (Example 1) t 0.25 h or 15 min
Suppose the family car gets 25 miles to a gallon of gasoline and you want to calculate how many gallons of gasoline you will need for a trip that is more than 300 miles long. Let g the number of gallons of gasoline you will need. You can represent this situation with the inequality 25g 300. You can use the Multiplication and Division Properties for Inequalities to solve inequalities involving multiplication or division. The properties are summarized below. They are also true for and .
3. if c is
2 5
9. a 12
(Examples 1 & 2)
{xx 20} {rr 36} 10. Time Cherise drives to the restaurant where she works by different
Solving Multiplication and Division Inequalities
2. if c is
true
The solution is {xx 18}.
Guided Practice
6
Study Guide Masters, p. 74 12–3
x 3 21 ? 6 3
6
Check for Understanding
Answers
2
Let x 21.
x 6 3 18 ? 6 3
Algebra: Concepts and Applications
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Exercises
•
•
•
•
•
•
•
•
•
•
•
•
Homework Help For Exercises 11–13, 17–19, 23–25, 29–31, 35–38
See Examples 1, 2
14–16, 20–22, 26–29, 32–34
3
Extra Practice See page 716.
Applications and Problem Solving rld
al Wo
11. 6h 12
12. 9n 18
13. 8d 24
a 14. 5
15. 12
16. 5
17. 7x 49
6
g 3
p 20. 1 2
18. 4y 40
19. 3z 9
c 21. 6 9
22. 4
23. 3k 5
24. 2t 11
25. 3 6w
3 26. x 3 4
5 27. 5 y 6
28. v 20
29. 5.3v 10.6
30. 4.1x 6.15
31. 28 0.1s
h 32. 2 3.8
n 33. 10 10.5
34. 7
a 8 2 5
x 0.5
35. Geometry An acute angle has a measure less than 90°. If the measure of an acute angle is 2x, what is the value of x? x 45
2 x˚
37. Budgeting Jenny mows lawns to earn money. She wants to earn at least $200 to buy a new stereo system. If she charges $12 a lawn, at least how many lawns does she need to mow? at least 17 lawns 38. Critical Thinking Use a counterexample to show that if x y, then x2 y2 is not always true. Sample answer: x 3, y 2
Solve each inequality. Check your solution. (Lesson 12–2) 39. z 1 5 {zz 4} 40. 3 b 11
{bb 14}
41. Contests At a beach museum in San Pedro, California, more than 600 people built a lifesize sand sculpture of a whale. Use an inequality to represent the number of people who built the sculpture. (Lesson 12–1)
p 600
Basic: 11–37 odd, 38–46 Average: 12–34 even, 35–46 All: Quiz 1, 1–10
b 8
36. Production The ink cartridge that Bill just bought for his printer can print up to 900 pages of text. Bill is printing handbooks that are 32 pages each. How many complete handbooks can he print with this cartridge? at most 28 handbooks
Mixed Review
Assignment Guide
•
Solve each inequality. Check your solution. 11–34. See margin.
Practice
Re
• • • • •
Answers 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
{h  h 2} {n n 2} {d  d 3} {a a 30} { g  g 36} {b b 40} {x x 7} { y y 10} {z z 3} { p p 2} {c c 54} {a a 32}
23. k  k 24. t  t
5 3
25. w w 26. 27. 28. 29. 30. 31. 32. 33. 34.
11 2 1 2
{x x 4} { y  y 6} {v  v 50} {v  v 2} {x x 1.5} {s s 280} {h h 7.6} {n n 105} {x x 3.5}
Practice Masters, p. 74 12–3
NAME
DATE
PERIOD
Practice
Student Edition Pages 514–518
Solving Multiplication and Division Inequalities Solve each inequality. Check your solution. 1. 4y 16
2. 3q 18
{ y  y 4}
Whale sculpture in San Pedro, California 4.
Lesson 12–3 Solving Multiplication and Division Inequalities 517
p 5
5
5.
{ p p 25} 7. 6x 30
10.
3
16. 4
{u  u 36} 19. 4k 6
{k k 1.5}
22.
2 y 3
6
{ y  y 9}
©
Glencoe/McGrawHill
6. 7 7
{m m 49} 9. 16 2e
{z  z 7} 11. 4
f 6
{e e 8} w
12. 8 5
{f  f 24} 14. 6r 42
{v  v 9} u 9
m
4
8. 4z 28
{n n 9} 13. 81 9v
a 2
{g g 3}
{a  a 8}
{x  x 5} n 3
3. 9g 27
{q  q 6}
{w w 40} 15. 12a 60
{r  r 7}
17.
d 6
8.1
{d  d 48.6} 20. 0.9b 2.7
{b  b 3} 3
23. c 15 5
{c  c 25}
T74
{a  a 5}
18.
l 8
8
{l  l 64} 21. 1.6 4t
{t  t 0.4} 5
24. j 10 8
{ j  j 16}
Algebra: Concepts and Applications
Lesson 12–3
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42. Solve x2 5x 4 0 by factoring.
4 ASSESS
4, 1
(Lesson 11–3)
(Lesson 9–5) 4v 2 1
43. Find the product of 2v 1 and 2v 1.
OpenEnded Assessment Speaking Write several onestep inequalities on the board or overhead. Ask students to describe how they would solve the inequality, including whether or not they would reverse the inequality symbol. Include both addition and subtraction inequalities and multiplication and division inequalities to test students’ understanding.
44. Earthquakes In a recent year, the state of Illinois experienced an earthquake tremor. It measured 3.5 on the Richter scale, releasing about 1.6 107 Joules of energy. The largest earthquake ever recorded in Illinois measured 5.5 on the Richter scale, releasing about 5.7 1011 Joules of energy. How many times as strong was the largest earthquake as the tremor? (Lesson 8–4) 3.5625 104
times stronger
45. Short Response There are 20 students in a class. Each student’s name is written on a separate slip of paper and placed in a box. A name is randomly drawn to determine who will read the daily announcements. The slip of paper is then returned to the box. What is the probability that the same name is drawn two days in a row? (Lesson 5–7) 1 400 46. Multiple Choice If the figure at the right is a square, then what is the value of x? (Lesson 4–4) B 3x A 3 B 4 C 12 D 36
Standardized Test Practice
Quiz 1 The Quiz provides students with a brief review of the concepts and skills in Lessons 12–1 through 12–3. Lesson numbers are given to the right of the exercises or instruction lines so that students can review concepts not yet mastered. Chapter 12, Quiz A (Lessons 12–1 through 12–3) is available in the Assessment and Evaluation Masters, p. 231.
12
Quiz 1
>
Lessons 12–1 through 12–3
1. Write an inequality for the graph. (Lesson 12–1) 3 2 1
0
x 1
1
Solve each inequality. Graph the solution. (Lesson 12–2) 2. a 2 10
3. 4 x 1
{aa 8}
6
7
8
9
10
{xx 5}
3
4. 5 s (3)
5. 3 13z 14z
{ss 8}
{zz 3}
10 9 8 7 6
4
5
6
7
5 4 3 2 1
Solve each inequality. Check your solution. (Lesson 12–3) 6. 3b 30
Enrichment Masters, p. 74 12–3
NAME
DATE
Enrichment
k 2
8. 8
{bb 10} {kk 16}
7. 5x 25 {xx 5} 4 9. v 6 5
PERIOD Student Edition Pages 514–518
Some Properties of Inequalities The two expressions on either side of an inequality symbol are sometimes called the first and second members of the inequality. If the inequality symbols of two inequalities point in the same direction, the inequalities have the same sense. For example, a b and c d have the same sense; a b and c d have opposite senses. In the problems on this page, you will explore some properties of inequalities.
10. Weather The record high temperature in Jackson, Mississippi, is 100° F. Suppose a recent temperature in Jackson is 84° F. How many degrees must the temperature rise to break the record? (Lesson 12–2) t 16° F
518 Chapter 12 Inequalities
Three of the four statements below are true for all numbers a and b (or a, b, c, and d). Write each statement in algebraic form. If the statement is true for all numbers, prove it. If it is not true, give an example to show that it is false. 1. Given an inequality, a new and equivalent inequality can be created by interchanging the members and reversing the sense.
If a > b, then b < a. a > b, a b > 0, b > a, (1)(b) < (1)(a), b < a
2. Given an inequality, a new and equivalent inequality can be created by changing the signs of both terms and reversing the sense.
If a > b, then a < b. a > b, a b > 0, b > a, a < b
3. Given two inequalities with the same sense, the sum of the corresponding members are members of an equivalent inequality with the same sense.
If a > b and c > d, then a c > b d. a > b and c > d, so (a b) and (c d) are positive numbers, so the sum (a b) (c d) is also positive. a b c d > 0, so a c > b d.
4. Given two inequalities with the same sense, the difference of the corresponding members are members of an equivalent inequality with the same sense.
If a > b and c > d, then a c > b d. The statement is false. 5 > 4 and 3 > 2, but 5 3 >/ 4 2.
©
Glencoe/McGrawHill
518 Chapter 12
vv 125
T74
Algebra: Concepts and Applications
Extra Credit Write a realworld problem that involves a multiplication or division inequality. Define a variable and write the inequality. Then solve the inequality and graph its solution. See students’ work.
www.algconcepts.com/self_check_quiz
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Solving MultiStep Inequalities
Lesson 12–4
You’ll learn to solve inequalities involving more than one operation.
Some inequalities involve more than one operation. The best strategy to solve multistep inequalities is to undo the operations in reverse order. In other words, work backward just as you did to solve multistep equations. For example, 2x + 5 > 11 is a multistep inequality. You can solve this inequality by following these steps.
Why It’s Important
Step 1: Undo addition.
What You’ll Learn
School You can determine what score is needed to receive a certain class grade. See Example 5.
Step 2: Undo multiplication.
Divide each side by 2.
2x 6 2 2
2.
Solving MultiStep Equations: Lesson 4–5
3x 18 3 3
b 4
5 b 20 3
3. y 9 y 12 4
Solve 9 3x 27. Check your solution. 9 3x 27 9 3x – 9 27 – 9 3x 18
Lesson 12–3 Solve each inequality. Check your solution. 1. 3r 15 r 5
2x 5 5 11 5 2x 6
1
5Minute Check
Subtract 5 from each side.
x3
Example
1 FOCUS
4. A toner cartridge for a copy machine can print up to 10,000 copies. A small business makes an average of 250 copies per day. How many days can the company make copies using a single cartridge? at most 40 days 5. You want to build a patio with an area of at least 300 square feet along the back of your house as shown. What must be the width w of the patio?
Subtract 9 from each side. Divide each side by 3.
x6 Check: Substitute 0 and 6 into the inequality. Let x 0. 9 3x 27 ? 9 3(0) 27 ? 9 0 27 9 27 true
Let x 6. 9 3x 27
32 ft Patio
?
9 3(6) 27 ? 27 9 18 27 27 false
w
House
w 9.375 ft
The solution is {xx 6}.
Your Turn
Motivating the Lesson
Solve each inequality. Check your solution.
RealWorld Connection Explore Celsius and Fahrenheit temperatures with students. Ask who knows the boiling and freezing temperatures of water in degrees Fahrenheit and degrees Celsius. Point out that most of the world uses Celsius temperatures because of their ease of use. Ask students to estimate the inside and outside temperatures in degrees Fahrenheit, and then ask if any of them can estimate the equivalent temperature in degrees Celsius. This can then lead to a discussion of the formula for FahrenheittoCelsius temperature conversion.
a. 4 2x 12 {xx 4}
b. 8x – 5 11
{xx 2}
Lesson 12–4 Solving MultiStep Inequalities 519
Resource Manager Reproducible Masters • Study Guide, p. 75 • Practice, p. 75 • Enrichment, p. 75 • Graphing Calculator, p. 33 • HandsOn Algebra, p. 135 • Assessment and Evaluation, p. 230
Transparencies • 5Minute Check, 12–4 • Teaching, 12–4 • Answer Key, 12–4 Technology/Multimedia • AlgePASS, Lesson 29
Lesson 12–4
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Example
2
Weather Link
InClass Examples
5 (F 32) 30 9 9 5 9 (F 32) 30 5 9 5
Example 1 Solve 5x 9 21. Check your solution. {x  x 6}
Remember to reverse the inequality symbol if you multiply or divide by a negative number.
Example
3
Solve 4x 3 23 6x. Check your solution. 4x 3 23 6x 4x 3 6x 23 6x 6x 10x 3 23 10x 3 3 23 3 10x 20
Subtract 6x from each side. Subtract 3 from each side.
20 10x 10 10
Divide each side by –10 and reverse the symbol.
x 2 The solution is {xx 2}.
Your Turn
InClass Examples
Check your solution.
Solve each inequality. Check your solution.
c. 10 5x 25 {xx 3}
Example 3 Solve 16 2x 3x 1. Check your solution. {x  x 3}
d. 3x 1 17 {xx 6}
To solve inequalities that contain grouping symbols, you may use the Distributive Property first.
Example
4
Solve 3(x 2) 75. Check your solution. {x  x 27}
Solve 8 2(x 5). Check your solution. 8 2(x 5) 8 2x 10 Distributive Property 8 10 2x 10 10 Subtract 10 from each side. 2 2x 2 2x Divide each side by 2 and 2 2 reverse the symbol. 1x The solution is {xx 1}.
520 Chapter 12 Inequalities
Check your solution.
www.algconcepts.com/extra_examples
Talk with family members about relatives, ancestors, or friends who lived or who live in different countries. Do some research to find out the current range of typical temperatures in the areas where they are from. Find out what temperature scale people of that country usually use. Give the temperature ranges both in Fahrenheit and Celsius temperatures. A formula that converts 9 degrees Celsius to degrees Fahrenheit is F C 32. 5
520 Chapter 12
5 9
Therefore, Rafael can expect it to be warmer than 86°F in Mexico.
Teaching Tip Stress the paragraph between Examples 2 and 3. Also, as you discuss Example 3, ask students how they could solve the inequality without having to divide each side by a negative number.
Example 4
9 5
Multiply each side by , the reciprocal of .
F 32 54 F 32 32 54 32 Add 32 to each side. F 86 Check your solution.
Example 2 Kyle is tiling his shower walls with 4inch square tiles. The expression 9x 20 estimates the number of tiles he needs to tile x square feet of shower walls. Kyle has 560 tiles. Solve the inequality 9x 20 560 to find how many square feet Kyle can tile. x 60 square feet
During Rafael’s trip to Mexico, the temperature was always warmer 5 than 30º Celsius. Use the formula (F – 32) C, where F is degrees 9 Fahrenheit and C is degrees Celsius, to write and solve an inequality for the temperature in degrees Farenheit.
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Your Turn
Solve each inequality. Check your solution.
e. 2 (x 7) {xx 9}
f. 3(x 4) x 5 {xx 3.5}
You can use a graphing calculator to solve multistep inequalities.
Graphing Calculator Tutorial See pp. 724– 727. Solve 5 8x 43 by using a graphing calculator.
Teaching Tip You might want to have students rework the problem in Example 5 with different target mean scores so that students can see situations in which there is an applicable solution. You may also want to have students substitute 100 for s in the mean computation to see what the highest possible mean is that Hannah can have for the four tests.
Step 1
Clear the list to enter the inequality 5 8x 43. (The symbol is item 4 on the TEST menu.)
InClass Example
Step 2
Press ZOOM 6. Use the
Example 5 Karl’s point totals in the first four of five basketball games were 15, 12, 19, and 18. How many points t must he score in the fifth game to have a mean point total of more than 16? t 16
TRACE and arrow keys to
to move the cursor along the graph. You should see a line above the xaxis for values of x that are less than or equal to 6. This represents the solution {xx 6}.
1. 9x 2 20 {xx 2}
al Wo
2. 5(x 4) 3(x 4) {xx 1}
rld
Re
Try These Solve each inequality. Check your solution with a graphing calculator.
Example School Link
5
Hannah’s scores on the first three of four 100point tests were 85, 92, and 90. What score must she receive on the fourth test to have a mean score of more than 92 for all tests? Explore Let s Hannah’s score on the fourth test. Plan
Solve
The sum of Hannah’s four test scores, divided by 4, will give the mean score. The mean must be more than 92. 85 90 92 s 92 4 85 90 92 s 4
4 4(92)
Multiply each side by 4.
85 90 92 s 368
Study Guide Masters, p. 75
267 s 368 267 s 267 368 267 Subtract 267 from each side. s 101 (continued on the next page)
Lesson 12–4 Solving MultiStep Inequalities 521
12–4
NAME
DATE
PERIOD
Study Guide
Student Edition Pages 519–523
Solving MultiStep Inequalities Solving inequalities may require more than one operation. The best strategy to use is to undo the operations in reverse order. In other words, first undo addition or subtraction and then undo multiplication or division, just as you did in solving equations with more than one operation. Remember that multiplying or dividing by a negative number reverses the inequality symbol. Example 1: Solve 6 4x 18. 6 4x 18 6 4x 6 18 6 4x 12 4x 4
Subtract 6 from each side.
12 4
Divide each side by 4.
x 3 or {xx 3} Example 2: Solve 4 3x 8 x. 4 3x 8 x 4 3x 4 8 x 4 3x 12 x 3x x 12 x x 4x 12 4x 4
Subtract 4 from each side. Subtract x from each side.
12 4
Divide each side by 4. Reverse the symbol.
x 3 or { xx 3}
It is often difficult to use a calculator graph to trace to the exact spot where the solution set of an inequality begins. Zooming in close to this spot, however, can help you make an educated guess. In this case, it appears that the first value of x that makes 5 8x 43 true is 6. Substitution confirms that the value of 5 8x is exactly 43 when x 6. Students can also confirm this using the TABLE feature. At x 6, the value for Y1 changes from 0 to 1. Zero indicates a false inequality, whereas one indicates a true inequality.
Solve each inequality. Check your solution. 1. 2n 8 26 {n n 9}
2. 6x 12 48 {x  x 10}
3. 12 4y 16 { y  y 7}
4. 3x 1 9 x {x  x 2}
5. 8
t 2
{
2 {t  t 20}
7. 2 y 1.6 { y  y 3.6} 9. y 3 2y 3.1 { y  y 0.1} 11. 6y 8.2 36.8 { y  y 7.5}
©
Glencoe/McGrawHill
1 3
6. 3 3p 2 p  p
}
8. 2x 8 4.2 {x  x 6.1} 10. 3.2x 16 3.2 {x  x 4}
{
1 2
12. 1 2x 2 x  x 1
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Algebra: Concepts and Applications
Lesson 12–4
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Examine Substitute a number greater than 101, such as 102, into the original problem. Hannah’s average would be
3 PRACTICE/APPLY
85 90 92 102 4
Error Analysis Watch for students who try to begin solving Exercise 12 by dividing each side by 3. Prevent by stressing that although solving the inequality this way is mathematically correct, it is not efficient because it will introduce fractional values. Stress that it is often best to first use the Distributive Property to eliminate grouping symbols, though there may be exceptions to this. For example, in Exercise 30, the most efficient solution method is to first multiply each side by 6.
statement, the solution is correct. Hannah must score more than 101 points out of a 100point test. Without extra credit, this is not possible. So, Hannah cannot have a mean score over 92.
Check for Understanding 1. Name the operations used to solve 5 2x 9. 1–2. See margin. 2. Write an inequality requiring more than one operation to solve.
Communicating Mathematics
Sample: 12x 4 20 3. 15z 7 10
Practice
For Exercises 14–25, 29, 30, 32–36
Practice Masters, p. 75
26–28, 31, 37 PERIOD
Practice
Student Edition Pages 519–523
Solve each inequality. Check your solution. 1. 3x 5 14
2. 3t 6 15
{x  x 3}
{t  t 7}
{ y  y 6}
5. 6 4a 10
6. 28 7 7w
{n n 4}
{a a 1}
{w  w 5}
7. 5 1.3z 31
8. 1.7b 1.1 2.3
k 2
1
13.
2n 6 4
8
c
11. 93 6
{c  c 36}
14.
6 3n 6
5
{n n 13}
{n n 12}
16. 7p 4 3p 12
17. 2f 5 4f 13
{ p  p 4}
19. 2(q 2) 3(q 6)
{q  q 22}
©
3. 5y 2 32
{f  f 9}
20. 3(h 5) 6(h 4)
{h h 1}
Glencoe/McGrawHill
522 Chapter 12
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• • • • •
•
•
•
3
Extra Practice
14. 16. 18. 20. 22.
4t 8 16 {tt 6} 2 3n 17 {nn 5} 5 7c 2 {cc 1} 12 11r 54 {rr 6} 7 0.1a 9 {aa 20}
s 24. 6 11 4
{ss 20}
26. 3h 5 2h 4 {hh 9} 28. 6j 9 j 6 {jj 3} 1 6
30. (z 2) 1 {zz 4}
522 Chapter 12 Inequalities
9. 6.4 8 2g
{b b 2}
{k k 10}
See Examples 1, 2, 4
See page 716.
Solving MultiStep Inequalities
9
{g  g 0.8}
12.
5m 5 3
15
{m  m 8}
15. 9 5j j 3
{ j  j 2}
18. 5(7 2a) 15
{a  a 5}
21. 2(b 3) 4(b 9)
{b b 7}
Algebra: Concepts and Applications
9
•
•
•
•
•
•
•
•
•
•
Solve each inequality. Check your solution.
Homework Help
10. 6
n3 6. 11
13. School Kira wants her average math grade to be at least 90. Her test scores are 88, 93, and 87. What score does she need on her fourth test to earn an average score of at least 90? (Example 5) s 92
Exercises
{z  z 20}
4. 24 8b 3 3
7. 2y 4 12 {yy 4} 8. 8 3h 20 {hh 4} 9. 5 2x 7 {xx 1} 10. 7z 4 10 {zz 2} 11. 10 5w w 22 {ww 2} 12. 3(n 4) 2(n 6) {nn 24}
1. subtraction, then division 2. Sample answer: 2x 6 4
4. 2n 3 11
7
Solve each inequality. Check your solution. (Examples 1–4)
Answers
DATE
Solution: Subtract 4 from each side.
5. 4.5a (3.1) 8.2 3.1
Basic: 15–37 odd, 38–45 Average: 14–34 even, 35–45
NAME
State which operation you would perform first to solve the inequality.
Getting Ready
Guided Practice
Assignment Guide
12–4
or 92.25. Since 92.25 92 is a true
Reteaching Activity Interpersonal Learners Have each student work with a partner. On the board or overhead, write several inequalities. For each inequality, one student should solve the inequality algebraically while the other solves it using a graphing calculator. The partners should compare their work and then switch roles for the next inequality.
15. 17. 19. 21. 23.
3b 9 45 {bb 12} 20 8 7x {xx 4} 6g 1 13 {gg 2} 8 4v 6 {vv 0.5} 0.3m 2.1 3.0 {mm 3}
11 6d 25. 1 {dd 1} 5
27. 5x 3 2x 9 {xx 2} 29. 2(7 2y) 10 {yy 1} 2 1 31. (b 1) (b 5) {bb 11} 3
2
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Write and solve an inequality for each situation. 32. The sum of twice a number and 17 is no greater than 41. {xx 12} 33. Five times the sum of a number and 6 is less than 35. {xx 1} 34. Two thirds of a number decreased by 7 is at least 9. {xx 24}
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Applications and Problem Solving
36a. 10 rides or less
36b. more than 6 rides
35. Employment Jeremy is a receptionist at a hair salon. He earns $7.00 an hour, plus 10% of any hair products he sells. Suppose he works 22 hours a week. How much money in hair products must he sell to earn at least $180? at least $260 36. Recreation The admission fee to a state fair is $5.00. Each ride costs an additional $1.50. a. Suppose Pilar does not want to spend more than $20. How many rides can she go on? b. The fair has a special admission price for $14, which includes unlimited rides. For how many rides is this a better deal than paying for each ride separately?
4 ASSESS OpenEnded Assessment Writing Have students give an example of an inequality that relates two binomials of degree one. Then have them write a brief summary of the methods and properties they can use to solve the inequality. MidChapter Test (Lessons 12–1 through 12–4) is available in the Assessment and Evaluation Masters, p. 230.
37. Finance At a bank, an advertisement reads, “In one year, your earnings will be greater than your original investment plus 6% of the investment.” Suppose Diego invests $1400. How much money can he expect to have at the end of the year? x $1484 38. Critical Thinking Would the solution of x 2 4 be x 2? Justify your answer. No; the solution also includes numbers less than 2.
Mixed Review
Solve each inequality. Check your solution. (Lesson 12–3) 39. 5p 35 {pp 7}
40. 24 8v {vv 3}
x 41. 10 4
2 42. z 6
{xx 40}
5
{zz 15}
43. Budgeting Haley earned $36 babysitting. She plans to buy two books that cost $8.25 each. With the rest of the money, she plans to go to dinner and see a movie with her friends. At most, how much money can she spend on a movie and dinner? (Lesson 12–2)
m $19.50
Standardized Test Practice
44. Grid In Find the value of c that makes x 2 6x c a perfect trinomial square. (Lesson 11–5) 9 x 2 3x 10
45. Multiple Choice If x 5, then is equivalent to which of x5 the following? (Lesson 10–3) A A x2 B x3 C x5 D x 10
www.algconcepts.com/self_check_quiz
Enrichment Masters, p. 75 12–4
NAME
DATE
Enrichment
PERIOD Student Edition Pages 519–523
Consecutive Integer Problems Many types of problems and puzzles involve the idea of consecutive integers. Here is an example. Find four consecutive odd integers whose sum is 80. An odd integer can be written as 2n 1, where n is any of the numbers 0, 1, 2, 3, and so on. Then, the equation for the problem is as follows. (2n 1) (2n 3) (2n 5) (2n 7) 80
Lesson 12–4 Solving MultiStep Inequalities 523
Solve these problems. Write an equation or inequality for each. 1. Complete the solution to the problem 2. Find three consecutive even integers in the example. whose sum is 132.
n 12; The integers are 23, 21, 19, 17.
Extra Credit Amaar has an “A” average, or an average at or above 90 points out of 100, on his first three history quizzes. Amaar scored 3 points higher on the second quiz than on the first one, and 6 points higher on the third quiz than on the second one. What are Amaar’s possible scores s on the first quiz? s 86 and s 91 (or 86 s 91)
2n (2n 2) (2n 4) 132; n 21; Integers are 42, 44, 46.
3. Find the two least consecutive integers whose sum is greater than 20.
4. Find the two greatest consecutive integers whose sum is less than 100.
5. The lesser of two consecutive even integers is 10 more than onehalf the greater. Find the integers.
6. The greater of two consecutive even integers is 6 less than three times the lesser. Find the integers.
n (n 1) > 20; n > 9.5; Integers are 10 and 11.
n (n 1) < 100; n < 49.5; Integers are 49 and 50.
2n 2 3(2n) 6; n 2; Integers are 4 and 6.
1 2 (2n
2n 10 2 ; n 11; Integers are 22 and 24.
7. Find four consecutive integers such that twice the sum of the two greater integers exceeds three times the first by 91.
2[(n 2) (n 3)] 3n 91; n 81; Integers are 81, 82, 83, 84.
©
Glencoe/McGrawHill
8. Find all sets of four consecutive positive integers such that the greatest integer in the set is greater than twice the least integer in the set.
n 3 > 2n; n < 3; The two sets are {1, 2, 3, 4} and {2, 3, 4, 5}.
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Lesson 12–4
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Lesson 12–5
12–5
1 FOCUS
What You’ll Learn
Solve each inequality. Check your solution. 1. 8 4(x 6) {x  x 8} 2. 0.5w 3.7 2.0 {w  w 11.4} 2 x 3
Nutrition Pharmacists use inequalities to write prescriptions. See Example 2.
Another way to write this information is to use an inequality. Let w represent the weight that requires 2 tablets.
Why It’s Important
Weight is greater than 25. w 25
4 12 {x  x 12}
Test 1 2 3 4 Score 78 95 88 92 s 97
Motivating the Lesson HandsOn Activity Pick some readily observable distinctions among students in your class, such as eye color, gender, height, shirt color, and so on. By having students who match one or more criteria raise their hands, you can model the use of and and or as in compound inequalities. For example, have students with brown eyes raise their left hands and boys raise their right hands. Students with both hands raised satisfy the and condition, while any student with a hand raised satisfies the or condition.
524 Chapter 12
and
Method 1 25 w 50 This can be read as 25 is less than w, which is less than or equal to 50.
w–5
5. Leticia’s scores on the first four of five 100point tests are shown below. What score s on the fifth test will give her a mean score of at least 90 for all five tests?
Dog’s Weight (pounds)
1 tablet
25 or less greater than 25 and less than or equal to 50 more than 50
2 tablets 3 tablets
w 50
These two inequalities form a compound inequality. The compound inequality w 25 and w 50 can be written without using the word and.
4. The perimeter of the rectangle below is at most 32 meters. What is the value of w? w 9 m
w+3
Daily Dose
Weight is less than or equal to 50.
Lesson 12–4
Solving Compound Inequalities Lamar is buying vitamins for his dog. The daily dose for the vitamins is based on the dog’s weight. Lamar’s dog weighs 32 pounds. Since 32 is greater than 25, but less than or equal to 50, he will give his dog 2 tablets.
You’ll learn to solve compound inequalities.
5Minute Check
3.
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Method 2 50 w 25 This can be read as 50 is greater than or equal to w, which is greater than 25.
Note that in each, both inequality symbols are facing the same direction.
Example
1
Write x 2 and x 7 as a compound inequality without using and. x 2 and x 7 can be written as 2 x 7 or as 7 x 2.
Your Turn
a. 4 x 10 or 10 x 4
a. x 10 and x 4
b. x 6 and x 2
b. 2 x 6 or 6x2
A compound inequality using and is true if and only if both inequalities are true. Thus, the graph of a compound inequality using and is the intersection of the graphs of the two inequalities. Consider the inequality 2 x 3. It can be written using and: x 2 and x 3. To graph, follow the steps below. Step 1
Graph x 2.
Step 2
Graph x 3.
5 4 3 2 1 0 1 2 3 4 5 5 4 3 2 1 0 1 2 3 4 5
Step 3 Find the intersection of the graphs.
5 4 3 2 1 0 1 2 3 4 5
The solution, shown by the graph of the intersection, is {x2 x 3}.
524 Chapter 12 Inequalities
Resource Manager Reproducible Masters • Study Guide, p. 76 • Practice, p. 76 • Enrichment, p. 76 • HandsOn Algebra, pp. 136–137
Transparencies • 5Minute Check, 12–5 • Teaching, 12–5 • Answer Key, 12–5
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Example
2
Pet Care Link
Refer to the application at the beginning of the lesson. Graph the solution of 25 w 50. Rewrite the compound inequality using and. 25 w 50 is the same as w 25 and w 50.
Most of the time, or symbols are used with compound inequalities.
Step 1
Graph w 25.
Step 2
Graph w 50.
Step 3
Find their intersection.
10 15 20 25 30 35 40 45 50 55 60 10 15 20 25 30 35 40 45 50 55 60 10 15 20 25 30 35 40 45 50 55 60
The solution is {w25 w 50}.
Your Turn c. Graph the solution of 3 x 5.
2 TEACH Teaching Tip When discussing the two ways to write the compound inequality for the weight requirement in the opening scenario, emphasize that the direction of the inequality symbols in relation to the numbers and variable stays the same. In both inequalities, the point of the or symbol is directed towards w, and the open part of the or symbol is directed towards w.
0 1 2 3 4 5 6 7 8
InClass Examples Example 1 Often, you must solve a compound inequality before graphing it.
Example
3
Solve 4 x 3 12. Graph the solution. Step 1
Step 2
Step 3
Rewrite the compound inequality using and. 4 x 3 12 x34 and x 3 12 Solve each inequality. x34 and x3343 x1
x 3 12 x 3 3 12 3 x9
Rewrite the inequality as 1 x 9.
The solution is {x1 x 9}. The graph of the solution is shown at the right.
Your Turn
2x6
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8
d. Solve 2 x 4 2. Graph the solution.
Another type of compound inequality uses the word or. This type of inequality is true if one or more of the inequalities is true. The graph of a compound inequality using or is the union of the graphs of the two inequalities.
www.algconcepts.com/extra_examples
Lesson 12–5 Solving Compound Inequalities 525
Inclusion Strategies
Write x 0 and x 3 as a compound inequality without using and. 3 x 0 or 0 x 3
Example 2 A veterinarian has a scale for weighing dogs and cats that weigh more than 10 pounds but no more than 65 pounds. The weights w that can be measured on this scale can be written as 10 w 65. Graph the solution of this inequality. 0 10 20 30 40 50 60 70
Teaching Tip In Example 3, the graphing is completed in a single step. Encourage students to use the three graphing steps shown in Example 2 until they are comfortable graphing a compound inequality in one step.
InClass Example Example 3 Solve 6 x 3 1. Graph the solution. {x  3 x 2} –4 –3 –2 –1 0 1 2 3 4
The problems in this lesson have more steps than in many previous lessons. Students with behavioral difficulties may have trouble staying focused through these problems. You might try pairing these students with stronger students, having each partner take portions or steps of a problem to solve. For example, partners could each solve and graph one of the simple inequalities in a compound inequality and then compare solutions to arrive at the solution to the compound inequality. Lesson 12–5
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Example
4
Example 4 Graph the solution of x 7 or x 3.
Graph the solution of x 0 or x 1. Step 1
Graph x 0.
Step 2
Graph x 1.
5 4 3 2 1 0 1 2 3 4 5
1 2 3 4 5 6 7 8 9
Teaching Tip In Example 5, students may be confused by the fact that the solution of the compound inequality is the same as the solution of one of its simple inequality parts. Explain that this is because every point in the solution of x 5 is also a solution of x 2, that is, the solutions of x 5 are a subset of the solutions of x 2.
Step 3 Find the union of the graphs.
Your Turn
9 8 7 6 5 4 3 2 1 0 1
Sometimes you must solve compound inequalities containing the word or before you are able to graph the solution.
Example
5
Solve 3x 15 or 2x 4. Graph the solution. or
Example 5
x5
4 or 5x 20.
2x 4 4 2x 2 2
3x 15 3 3
InClass Example Solve
5 4 3 2 1 0 1 2 3 4 5
d. Graph the solution of x 3 or x 5.
3x 15
2 x 3
5 4 3 2 1 0 1 2 3 4 5
x 2
Now graph the solution.
Graph the solution. {x x 4}
Step 1
Graph x 5.
Step 2
Graph x 2
3 2 1 0 1 2 3 4 5 6 7
–5 –4 –3 –2 –1
Answers
Step 3 Find the union of the graphs.
1. an inequality made of two inequalities
3 2 1 0 1 2 3 4 5 6 7
3 2 1 0 1 2 3 4 5 6 7
The last graph shows the solution {xx 2}.
Page 527 Your Turn
9. 21 0 1 2 3 4 5 6 7 8
{xx 3}
5 4 3 2 1 0 1 2 3 4 5
e. Solve 6x 18 or x 2 1. Graph the solution.
10. 7654321 0 1 2 3
11. 0 1 2 3 4 5 6 7 8 9 10
Check for Understanding
12. 10987654321 0
13. 321 0 1 2 3 4 5 6 7
14. 0 1 2 3 4 5 6 7 8 9 10
Communicating Mathematics
1. Define compound inequality in your own words.
See margin. 2. Write a compound inequality for x is greater than 3 and less than or equal to 5.
526 Chapter 12 Inequalities
3 x 5 or 5 x 3
15. 54 321 0 1 2 3 4 5
16. 54 321 0 1 2 3 4 5
Reteaching Activity
17. 8
9
526 Chapter 12
10
11
12
13
Visual/Spatial Learners Distribute overhead transparency sheets that have permanent number lines drawn on them to students. Students can graph solutions to each part of an assigned compound inequality on the same number line using different colors of washable markers. The solution to an and compound inequality is the portion of the number line where the two colors are blended; the solution to an or compound inequality is the portion of the number line marked with either or both colors.
compound inequality intersection union
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Getting Ready
State whether to find the intersection or union of the two inequalities to graph the solution.
Sample 1: 8 x 4 12
Solution: Find the intersection since the inequalities can be joined by the word and. Solution: Find the union since the inequalities are joined by the word or.
Sample 2: x 9 or x 5
3. x 10 and x 3 intersection 4. x 7 or x 4 union 5. x 5 2 or x 15 union 6. 13 9x 11 intersection Write each compound inequality without using and. (Example 1)
5 10
7. ᐉ 5 and ᐉ 10
8. b 3 and b 2 2 b 3
Graph the solution of each compound inequality. (Examples 2 & 4)
9–16. See margin for graphs. 12. {s7 s 2}
9. y 5 and y 0
10. x 3 or x 7
Solve each compound inequality. Graph the solution. (Examples 3 & 5) 11. 10 c 2 5 {c3 c 8} 13. 0 2v 8 {v0 v 4} 15. j 6 6 or 4 j 4
12. 2 s 4 3 14. 5x 10 or 7x 28 {xx 4} 16. 1 r 4 or 1 r 3
{jj 1 or j 0}
{rr 3 or r 4}
17. Construction Odyssey of the Mind competitions encourage students to use creativity in solving difficult problems. One year, students had to construct a balsawood structure. The structure needed to be at least 9 inches tall and no more than 11.5 inches tall. Write a compound inequality describing the height of the structure. Graph the solution. (Example 2) 9 t 11.5; See margin for graph.
Exercises
3 PRACTICE/APPLY Error Analysis Watch for students who reverse the use of union and intersection when graphing compound inequalities such as those in Exercises 9 and 10. Prevent by having students pick points in different portions of their graphs to test in the original inequalities. For an and compound inequality, any point must satisfy both simple inequalities. For an or compound inequality, any point has to satisfy only one of the simple inequalities. Unless the two simple inequalities in the original compound inequality are the same, there will always be points that satisfy one but not both of the simple inequalities. Assignment Guide Basic: 19–59 odd, 60–71 Average: 18–54 even, 56–71
Answers • • • • •
•
•
•
•
•
•
•
•
•
•
•
•
•
24. 7654 321 0 1 2 3
Practice
Write each compound inequality without using and.
Homework Help For Exercises 18–23 56–59 24–29
See Examples 1 2, 3 2, 4
30–35
3
36–55
3, 4, 5
Extra Practice See page 717.
34. {r8 r 4}
18. b 0 and b 5 0 b 5 19. h 8 and h 8 8 h 8 20. y 4 and y 1 1 y 4 21. g 2 and g 5 2 g 5 22. 2 r and 1 r 2 r 1 23. 6 x and x 8 6 x 8 Graph the solution of each compound inequality. 24. x 0 or x 4 26. k 7 and k 5 28. b 4 or b 0
25. z 2 and z 2 27. y 16 and y 21 29. m 10 or m 6
24–35. See margin for graphs. Solve each compound inequality. Graph the solution. 30. 2 a 3 7 {a1 a 4} 31. 9 x 1 5 {x4 x 8} 32. 16 8s 16 {s2 s 2} 33. 9 3w 0 {w0 w 3} 34. 6 r 2 10 35. 2 h 5 8 {h3 h 3}
Lesson 12–5 Solving Compound Inequalities 527
25. 54 321 0 1 2 3 4 5
26. 0 1 2 3 4 5 6 7 8 9 10
27. 13 14 15 16 17 18 19 20 21 22 23
Study Guide Masters, p. 76 NAME
12–5
DATE
PERIOD
Study Guide
Student Edition Pages 524–529
Solving Compound Inequalities In a doctor’s office, you may see a sign that displays the normal weight range for a person of your age and height. For example, if you are a 14yearold girl who is 5 foot 2 inches tall, it may say that your normal weight is between 100 and 120 pounds, inclusive. Another way to write this information is to use an inequality. If w represents weight, then 100 w 120 is a compound inequality that represents this situation. Another way to write the inequality is to write two inequalities using the word and: 100 w and w 120. A compound inequality using and is true if and only if both inequalities are true. The graph of a compound inequality using and is the intersection of the graphs of the inequalities, as shown below. Example 1: Graph 100 w and w 120.
29.
54 321 0 1 2 3 4 5
30.
54 321 0 1 2 3 4 5
Step 1 Graph x 2.
–4 –3 –2 –1
0
1
2
3
Step 2 Graph x 3.
–4 –3 –2 –1
0
1
2
3
Step 3 Find the union of the graphs.
–4 –3 –2 –1
0
1
2
3
Graph the solution of each compound inequality. 1. n 2 and n 6
34.
1
10987654321 0
321 0 1 2 3 4 5 6 7
2
3
4
5
2. x 2 or x 1
6
7
–3 –2 –1 0
3. y 2 and y 6
35. 0 1 2 3 4 5 6 7 8 9 10
–2 –1 0
5. 2 y or y 1
54 321 0 1 2 3 4 5
–3 –2 –1 0
©
1
Glencoe/McGrawHill
1
2
3
4. 1 p and p 0
–7 –6 –5 –4 –3 –2 –1
31.
90 100 110 120 130 140
Example 2: Graph the solution of x 2 or x 3.
33. 2 3 4 5 6 7 8 9 10 11 12
90 100 110 120 130 140
Another type of compound inequality uses the word or. A compound inequality using or is true if and only if either or both inequalities are true. Its graph is the union of the graphs of the inequalities, as shown below.
32. 321 0 1 2 3 4 5 6 7
90 100 110 120 130 140
Step 2 Graph w 120 Step 3 Find the intersection of the graphs.
Answers 28.
Step 1 Graph 100 w.
1
2
3
4
6. h 8 and h 10 2
3
6
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8
9 10 11 12
Algebra: Concepts and Applications
Lesson 12–5
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Answers 36. 10987654321 0
37. 10987654321 0
38. 7654 321 0 1 2 3
39. 12108 642 0 2 4 6 8
40. 13 14 15 16 17 18 19 20 21 22 23
41. 54 321 0 1 2 3 4 5
42. 0 1 2 3 4 5 6 7 8 9 10
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36–47. See margin for graphs. 36. {y4 y 3} 37. {c5 c 4} 38. {x5 x 1} 39. {zz 7 or z 4} 40. {vv 16 or v 20} 41. {rr 1 or r 0} 42. {jj 6} 43. {pp 1} 44. {cc 8.8 or c 10} 45. {dd 1.5}
43. 54 321 0 1 2 3 4 5
44.
8.6
9.0 8.8
8.4
9.4 9.2
9.8
3
4
Applications and Problem Solving
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5
46. 0
5 y 1 4 6 x (5) 0 v 8 12 or v 4 20 3j 18 or j 3 5 c 2.4 7.6 or c 8.8 0 3 4
37. 39. 41. 43. 45.
16 4c 20 z 3 7 or z 5 12 r 1 or 8r 0 p 5 3 or 2p 2 d 2.1 3.6 or d 4 0.5
w 47. 2 or w 2
46. 8x 4 or x 1
3
1 1 {ww 6 or w 2} xx or x 4 2 Write a compound inequality for each solution shown below.
48.
49. 5 4 3 2 1 0 1 2 3 4 5
5 4 3 2 1 0 1 2 3 4 5
2 x 3
x 3 or x 3
50.
51. 8 7 6 5 4 3 2 1 0 1 2
4 3 2 1 0 1 2 3 4 5 6
x 4 or x 1
3 x 5 Solve each compound inequality. 52–55. See margin. 53. 4x 7 5 or (x 8) 1 55. 10 2(k 6) 14
10.0 10.4
9.6
2
1
36. 38. 40. 42. 44.
52. 2 3y 2 13 54. 2x 1 5 or 3(x 1) 6
10.2
45. 0
Solve each compound inequality. Graph the solution.
56. Taxes Matthew Brooks is single and has a parttime job while attending college. Last year, he paid $649 in federal income tax. Write an inequality for his taxable income. Use the table that describes the different tax brackets. $4300 I $4350
2 If Form 1040A, line 24, is —
47. 7654321 0 1 2 3
52. { y  5 y 0} 53. { x  x 3 or x 9} 54. { x  x 2 or x 3} 55. { k  11 k 13} 57b.
And you are —
At least
But less than
Single
4,200 4,250 4,300 4,350 4,400 4,450 4,500 4,550
4,250 4,300 4,350 4,400 4,450 4,500 4,550 4,600
634 641 649 656 664 671 679 686
Married filing jointly
Married filing separately
Your tax is — 634 634 641 641 649 649 656 656 664 664 671 671 679 679 686 686
Head of a household
634 641 649 656 664 671 679 686
8 9 10 11 12 13 14 15 16 17 18 Source: Ohio Department of Taxation
57a. Sample answer: 11 h 14
Practice Masters, p. 76 NAME
12–5
DATE
PERIOD
Practice
Student Edition Pages 524–529
Solving Compound Inequalities Write each compound inequality without using and. 1. a 2 and a 7
2. b 9 and b 6
2a7
3. w 4 and w 3
6b9
4. k 4 and k 1
3 w 4
5. z 0 and z 6
4 k 1
6. p 8 and p 5
6 z 0
8 p 5
Graph the solution of each compound inequality. 7. f 1 and f 5 –3 –2 –1 0
1
2
4
5
9. y 3 or y 1 –5 –4 –3 –2 –1 0
1
6
2
3
4
5
6
7
8
9 10
10. h 3 or h 2 1
2
3
–7 –6 –5 –4 –3 –2 –1 0
4
1
2
5
6
Solve each compound inequality. Graph the solution. 11. 4 c 6 2
12. 6 u 5 0
{c  4 c 2}
{u  1 u 5}
–7 –6 –5 –4 –3 –2 –1 0
1
–3 –2 –1 0
2
13. 6 2m 10
t 3
–1
1
2
1
2
3
5
6
7
0
–2 –1 0
8
1
2
3
0
20.
{ y  y 1.5 or y 2} 1
2
3
2
3
4
5
6
7
{a  a 6}
–9 –8 –7 –6 –5 –4 –3 –2 –1 0
19. 4y 6 or 2.5y 5
1
18. a 5 3 or 5a 30
{v  v 6 or v 5}
0
4
{r  r 1 or r 5} 4
17. v 2 4 or v 7 2
©
3
16. r 2 3 or 5r 25
{t  0 t 6} –1 0
2
{n  0.5 n 2.5}
–8 –7 –6 –5 –4 –3 –2 –1 0
15. 0
1
14. 10 4n 2
{m  5 m 3}
w 2
1
2
3
7
8
9
–8 –7 –6 –5 –4 –3 –2 1
0
1
1 or
4
w 3
5
6
2
{w  w 2} 4
Glencoe/McGrawHill
528 Chapter 12
T76
58. Geometry To construct any triangle, the sum of the lengths of two sides must be greater than the length of the third. Suppose that two sides of a triangle have lengths of 4 inches and 12 inches. What are the possible values for the length of the third side? Express your answer as a compound inequality. 8 s 16
528 Chapter 12 Inequalities
8. x 7 and x 4 3
57. Cooking A box of macaroni and cheese lists two sets of directions for cooking. It says to heat the macaroni for 11 to 13 minutes on the stove or 12 to 14 minutes in the microwave. a. Write an inequality that represents possible heating times. b. Graph the solution. See margin.
Algebra: Concepts and Applications
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59. Chemistry Soil pH is measured on a scale from 0 to 14. The pH level describes the soil as shown below. pH SCALE 0
INCREASING ACIDITY 1 2 3 4 5
NEUTRAL 6 7 8
DECREASING ACIDITY 9 10 11 12 13
OpenEnded Assessment 14
Most plants grow best if the soil pH is between 6.5 and 7.2. Mr. Cohen took samples of soil from his garden and found pH values of 6.3, 6.4, and 6.7. What range of values must the fourth sample have if the soil pH is the best for growing plants? (Hint: Find the mean soil pH.)
6.6 p 9.4
60. Critical Thinking Graph each compound inequality. Then state the solution. a–b. See margin for graphs. a. x 4 or x 4 b. 2 3z 2 14
{xx is a real number}
Mixed Review
Solve each inequality. Check your solution. (Lessons 12–3 & 12–4) 61. 2y 4 4 {yy 0}
62. 3n 8 22 {nn 10}
63. 2 0.6x 5 {xx 5}
64. p 3 7 {pp 15} 3
2
65. 10b 60 {bb 6}
66. 3r 12
4 ASSESS
{rr 4}
67. Welding Maxwell is welding two pieces of iron together. During this process, the iron melts and begins to boil. Small droplets erupt and follow the paths of parabolic arcs. The paths can be modeled by the quadratic function h(d) d2 4d 30, where h(d) represents the height of the arc above the ground at any horizontal distance d from the two pieces of iron. All measures are in inches. (Lesson 11–1) a. Graph the function. See margin. b. How high above the pieces of iron do the iron droplets jump?
Act It Out Following the order of class seating, assign each student a consecutive integer from a set containing both positive and negative integers. Have students write their numbers in large print on a piece of paper. Then write a pair of simple inequalities for each student to solve. Tell students whether you want to form an and or an or compound inequality with the simple inequalities. Have each student decide whether her or his number belongs to the solution set. If it does, have those students hold up their numbers. Then have the class decide whether the appropriate students have held up their numbers.
Answers 60a. 54 321 0 1 2 3 4 5
60b. 54 321 0 1 2 3 4 5
67a.
h (d )
34 in. 68. Find the GCF of 8x2, 2x, and 4xy. 69. Simplify 3(b 6).
Standardized Test Practice
(Lesson 10–1)
2x
(Lesson 9–3) 3b 18
70. Short Response Simplify (c2 5) (c2 8c 1).
8c 4
O
16128
28 24 20 16 12 8 4
d
4
12 16
Enrichment Masters, p. 76 (Lesson 9–2)
71. Multiple Choice Emily drove 8 miles in 12 minutes. At this rate, how many miles will she drive in 1 hour? (Lesson 5–1) C A 4 mi B 20 mi C 40 mi D 56 mi E 96 mi
12–5
NAME
DATE
PERIOD
Enrichment
Student Edition Pages 524–529
Precision of Measurement The precision of a measurement depends both on your accuracy in measuring and the number of divisions on the ruler you use. Suppose you measured a length of wood to the nearest oneeighth 5 of an inch and got a length of 68 in.
5
6 –8
www.algconcepts.com/self_check_quiz
Lesson 12–5 Solving Compound Inequalities 529
5
6
7
8
The drawing shows that the actual measurement lies somewhere 9 11 between 6 16 in. and 6 16 in. This measurement can be written using the symbol , which is read “plus or minus.” It can also be written as a compound inequality. 5
1
68 16 in.
Extra Credit Using only two different integers, write a compound inequality that has no solution, one that has exactly one solution, one whose graph is two separate rays, and one whose graph is the entire real number line. Sample answers: x 2 and x 2; x 2 and x 2; x 2 or x 2; x 2 or x 2
9
11
6 16 in. m 6 16 in.
1 In this example, 16 in. is the absolute error. The absolute error is
onehalf the smallest unit used in a measurement.
Write each measurement as a compound inequality. Use the variable m. 1
1
1. 32 4 in. 1
2. 9.78 0.005 cm 3
3 4 in. <_ m <_ 3 4 in. 1
4. 28 2 ft 1 1 27 2 ft <_ m <_ 28 2 ft
3. 2.4 0.05 g
9.775 cm <_ m <_ 9.785 cm
2.35 g <_ m <_ 2.45 g 11
5. 15 0.5 cm
1
6. 16 64 in.
14.5 cm <_ m <_ 15.5 cm
43 64
in. <_ m <_
45 64
in.
For each measurement, give the smallest unit used and the absolute error. 1 3 8. 128 in. m 128 in.
7. 12.5 cm m 13.5 cm
1 4
1 cm, 0.5 cm 1
1
9. 562 in. m 572 in.
1 in., ©
1 2
in.
Glencoe/McGrawHill
in.,
1 8
in.
10. 23.05 mm m 23.15 mm
0.1 mm, 0.05 mm T76
Algebra: Concepts and Applications
Lesson 12–5
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Lesson 12–6
12–6
1 FOCUS
What You’ll Learn
5Minute Check Lesson 12–5 1. Write the compound inequality x 0 and x 7 without using the word and. 0 x 7, or 7 x 0
You’ll learn to solve inequalities involving absolute value.
Solving Inequalities Involving Absolute Value There are three types of open sentences that can involve absolute value. They are listed below. Note that in each case, n is nonnegative since the absolute value of a number can only equal 0 or a positive number. x n
Why It’s Important Manufacturing Employees use inequalities involving absolute value to determine tolerances. See Example 3.
Types of Open Sentences with Absolute Value x 3 3 units
–10 –9 –8 –7 –6 –5 –4 –3 –2 x 6
The distance from 0 is 3. So, x 3 or x 3.
{x  x 6 or x 2}
x 3
0 1 2 3 4 5 6 7 8
Age Under 3 3 to 12 over 12
3 units
Absolute Value: Lesson 3–7
5 432 1 0 1 2 3 4 5
x 3 3 units
3 units
5 432 1 0 1 2 3 4 5
The distance from 0 is greater than 3. So, x 3 or x 3.
5. Write a compound inequality for the solution shown below. x 1 or x 2
3 units
The distance from 0 is less than 3. So, x 3 and x 3, or 3 x 3.
Admission Free $4.00 $6.00
–3 –2 –1 0 1 2 3 4 5
3 units
5432 1 0 1 2 3 4 5
1 or x 6 8
4. The table below gives admission prices for a museum. Write an inequality to represent the age a of people charged $4.00 for admission. 3 a 12
x n
You have already studied equations involving absolute value. Inequalities involving absolute value are similar. Consider the graphs and solutions of the three open sentences below.
Solve each compound inequality. Graph the solution. 2. 8 2x 18 {x  9 x 4}
3.
x n
For both equations and inequalities involving absolute value, there are two cases to consider. Case 1 The value within the absolute value symbols is positive. Case 2 The value within the absolute value symbols is negative.
Motivating the Lesson RealWorld Connection Most political polls are reported with a margin of error. For example, a poll may predict that a candidate will receive 58% of the vote in an election with a margin of error of
3%. Ask students if any of them knows what this means. Explain that it means that in an actual election the candidate is very likely to receive between 55% and 61% of the vote. Explain to students how the absolute value inequality  x 58  3 represents this situation.
530 Chapter 12
530 Chapter 12 Inequalities
Resource Manager Reproducible Masters • Study Guide, p. 77 • Practice, p. 77 • Enrichment, p. 77 • HandsOn Algebra, p. 138
Transparencies • 5Minute Check, 12–6 • Teaching, 12–6 • Answer Key, 12–6
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Solve x 5 2. Graph the solution. Case 1 x 5 is positive. x52 x 5 5 2 5 Add 5. x7
Case 2 x 5 is negative. (x 5) 2 Multiply by –1 (x – 5)(1) 2(1) and reverse the symbol. x – 5 2 5 5 2 5 x3
Add 5.
So, the solution is {x3 x 7}. 0 1 2 3 4 5 6 7 8 9 10
The solution makes sense since 3 and 7 are at most 2 units from 5.
Your Turn
2 3 4 5 6 7 8 9 10 11 12
a. Solve x 7 4. Graph the solution. {x3 x 11}
As in Example 1, when solving an inequality involving absolute value and the symbols or , the solution can be written as an inequality using and. However, when solving an inequality involving absolute value and the symbols or , the solution can be written as an inequality using or.
Example
2
Solve 6x 18. Graph the solution. Case 1 6x is positive. 6x 18 6x 18 6 6
Divide by 6.
x3
Case 2 6x is negative. 6x 18 18 6x Divide by 6 and 6 6 reverse the symbol. x 3
2 TEACH Teaching Tip When discussing Example 1, some students may be puzzled that the expression inside the absolute value sign can be negative. Remind them that this expression can be either positive or negative. It is only after the absolute value is taken that the result must be positive. Have students substitute numbers from the solution set into the original inequality to see how the expression inside the absolute value sign can be either positive or negative. Teaching Tip Interpreting an absolute value inequality in terms of distance, as in the last sentence of Example 1, will bring clarity to some students. Point out that  x b  is the distance of a point x from b. When students encounter an expression of the form x b, as in InClass Example 1 below, they can see by rewriting it as  x (b) that it expresses the distance of a point x from b.
InClass Examples Example 1 Solve  x 3  4. Graph the
So, the solution is {xx 3 or x 3}.
solution. {x  7 x 1}
54 321 0 1 2 3 4 5
–7 –6 –5 –4 –3 –2 –1 0 1
Your Turn
32 1 0 1 2 3 4 5 6 7
b. Solve x 2 3. Graph the solution.
{xx 1 or x 5}
Example 2 Solve  4x  16. Graph the
solution. {x  x 4 or x 4} –4 –3 –2 –1 0 1 2 3 4
Inequalities involving absolute value are often used to indicate tolerance. Tolerance is the amount of error or uncertainty that is allowed when taking measurements.
www.algconcepts.com/extra_examples
Lesson 12–6 Solving Inequalities Involving Absolute Value 531
Lesson 12–6
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Teaching Tip When discussing the Examine portion of Example 3, point out to students that the phrase from 0.495 inch to 0.505 inch, inclusive includes not only all measurements between 0.495 inch and 0.505 inch, but the measurements 0.495 inch and 0.505 inch as well.
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Example
1
When producing inch bolts for bicycle parts, the tolerance is 2 0.005 inch. What is the range of acceptable bolt measures?
3
Measurement Link
Explore
The difference in the actual size of the bolt and its expected size has to be less than or equal to 0.005 inch.
Plan
Let m the actual measure of the bolt. 1 2
Then, m 0.005. Solve
1 2
m 0.5 0.005 Write as 0.5. Case 1 m 0.5 is positive.
InClass Example
(m 0.5) 0.005
m – 0.5 0.5 0.005 0.5
(m 0.5)(1) 0.005(1)
Example 3 A lumber company makes 1 3 foot railing posts to use in 4 making decks. The tolerance for the posts is 0.025 foot. What is the range of acceptable post lengths? {  3.225 ft 3.275 ft}
m 0.505
m 0.495 The solution is {m0.495 m 0.505}. Examine An acceptable bolt must measure from 0.495 inch to 0.505 inch, inclusive. To check the solution, choose a value for m within this range and one outside of this range. Substitute them into the original problem. Which value results in a true inequality?
Answers
Check for Understanding Communicating Mathematics
2. intersection, x 2 1; union, x 2 1
Study Guide Masters, p. 77 NAME
DATE
1. Compare and contrast the graphs of the solutions for x 7 and x 7. See margin. 2. Graph the solutions for x 2 1 and x 2 1. Which inequality is the intersection of the graphs of two inequalities? Which inequality is the union of the graphs of two inequalities? 3.
Write two inequalities to describe the solution.
Sample 1: x 5 Solution: x 5 and x 5
Student Edition Pages 530–534
Solving Inequalities Involving Absolute Value You have already studied equations of the form x n involving absolute value, where n is a nonnegative number. Inequalities involving absolute value are similar. They are of the form x n or x n, where n is a nonnegative number.
Madison says that the solution for x 0 is the same as the solution for x 0. Mia says it is not. Who is correct? Explain. See margin.
Getting Ready
Guided Practice PERIOD
Study Guide
m 0.5 0.005 m 0.5 0.5 0.005 0.5
1.  x  7 includes all numbers between 7 and 7.  x  7 includes all numbers to the left of 7 and to the right of 7. 3. Mia;  x  0 includes only 0.  x  0 includes all real numbers.
12–6
Case 2 m 0.5 is negative.
m 0.5 0.005
4. x 10 x 10 and x 10 6. x 2 x 2 or x 2
To solve both equations and inequalities involving absolute value, there are two cases to consider. Case 1 The value of the expression within the absolute value symbol is positive.
Sample 2: x 1 Solution: x 1 or x 1 5. x 3 x 3 and x 3 7. x 8 x 8 or x 8
532 Chapter 12 Inequalities
Case 2 The value of the expression within the absolute value symbol is negative. Example 1: Solve x 4 2. Graph the solution. Case 1 x 4 is positive.
Case 2 x 4 is negative.
x42 x4424 x6
(x 4) (x 4)(1) x4 x44 x The solution is { x2 x 6}. 1
2
3
4
5
2 2(1) 2 2 4 2 6
7
Reverse the symbol.
Reteaching Activity
8
Example 2: Solve x 1 4. Graph the solution. Case 1 x 1 is positive.
Case 2 x 1 is negative.
x14 x1141 x3
(x 1) (x 1)(1) x1 x11 x The solution is {xx 3 or x 5}.
4 4(1) 4 4 1 5
Reverse the symbol.
–6 –5 –4 –3 –2 –1 0 1 2 3 4
Solve each inequality. Graph the solution. 1. n 5 {n 5 n 5} –6 –5 –4 –3 –2 –1 0
1
2
2. 4x 12 {x 3 x 3} 3
4
5
6
3. y 1 2 { y y 1 or y 3} –6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
5. t 3 2 {t  t 1 or t 5} –6 –5 –4 –3 –2 –1 0 ©
1
2
3
4
5
–6 –5 –4 –3 –2 –1 0
532 Chapter 12
2
3
4
5
6
–6 –5 –4 –3 –2 –1 0
1
2
3
4
5
6
3
4
6. h 2 6 {h 8 h 4}
6
Glencoe/McGrawHill
1
4. 6p 2.4 { p p 0.4 or p 0.4}
–8 –7 –6 –5 –4 –3 –2 –1 0
T77
1
2
Algebra: Concepts and Applications
Naturalist Learners Have students bring in natural items such as flowers or leaves that are hard to measure exactly. Point out that there is always some degree of uncertainty in measurement. Have students measure their items, also giving a value to how precisely they think they can find a reliable measure. For example, a student may think her measurement of a leaf is reliable to the nearest eighth of an inch. Then have students write and solve absolute value inequalities to find the expected range of actual measurements.
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Solve each inequality. Graph the solution. (Examples 1 & 2)
8–13. See margin for graphs. 13. {ss 1 or s 7}
8. n 4 5
{n1 n 9}
10. 3j 12 {j4 j 4}
11. t 5 3 {tt 8 or t 2}
12. 2y 2 {yy 1 or y 1} 13. s 4 3 14. Measurement Refer to Example 3. What are the possible measures for the bolt if the tolerance is 0.05 inch? Does a lesser or greater tolerance ensure more accurate measurements? Explain. (Example 3)
0.45 m 0.55; lesser tolerance
Exercises
• • • • •
•
•
•
•
•
•
•
•
•
•
•
•
•
Solve each inequality. Graph the solution. 15–32. See Solutions Manual.
Practice
Homework Help For Exercises
See Examples 1, 3
15, 17, 18–20 22, 23, 25, 26, 28–35, 38–43 18, 21, 24, 27
2
Extra Practice See page 717.
33. s 90 4
3 PRACTICE/APPLY
9. x 2 6 {x4 x 8}
15. m 1 5
16. 3v 15
17. z 7 2
18. x 3 8
19. p 1 2
20. r 4 4
21. 7t 14
22. a 3 4
23. k 2 3.5
24. 5n 30
25. y 3 6
26. z 1 1
27. 9x 18
28. w 2 5
29. r 4 1
30. a 8 3
31. h 3 9
32. d 9 0.2
Error Analysis Watch for students who assign positive and negative values to the side of the inequality opposite the variable expression, such as writing the inequalities n 4 5 and n 4 5 for Exercise 6, thus forgetting to change the direction of the second inequality. Prevent by encouraging students to set up their work explicitly as shown in Examples 1–3 to avoid any misunderstanding or errors. Assignment Guide Basic: 15–43 odd, 44–52 Average: 16–40 even, 42–52 All: Quiz 2, 1–10
Write an inequality involving absolute value for each statement. Do not solve. 33. Quincy’s golf score s was within 4 strokes of his average score of 90. 34. The measure m of a board used to build a cabinet must be within 1 1 inch of 46 inches, inclusive, to fit properly. m 46 4 4 35. The cruise control of a car set at 65 mph should keep the speed s within 3 mph, inclusive, of 65 mph. s 65 3
Answers 8. 1 0 1 2 3 4 5 6 7 8 9
9. 864 2 0 2 4 6 8 10 12
10. 54 321 0 1 2 3 4 5
For each graph, write an inequality involving absolute value. 36. 5 4 321 0 1 2 3 4 5
0 1 2 3 4 5 6 7 8 9 10 5 4 321 0 1 2 3 4 5
x 1 1 Solve each inequality. Graph the solution. 38–41. See margin.
12.
38. 2x 11 7
39. 3x 12 12
13.
40. 4x 3 8
41. 10x 1 90
x 4
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Applications and Problem Solving
43. m 3.25 0.05; 3.20 m 3.30
11.
37.
54 321 0 1 2 3 4 5 98 7654321 0 1
42. b 8758.20 2; 8756.20 b 8760.20
For Exercises 42–43, write and solve an absolute value inequality. 42. Finance Ms. Gibson is a bank teller. She must balance her drawer and be within $2 of the expected balance. If Ms. Gibson’s expected balance is $8758.20, what are acceptable balances for her drawer? 43. Chemistry For a chemistry project, Marvin must pour 3.25 milliliters of solution into a beaker. If he does not pour within 0.05, inclusive, of 3.25 milliters, the results will be inaccurate. How many milliliters of solution can Marvin use?
Practice Masters, p. 77 NAME
12–6
DATE
Student Edition Pages 530–534
Solving Inequalities Involving Absolute Value Solve each inequality. Graph the solution. 1. k 2 1
2. m 7 4
{k  3 k 1}
{m  11 m 3}
–8 –7 –6 –5 –4 –3 –2 –1 0
–12 –11 –10 –9 –8 –7 –6 –5 –4 –3 –2
1
3. 4p 16
4. w 3 3
{ p  4 p 4}
Lesson 12–6 Solving Inequalities Involving Absolute Value 533
{w  0 w 6} –2 –1 0
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6
5. a 5 4
9. b 8 2
0 1 2 3 4 5 6 7 8 9 10
39. {x  0 x 8} 1 0 1 2 3 4 5 6 7 8 9
8 7654321 0 1 2
41. {x  x 10 or x 8} 108642 0 2 4 6 8 10
6
–1
7
8
3
4
5
0
1
2
3
4
5
1
2
3
4
{z  z 5 or z 9}
13. 5c 25
–11 –10 –9 –8 –7 –6 –5 –4 –3 –2
14. 2g 2
{c c 5 or c 5}
{g  g 1 or g 1} –5 –4 –3 –2 –1 0
–6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6
15. f 5 2
4
Glencoe/McGrawHill
1
2
3
4
16. s 6 1.5
{f  f 7 or f 3}
©
2
12. z 7 2
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2
3
1
–5 –4 –3 –2 –1 0
9 10 11 12
{x  x 0 or x 8}
2
7
{ y  y 2 or y 4}
11. x 4 4
1
6
10. y 1 3
{b  b 10 or b 6} 5
5
{q  0.5 q 4.5}
–14 –13 –12 –11 –10 –9 –8 –7 –6 –5
4
4
8. q 2 2.5
{v  12 v 6}
3
3
–4 –3 –2 –1 0
–2 –1 0 1 2 3 4 5 6 7 8 9 10
40. {x  5 x 1}
2
{t  2 t 2}
7. v 9 3
Answers
1
6. 6t 12
{a  1 a 9}
38. {x  x 2 or x 9}
PERIOD
Practice
5
6
7
{s s 7.5 or s 4.5} 8
4
9 10
T77
5
6
7
8
Algebra: Concepts and Applications
Lesson 12–6
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44. Critical Thinking The percent of error is the ratio of the greatest possible error (tolerance) to a measurement. You can find the percent of error using this formula.
4 ASSESS OpenEnded Assessment
greatest possible error
percent of error 100 measurement
Writing Have students write a summary explaining the similarities and differences between solving inequalities that do and don’t involve absolute value.
One rating system for inline skate Rating bearings is based on tolerances. The table shows tolerances for the outside diameter 1 of bearings measuring 22 millimeters. 3 a. Find the percent of error for each rating 5 to the nearest hundredth. b. What can you conclude about the rating system?
Quiz 2 The Quiz provides students with a brief review of the concepts and skills in Lessons 12–4 through 12–6. Lesson numbers are given to the right of exercises or instruction lines so students can review concepts not yet mastered.
Tolerance (mm) 0.010 0.008 0.005
The higher the rating, the better the bearings. Mixed Review 44a. rating 1, 0.05%; rating 3, 0.04%; rating 5, 0.02%
Graph the solution of each compound inequality. (Lesson 12–5) 45. m 7 or m 0 47. y 5 and y 0
45–48. See margin.
49. Solve 1 4y 5.
Answers
46. x 2 and x 5 48. r 2 or r 2 (Lesson 12–4)
{yy 1}
50. Sales Grant bought a sweater for $44.52. This cost included 6% sales tax. What was the cost of the sweater before tax? (Lesson 5–5) $42
Quiz 2 4.
(Lesson 3–7) 2, 6
51. Solve x 2 4.
54 321 0 1 2 3 4 5
Standardized Test Practice
5. 7654321 0 1 2 3
5 8
52. Short Response Write three fractions whose sum is 1.
See students’ work.
(Lesson 3–2)
6. 654321 0 1 2 3 4
7. 2 3 4 5 6 7 8 9 10 11 12
8.
Quiz 2
1211109 8 765432
9.
>
54 321 0 1 2 3 4 5
Lessons 12–4 through 12–6
Solve each inequality. (Lesson 12–4) 1. 3x 8 11 {xx 1} 4–9. See margin for graphs.
2. 5 6n 19 {nn 4}
3. 9d 4 8 d {dd 1.2}
Solve each compound inequality. Graph the solution. (Lesson 12–5) 4. 1 x 4 or 1 x 4
{xx 3}
12–6
DATE
{n5 n 1}
6. 6 2f 10
Solve each inequality. Graph the solution. (Lesson 12–6)
Enrichment Masters, p. 77 NAME
5. 2 n 3 4
7. y 7 2
PERIOD
Enrichment
Student Edition Pages 530–534
8. a 8 1
{f5 f 3}
9. 5x 20
{y5 y 9} {aa 9 or a 7} {x4 x 4} 10. Travel Before the meter begins ticking, the charge for a taxi is $1.60. For each mile driven, there is an additional charge of 80 cents. What is the greatest distance you can travel in this taxi if you do not want to pay more than $10.00? (Lesson 12–4) 10.5 mi
Absolute Value Functions Some types of functions that occur frequently have special names. Absolute value functions are an example. Example: Graph y x 2. y x
y
4
2
3
1
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4 2
0
1
1
0
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1
3
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4
–4
–2
534 Chapter 12 Inequalities
2 x
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Complete the table for each equation. Then, draw the graph. 1. y x
2. y x 2
x
y
3
3 2 1 0 1 2 3
2 1 0 1 2 3
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–2
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3 2 1 0 1 2 3
1 0 1 2 3 4
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1 0 1 2 1 0 1
1 0
–2
1 2 3
3. y x 1 x
x 3 2
Extra Credit
O
–2
2
2
x
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2
4 3 2 1 0 1 2
1 –2
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4 x
0 1
–2
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45.
x
–2
4. y 2 x y
Answers
y 2
y 4 2
–2
O
2
4 x
–2
Algebra: Concepts and Applications
A bakery guarantees that a pan of their cinnamon rolls will weigh from 15.85 ounces to 16.15 ounces, inclusive. What is the bakery’s tolerance for the weight? Write an absolute value inequality for the acceptable weight. 0.15 oz;  w 16  0.15
8 7654321 0 1 2
46. 4 321 0 1 2 3 4 5 6
47. 7654321 0 1 2 3
48. 54 321 0 1 2 3 4 5
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12–7
Graphing Inequalities in Two Variables
What You’ll Learn
Mr. Wheat is planning to take the Jazz Club to a music festival. Lawn tickets cost $20, and pavilion tickets cost $30. If he plans to spend at most $300, how many of each ticket can Mr. Wheat purchase?
Lesson 12–6 Solve each inequality. 1.  3x  15 {x  x 5 or x 5} 2.  t 6  11 {t  5 t 17}
The cost of lawn tickets
Let x represent the number of lawn tickets. Let y represent the number of pavilion tickets. Then the inequality below represents the solution.
Budgeting By graphing inequalities, you can solve problems where there are many solutions. See Example 3.
5Minute Check
Why It’s Important
1 FOCUS
You’ll learn to graph inequalities on the coordinate plane.
Lesson 12–7
20x
30y
300
plus
the cost of pavilion tickets
3. Write an inequality involving absolute value for the graph.
is at most $300.
–1 0 1 2 3 4 5 6 7
The inequality is written in two variables. It is similar to an equation written in two variables. An easy way to show the solution of an inequality is to graph it in the coordinate plane. This problem will be solved in Example 3. The solution set of an inequality in two variables contains many ordered pairs. The graph of these ordered pairs fills an area on the coordinate plane called a halfplane. The graph of an equation defines the boundary or edge for each halfplane. Use these steps to graph y 3. y
Step 1 Determine the boundary by graphing the related equation, y 3. The related equation for y 3 is y 3.
y 3
Step 2 Draw a dashed line since the boundary is not part of the graph. Step 3 Determine which halfplane is the solution. To do this, substitute a point from each halfplane into the inequality. Find which point results in a true statement. Test the point at (5, 8). y3 8 3 Replace y with 8. true
O
x
5. A company’s tolerance for the diameter of the washer below is 0.003 inch. What is the range of acceptable washer diameters d? 3 4 in.
Test the point at (3, 1). y3 1 3 Replace y with 1. false
The halfplane that contains (5, 8) is the solution. Shade that halfplane. Any point in the shaded region is a solution of the inequality y 3.
Lesson 12–7 Graphing Inequalities in Two Variables 535
Resource Manager Reproducible Masters • Study Guide, p. 78 • Practice, p. 78 • Enrichment, p. 78 • Graphing Calculator, pp. 34–35 • HandsOn Algebra, p. 139 • Assessment and Evaluation, p. 231
Sample answer:  x 3 2 4. Write an inequality involving absolute value for the statement A box of cookies is acceptable to sell if its 1 weight is within ounce of 4 the stated weight of 1 18 ounces.  w 18  4
Transparencies • 5Minute Check, 12–7 • Teaching, 12–7 • Answer Key, 12–7 Technology/Multimedia • AlgePASS, Lesson 30
{d  0.747 in. d 0.753 in.}
Motivating the Lesson HandsOn Activity Have groups of students cut out ten squares of paper, labeling three Q for quarter and seven D for dime. Have groups form and record all combinations (Q, D) they can with the squares so that the indicated value is not more than 75¢. Point out that these are solutions of an inequality in two variables, 25Q 10D 75. Have students identify the ordered pairs that are not also solutions of 25Q 10D 75. Point out that these points are analogous to endpoints of the graph of an inequality in one variable. Lesson 12–7
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Budgeting Link
Example
2 TEACH
1
Graph y x 3. Step 1 Determine the boundary by graphing the related equation, y x 3.
InClass Example Example 1 Graph y x 2. y
x
O
x
x 3
y
2 1 0 1 2
(2) 3 (1) 3 (0) 3 1 3 2 3
5 4 3 2 1
y
x
O
Step 2 Draw a dashed line since the boundary is not included. Step 3 Test any point to find which halfplane is the solution. Use (0, 0) since it is the easiest point to use in calculations.
Teaching Tip Prior to discussing Examples 2 and 3, you may want to review graphing linear equations using slope and yintercept, and using x and yintercepts. In Example 2, explain that any point in one of the halfplanes can be chosen in Step 3; (0, 0) is chosen when possible because it is the easiest to substitute in the inequality.
y x 3 0 (0) 3 x 0, y 0 0 3 false y
Since (0, 0) does not result in a true inequality, the halfplane containing (0, 0) is not the solution. Thus, shade the other halfplane.
y x 3
O
InClass Example
Your Turn
Example 2
a. Graph y x 7. See margin.
Graph 3x y 1.
x
y
O
Answer a.
When graphing inequalities, the boundary line is not always dashed. Consider the graph of y 3. Since the inequality means y 3 or y 3, the boundary is part of the solution. This is indicated by graphing a solid line.
x
Example
2
y O
Graph 4x y 12. To make a table or graph for the boundary line, solve the inequality for y in terms of x.
x
4x y 12 4x y 4x 12 4x y 4x 12
yx7
Subtract 4x from each side. Rewrite 12 4x as 4x 12.
536 Chapter 12 Inequalities
www.algconcepts.com/extra_examples
From the Classroom of … Nicki Hudson West Linn High School West Linn, Oregon
When teaching students to graph y x 2, I like to make sure students understand that for every ordered pair (x, y) whose point is in the shaded region, the yvalue will be less than the xvalue plus 2.
536 Chapter 12
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Step 1 Determine the boundary by graphing y 4x 12.
y 4x y 12
Step 2 Draw a solid line since the boundary is included. Step 3 Test (0, 0) to find which halfplane contains the solution.
10864
12 10 8 6 4 2
O
2 4
x
4x y 12 4(0) 0 12 x 0, y 0 0 12 true The halfplane that contains (0, 0) should be in the solution, which is indicated by the shaded region.
Teaching Tip In Example 3, emphasize that the realworld solution contains only points in the shaded halfplane with whole number coordinates. A number of tickets cannot be negative, so these points are bounded by the x and yaxes. So, the graph of the realworld solution shows distinct points in the coordinate plane bounded by the xand yaxes and 2 the line y x 10. 3
InClass Example Example 3 Your Turn b. Graph 2x y 10. See margin.
al Wo
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When solving reallife inequalities, the domain and range of the inequality are often restricted to nonnegative numbers or whole numbers.
Example
3
Budgeting Link
Refer to the application at the beginning of the lesson. How many lawn and pavilion tickets can Mr. Wheat purchase? First, solve for y in terms of x.
Ms. Kwan sells computer systems. She makes a commission of $50 per home system sold and $150 per business system sold. The inequality 50x 150y 450 represents how many of each system she must sell to make at least $450 a week in commissions. Graph the inequality. Describe the numbers of systems Ms. Kwan can sell to meet her goal. y
20x 30y 300 20x 30y 20x 300 20x Subtract 20x from each side. 30y 20x 300
50x 150y 450
30y 20x 300 Divide each side by 30. 30 30 20x 300 y 30 30 2 y x 10 3
Step 1 Determine the boundary by 2 graphing y x 10. 3
Step 2 Draw a solid line since the boundary is included. Step 3 Test (0, 0) to find which halfplane contains the solution.
x O
The solutions are ordered pairs in the shaded halfplane that have nonnegative wholenumber coordinates, such as (3, 2).
y
16 14 12 10 8 6 4 2 20x 30y 300
O
Study Guide Masters, p. 78 x
2 4 6 8 10 12 14 16
(continued on the next page)
NAME
12–7
DATE
Student Edition Pages 535–539
Graphing Inequalities In Two Variables Inequalities, like equations, may have two variables instead of one. The solution of an inequality having two variables contains many ordered pairs. The graph of these ordered pairs fills an area of the coordinate plane called a halfplane. The graph of the related equation defines the boundary or edge for each halfplane. Graph y 2x 3.
Example:
Step 1 Determine the boundary by graphing the related equation, y 2x 3.
Lesson 12–7 Graphing Inequalities in Two Variables 537
Make a table of values. y
x 2x 3 y 2 2(2) 3 7 1 2(1) 3 5 0 2(0) 3 3 1 2(1) 3 1 2 2(2) 3 1
12 10
42 2 4
(0, 3) (1, 1)
O
x (2, –1)
Step 3 Use a point not on the boundary to find which halfplane is the solution. Use (0, 0).
y
y
y 2x 3 ? 0 2(0) 3 0 3 true
6 2x y 10 4 2 8
(–1, 5)
Step 2 Draw a dashed line because the boundary is not included. Note: If the inequality involved or , the boundary would be included, and you would make the boundary a solid line.
Answer b.
PERIOD
Study Guide
x
O
Since 0 3 is true, shade the halfplane containing (0, 0). Note: If the result were false, you would shade the other halfplane. Graph each inequality. 1. y x 2
2. y x 2
3. y 2x 2 y
y
y
O
2 4 6 8x
O
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Mr. Wheat cannot buy a negative number of tickets, nor can he buy portions of tickets. The solution is positive ordered pairs that are
3 PRACTICE/APPLY
2 3
whole numbers beneath or on the graph of the line y x 10.
Error Analysis Watch for students who believe that you always shade the halfplane above the boundary line for and inequalities or that you always shade the halfplane below the boundary line for and inequalities, which will lead to errors when the coefficient of y is negative as in Exercise 5. Prevent by encouraging students always to test (0, 0) or another point in one of the halfplanes before deciding which to shade.
One solution is (12, 2). This represents 12 lawn tickets and 2 pavilion tickets costing $300.
Check for Understanding 1. Explain how to determine whether the boundary is a solid line or dashed line when graphing inequalities in two variables.
Communicating Mathematics Math th Jo Journal
halfplane boundary
2. Describe how you could check whether a point is part of the solution of an inequality.
1–2. See margin.
Answers 1. Graph a solid line for and . Graph a dashed line for and . 2. Substitute the coordinates into the original inequality. If the statement is true, then the halfplane containing that point is the solution. 12. y
Getting Ready
Guided Practice
Test (0, 0) to find which halfplane is the solution of each inequality.
Sample: 3x y 4 Solution: 3x y 4 3(0) 0 4 0 4 false Shade the halfplane not containing the point at (0, 0).
y
x
O
200
175 150
3. x 2
5x 8y 1000
4. y 1
5. 5x y 5
y
y
y
100 75
O
50 25
50 25
O
75
100 150 125 175
x
O
x
O
x
x
Sample answer: 56 singles, 90 couples Graph each inequality. (Examples 1 & 2) 6–11. See Solutions Manual. 6. y 1
Practice Masters, p. 78 NAME
12–7
9. 2x y 0
DATE
Student Edition Pages 535–539
Graphing Inequalities in Two Variables Graph each inequality. 2. y x 3 y
3. y x 1 y
y
x
O
x
O
10. 4x y 8
5. x y 4 y
O
x
O
7. 2x y 10
8. 3x y 9
x
Reteaching Activity
9. x 2y 6 y
x
O
11. 2x 2y 6
y
O
x
O
Glencoe/McGrawHill
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x
12. 4x 2y 12 y
y
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10. x 4y 8
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6. 2x y 2 y
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11. x 3y 9
12. Sales Tickets for the winter dance are $5 for singles and $8 for couples. To cover the deejay, photographer, and decoration expenses, a minimum of $1000 must be made from ticket sales. Write and graph an inequality that describes the number of singles’ and couples’ tickets that must be sold. Name at least one solution. (Example 3) See margin.
538 Chapter 12 Inequalities 4. y 3x 3
8. y 3x 1
PERIOD
Practice
1. y 2
7. y x 7
x
Algebra: Concepts and Applications
Verbal/Linguistic Learners Have students work in pairs. Write several inequalities on the board or overhead. Have one student in each pair verbally guide the second student through the steps in solving and graphing the first inequality. After an inequality is successfully completed, have the students reverse roles.
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Exercises
•
•
•
•
•
•
•
•
•
•
•
•
•
Graph each inequality. 13–32. See Solutions Manual.
Practice
Homework Help For Exercises
See Examples 1
13–21, 27, 28, 32 22–26, 29–31, 33–34
2, 3
Extra Practice See page 717.
Applications and Problem Solving al Wo
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• • • • •
13. 16. 19. 22. 25. 28.
y7 y x 3 y x x y 5 3x y 1 3y 3(4x 3)
14. 17. 20. 23. 26. 29.
x5 xy8 y 2x x 2y 10 3x 2y 12 2(x y) 14
15. 18. 21. 24. 27. 30.
yx6 yx y 3x 4 2x y 6 y 2(2x 1) 3(5x y) 0
For Exercises 31–32, write an inequality and graph the solution. 31. The sum of two numbers is greater than four. 32. Twice a number is less than or equal to another number. 33. Animals Amara and Toshi have set a goal to find homes for more than twelve pets through the Humane Society. a. Write and graph an inequality to determine how many homes each girl must find to reach the goal. b. List three of the solutions. c. Describe the limitations on the solution set.
33–34. See Solutions Manual. 34. Airlines For each mile that Mr. Burnett flies, he earns one point toward a free flight. His credit card company is associated with the airline, and he earns onehalf point for each dollar charged. Suppose he needs at least 20,000 points for a free flight. Show with a graph the miles that Mr. Burnett must fly and the money he must charge to get a free flight. 35. Critical Thinking Graph the intersection of the solutions of 4x 2y 8 and y x. See Solutions Manual.
Mixed Review 36–39. See margin.
OpenEnded Assessment Speaking Write several inequalities on the board or overhead. Have students take turns describing how they would graph the boundary line (using slope and yintercept or x and yintercept) and whether the boundary line is solid or dashed. Then have them describe how to determine which halfplane to shade. Chapter 12, Quiz B (Lessons 12–4 through 12–7) is available in the Assessment and Evaluation Masters, p. 231. Assignment Guide Basic: 13–33 odd, 35–42 Average: 14–32 even, 33–42
Answers 36. {t  t 7 or t 1} 98 7654321 0 1
37. {h 5 h 9} 2 3 4 5 6 7 8 9 10 11 12
38. { p  2 p 5} 321 0 1 2 3 4 5 6 7
Solve each inequality. Graph the solution. (Lessons 12–5 & 12–6) 36. t 4 3 38. 1 p 1 6
37. h 7 2 39. 5b 10 or b 4 5
40. Travel Ben and Pam left the park at the same time. Ben traveled north at 45 miles per hour. Pam traveled east at 60 miles per hour. After 1 hour, how far apart are Ben and Pam? (Lesson 8–7) 75 mi
Standardized Test Practice
4 ASSESS
41. Short Response 848.3 in scientific notation. (Lesson 8–4)
41. 8.483 102
42. Multiple Choice Suppose you toss 2 coins at the same time. What is the probability that both land heads up? (Lesson 5–6) B A 0 B 0.25 C 0.50 D 0.75 E 1
www.algconcepts.com/self_check_quiz
Lesson 12–7 Graphing Inequalities in Two Variables 539
39. {b  b 2 or b 1} 54 321 0 1 2 3 4 5
Enrichment Masters, p. 78 NAME
12–7
DATE
PERIOD
Enrichment
Student Edition Pages 535–539
Inequalities with Triangles Recall that a line segment can be named by the letters of its endpoints. Line segment AB (written as “AB”) has points A and B for endpoints. The length of AB is written without the bar as AB. AB BC
A B
The statement on the left above shows that AB is shorter than BC. The statement on the right above shows that the measure of angle A is less than that of angle B. These three inequalities are true for any triangle ABC, no matter how long the sides are.
A
a. AB BC AC b. If AB AC, then C B. c. If C B, then AB AC.
B
C
Use the three triangle inequalities for these problems. 1. List the sides of triangle DEF in order 2. In the figure below, which line segment of increasing length. is the shortest?
F D , DE , E F
K
85°
Extra Credit Guillermo has a package that will cost $3.20 to mail. He finds some old sheets of 21¢ and 9¢ stamps in a file cabinet. Write an inequality for the numbers of 21¢ stamps x and 9¢ stamps y he can use to mail the package. Then give an example of a solution that uses at least fifteen 9¢ stamps that does not exceed the required postage by more than 1¢. 21x 9y 320; (2, 31), (5, 24), or (8, 17)
M L
D
F
60°
65° 35°
55° 60° L 65°
E
J
3. Explain why the lengths 5 cm, 10 cm, and 20 cm could not be used to make a triangle. 5 10 is not greater
50°
65° M
4. Two sides of a triangle measure 3 in. and 7 in. Between which two values must the third side be? 4 in. and
than 20.
10 in.
5. In triangle XYZ, XY 15, YZ 12, and XZ 9. Which is the greatest angle? Which is the least?
Z; Y
6. List the angles A, C, ABC, and ABD, in order of increasing size.
ABD, A, ABC, C
C 13 B
12 15
5 D 9 A
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Investigation Chapter 12
Investigation
PREPARE This optional investigation is designed to be completed by pairs of students over 1–2 days.
Objective Students graph quadratic equations. By evaluating the quadratic equations for points on, inside, and outside of the resulting parabolas, they learn to identify solutions of quadratic inequalities. Students apply a quadratic inequality to a realworld situation and use the graph of its related function to identify solutions of the inequality. Students summarize what they have learned in a poster and make a brochure of how they can apply this knowledge to develop and solve an original realworld application.
Quadratic Inequalities
Materials grid paper ruler
In this investigation, you will learn how to solve and graph quadratic inequalities.
yellow and blue colored pencils
Investigate 1. Graph the quadratic equation y x 2 2x 3 on a piece of graph paper.
1a–d. See Solutions Manual.
a. With a yellow colored pencil, shade the region inside the parabola. b. Make a table like the one below. Fill in column 1 with five points that appear in the yellow region, such as (0, 0). Compare the value of the ycoordinate with the value of y x 2 2x 3 when it is evaluated at the xcoordinate. When the quadratic equation is evaluated at 0, the result is 3. The ycoordinate, 0, is greater than 3. Place the correct inequality symbol in column 3 for each of the other four points that fall in the yellow region.
Mathematical Overview The investigation utilizes the following concepts: • graphing quadratic equations, • evaluating quadratic equations, • identifying solution sets of inequalities, and • writing quadratic inequalities to represent realworld situations.
Ask students to think of realworld situations they have represented using linear inequalities. Explain that many realworld situations such as those involving area are represented by quadratic inequalities. Have students recall what they know about quadratic functions and their graphs. Ask them how they think a quadratic inequality will be represented.
540 Chapter 12
ycoordinate
or
y x 2 2x 3
(0, 0)
0
y (0)2 2(0) 3 or 3
c. Write a quadratic inequality that compares the points of the yellow region with the quadratic equation y x 2 2x 3. d. With a blue colored pencil, shade the region outside the parabola. Repeat parts b–c for the blue region. 2. Graph the quadratic equation y x 2 2x 3 on a separate piece of graph paper. a. See Solutions Manual. a. Shade the region inside the parabola yellow. Then shade the region outside the parabola blue. Make tables similar to those in Step 1. Write inequalities describing the yellow and blue regions.
Suggested Time Management Investigation 25–40 min Extension: Gathering Data 20–30 min Extension: Summarizing Data 35–50 min
Motivating the Lesson
Point
b. Suppose you wanted to include the values on the boundary line of a quadratic inequality. Explain how you would write the inequality to include the boundary line. Use the or symbols. c. Now suppose you did not want to include the boundary line of a quadratic inequality. Explain how you would draw the graph to show that the boundary line was not included in the solution.
540 Chapter 12 Inequalities
Draw the graph of the parabola with a dashed line.
Cooperative Learning This investigation offers an excellent opportunity for using cooperative groups. For more information on cooperative learning strategies and group management, see Cooperative Learning in the Mathematics Classroom, one of the titles in the Glencoe Mathematics Professional Series.
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3. Spring Town is building a community center, the Spring Town Pavilion. The building is to be 20 feet longer than it is wide and to have an area greater than or equal to 1500 square feet. a. See margin. a. Let w represent the width of the building. Then w 20 represents the length. The area of the building, w(w 20), must be greater than or equal to 1500 square feet. That is, w 2 20w 1500 0. Graph f(w) w 2 20w 1500.
MANAGE Teaching Tip Review graphing quadratic functions with students. The graphs of the functions in the body of the investigation can be sketched by finding the vertex and factoring to find the xintercepts. Point out the difference between the quadratic inequalities in two variables in Exercises 1 and 2, where the solution is a region of the coordinate plane, and the quadratic inequality in one variable in Exercise 3 solved by graphing the related function, in which the solution is a portion of a parabola.
w
w 20
b. Find the values for w that satisfy the inequality in part a. To do this, choose values for w both inside and outside of your parabola. {ww 50 or w 30} c. What restrictions are placed on your solution for w since it represents width? It must be positive. d. Use your graph to find two possible dimensions for the Spring Town Pavilion. Sample answer: 30 ft by 50 ft;
35 ft by 55 ft
Working in Pairs By working in pairs, one student can sketch a graph while the other tests points and records findings. The students can work together on exercise questions. Students can then switch roles for Exercise 2. Partners may want to divide up work on the brochure and poster to complete individually and then check and revise their work together.
In this extension, you will continue to investigate quadratic inequalities. Graph each quadratic inequality. 1–4. See Solutions Manual. 1. y x 2 4x 8 2. y x2 2x 15 3. y 3x2 3 4. The Spring Town Pavilion will have a deck for small outdoor gatherings. The length of the deck is to be 6 feet more than the width. The area of the deck is at least onefifth of the area of the community center building. Write and graph a quadratic inequality that describes this situation. Then give three possible dimensions for the new deck.
Working as a Class You may want to complete the realworld application in Exercise 3 as a class, since the concepts are not as straightforward as in the first two exercises. Make sure students understand the solutions of Exercise 3 before they go on to the Extending the Investigation.
Presenting Your Investigation Here are some ideas to help you present your conclusions to the class. • Make a poster that describes how to solve quadratic inequalities. Include at least two different inequalities. Show how the graphs are shaded to represent the solutions. • Write and solve a problem similar to the one in Step 3. Make a brochure that describes the problem, the graph of its solution, and a list of possible solutions.
ASSESS
Investigation For more information on graphing quadratic inequalities, visit: www.algconcepts.com
Chapter 12 Investigation Parabolas and Pavilions
Answer 3a. w2 20w 1500 1600
f (w)
800
541
Students’ work should show that they understand how the solution set of a quadratic inequality in two variables relates to the graph of the corresponding quadratic equation. They should understand how to solve an application of a quadratic inequality in one variable using the graph of the related quadratic function.
O 60 20 800
20
60
w
Students should add their posters and brochures to their portfolios at this time. Investigation
541
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Study Guide and Assessment Understanding and Using the Vocabulary This section provides a listing of the new terms, properties, and phrases that were introduced in this chapter. The exercises check students’ understanding of the terms by using a variety of verbal formats including matching, completion, and true/false. Glossary A complete glossary of terms appears on pages 728–734.
MindJogger Videoquizzes MindJogger Videoquizzes provide an alternative review of concepts presented in this chapter. Students work in teams to answer questions, gaining points for correct answers.
Answers 11. 5 4 3 2 1
12. 0
1
2
3
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CHAPTER
Study Guide and Assessment
12
Understanding and Using the Vocabulary
Review Activities For more review activities, visit: www.algconcepts.com
After completing this chapter, you should be able to define each term, property, or phrase and give an example or two of each. boundary (p. 535) compound inequality (p. 524) halfplane (p. 535)
intersection (p. 524) quadratic inequalities (p. 540)
setbuilder notation (p. 510) union (p. 525)
Complete each sentence using a term from the vocabulary list. 1. The solution of a compound inequality using or can be found by the ____?____ of the graphs of the two inequalities. union 2. Graph the related equation of an inequality to find the ____?____ of the halfplane. boundary 3. Use a test point from each ____?____ to find the solution of an inequality in two variables. halfplane 4. An inequality of the form x y z is called a(n) ____?____ . compound inequality 5. The ____?____ for the graph of an inequality in two variables will either be a dashed or solid line. boundary 6. The ____?____ of two graphs is the area where they overlap. intersection 7. A(n) ____?____ is an area on the coordinate plane representing the solution for an inequality in two variables. halfplane 8. 2x 3 5 or x 1 is an example of a(n) ____?____ . compound inequality 9. A solution written in the form {xx 3} is written in ____?____ . setbuilder notation 10. The solution of a compound inequality using and can be found by the ____?____ of the graphs of the two inequalities. intersection
4
Skills and Concepts
13. 4
5
1
0
3
14.
Objectives and Examples 2
• Lesson 12–1 Graph inequalities on a number line. Graph x 5 on a number line.
Review Exercises Graph each inequality on a number line. 11. x 3 12. z 2 1 13. 3.5 x 14. a 1 2
11–14. See margin.
The graph begins at 5, but 5 is not included. The arrow is to the left. The graph describes values that are less than 5.
Write an inequality for each graph. x 5 15. 7 6 5 4 3
x7
16. 2
3
4
5
6
7
5
542 Chapter 12 Inequalities
6
7
8
9
Study Guide
www.algconcepts.com/vocabulary_review
Resource Manager Reproducible Masters • Assessment and Evaluation, pp. 221–229, 232–234
542 Chapter 12
Technology/Multimedia • MindJogger Videoquizzes • TestCheck and Worksheet Builder
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Chapter 12 Study Guide and Assessment Objectives and Examples • Lesson 12–2 Solve inequalities involving addition and subtraction. Solve x 6 2. x62 x6626 x8
Review Exercises Solve each inequality. Check your solution. 17. x 3 7 {xx 4} 18. a 4 2 {aa 6} 2 1 19. y
Add 6 to each side.
The solution is {xx 8}.
3
2
yy 116
20. 12x 11x 5 3 {xx 2} 21. 3(x 1) 4x {xx 3}
Skills and Concepts The Objectives and Examples section reviews the skills and concepts of the chapter and shows completely worked examples. The Review Exercises provide practice for the corresponding objectives.
Answers • Lesson 12–3 Solve inequalities involving multiplication and division. Solve 4x 8. 4x 8 4x 8 Divide each side by 4 and 4 4 reverse the symbol. x2 The solution is {xx 2}.
• Lesson 12–4 Solve inequalities involving more than one operation. Solve 2(x 1) 4 3x. 2(x 1) 4 3x 2x 2 4 3x Distributive Property 5x 2 4 Add 3x to each side. 5x 2 Subtract 2 from each side. 2 5
x Divide each side by 5. The solution is xx . 2 5
• Lesson 12–5 Solve compound inequalities and graph the solution. Solve 3 x 2 8. x23 and x28 x2232 x2282 x1 x6
Solve each inequality. Check your solution. 22. 3y 12 {yy 4}
98 7654321 0 1
23. 6n 30 {nn 5} 24. 0.2w 1.8 {ww 9} 26. {tt 28} x t 25. 3 {xx 15} 26. 14
5 3 27. w 6 4
35.
2 h 28. 7 2.3
{ww 8}
{hh 16.1}
36. 54 321 0 1 2 3 4 5
37. 54 321 0 1 2 3 4 5
38. 0 1 2 3 4 5 6 7 8 9 10
Solve each inequality. Check your solution. 29. 2x 6 14 30. 9 0.2y 1
{xx 4}
2 31. t 4 2 3
{yy 50}
32. 3(4 n) 21
{tt 9}
{nn 3}
Write and solve an inequality. 33. Four times a number decreased by 3 is greater than 25. 4x 3 25; {xx 7} 34. Seven minus two times a number is no less than nine. 7 2x 9; {xx 1} Solve each compound inequality. Graph the solution. 35–38. See margin for graphs. 35. 4 y 3 2 {y7 y 1} 36. 2 3 t or 3 t 5 {tt 1} 37. 3a 6 or 5 a 6 {aa 1 or a 2} 38. 9 2x 1 13 {x4 x 6}
The solution is {x1 x 6}.
Chapter 12 Study Guide and Assessment 543
D
V A N TA
TestCheck and Worksheet Builder
A
G
E
GL
S
COE' EN
This stateoftheart networkable CDROM has three integrated modules. The Worksheet Builder creates customized worksheets, tests, and quizzes of freeresponse, multiplechoice, shortanswer, and openended items. The Student Module gives you the option of having students take tests onscreen and get immediate feedback on their performance. Use the optional Management System to keep detailed student records. Chapter 12
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Chapter 12 Study Guide and Assessment Applications and Problem Solving
Objectives and Examples
Answers 39. 4 321 0 1 2 3 4 5 6
40. 54 321 0 1 2 3 4 5
41. 8 7654321 0 1 2
42.
See pages 715–717.
Review Exercises
• Lesson 12– 6 Solve inequalities involving absolute value and graph the solution.
This section provides additional practice in solving realworld problems that involve the concepts of this chapter.
Extra Practice
Solve each inequality. Graph the solution. 39. t 1 5 {t4 t 6}
Solve x 1 1.
40. a 7 2
Case 1 x 1 is positive. x11 x 0 Subtract 1 from each side.
42. s 4 3 {ss 7 or s 1}
41. x 3 1 {x4 x 2}
Case 2 x 1 is negative. (x 1) 1 (x 1)(1) 1(1) Multiply by 1 and reverse the symbol. x 1 1 x 2 Subtract 1 from each side. The solution is {x2 x 0}.
98 7654321 0 1
43. y 2 0 {yy 2} 39–43. See margin for graphs.
Write an inequality involving absolute value for each statement. Do not solve. 44. Bianca’s guess g was within $6 of the actual value of $25. g 25 6 45. The difference between Greg’s score s on his final exam and his mean grade of 80 is more than 5 points. s 80 5
43. 54 321 0 1 2 3 4 5
• Lesson 12–7 Graph inequalities in the coordinate plane.
Graph each inequality. 46. y x 1 47. y 2x 4
y
Graph 2x y 5.
y 2
48. x 2
First, solve for y. 2x y 5 2x y 2x 5 2x y 2x 5
49. x y 3
2x y 5
46–49. See Solutions Manual. x
O
Graph the related function y 2x 5 as a dashed line and shade.
Applications and Problem Solving
Assessment and Evaluation Masters, pp. 223–224 12
NAME
DATE
50. Shipping An empty book crate weighs 30 pounds. The weight of a book is 1.5 pounds. For shipping, the crate can weigh no more than 60 pounds. What is the acceptable number of books that can be packed in a crate? (Lesson 12–4)
PERIOD
Chapter 12 Test, Form 1B
Write the letter for the correct answer in the blank at the right of each problem. 1. Which of the following inequalities describes a number that is more than 8? A. x 8 B. x 8 C. x 8 D. x 8 2. Which is the graph of x 2? A. B. –4 –3 –2 –1 0
C.
–4 –3 –2 –1 0
D. –4 –3 –2 –1 0
3. Which inequality matches the graph at the right? A. x 3 B. x 3 C. x 3
–4 –3 –2 –1 0
–8 –7 –6 –5 –4 –3 –2 –1 0
1.
A
2.
B
1
D. x 3
3.
A
4. Which is the solution of n 6 2? A. {nn 4} B. {nn 4} C. {nn 8}
D. {nn 8}
4.
A
5. Which is the solution of x 3 5? A. {xx 8} B. {xx 8} C. {xx 2}
D. {xx 2}
5.
A
6. Which is the graph of the solution of 1 2 x? A. B. C.
D.
6.
A
–1 0 1 2 3
–1 0 1 2 3
–5 –4 –3 –2 –1
–1 0 1 2 3
51. Geometry An obtuse angle measures more than 90° but less than 180°. If the measure of an obtuse angle is 3x, what are the possible values for x? (Lesson 12–5)
{x30 x 60}
3x˚
less than or equal to 20 books
544 Chapter 12 Inequalities
For Questions 7–13, solve each inequality. 7. 3x 12 A. {xx 4}
B. {xx 4}
C. {xx 15}
D. {xx 36}
7.
B
B. {xx 32}
C. {xx 2}
D. {xx 2}
8.
A
B. {xx 4}
C. {xx 4}
D. {xx 64}
B. {ww 3}
C. {ww 27} D. {ww 27}
x 8. 8 4
A. {xx 32} 9. 4x 16 A. {xx 64} 10. 9 3w A. {ww 3}
9.
C
10.
A
11. 2p 4 16 A. {pp 6}
B. {pp 6}
C. {pp 10}
D. { pp 10}
11.
D
12. 12 16 4x A. {xx 1}
B. {xx 7}
C. {xx 7}
D. {xx 1}
12.
D
13. 6 3x 3 A. {xx 1}
B. {xx 1}
C. {xx 3}
D. {xx 3}
13.
A
D. {xx 6}
14.
D
14. Which is the solution of 6 12 3x? A. {xx 2} B. {xx 6} C. {xx 6}
15. Write the compound inequality y 2 and y 3 without using the word and. Glencoe/McGrawHill 81 D. y 6 Geometry: Concepts and B Applications A.© 3 y 2 B. 3 y 2 C. y 2 15. ©
Glencoe/McGrawHill
544 Chapter 12
223
Algebra: Concepts and Applications
Assessment and Evaluation Four forms of Chapter 12 Test are available in the Assessment and Evaluation Masters. Chapter 12 Test, Form 1B, is shown at the left. Chapter 12 Test, Form 2B, is shown on the next page.
Form of Test
1A 1B 2A 2B
Multiple Choice Multiple Choice Free Response Free Response
pp. 221–222 pp. 223–224 pp. 225–226 pp. 227–228
Level
Average Basic Average Basic
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CHAPTER
Test
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Chapter Test Answers
1. List at least three verbal phrases that are used to describe inequalities.
1. Sample answer: at least, no more than, at most 2. Reverse the symbol. 3. whether or not the expression in the absolute value is positive or negative 16.
2. If you multiply or divide each side of an inequality by a negative number, what must happen to the symbol for the inequality to remain true? 3. Before solving an inequality involving absolute value, which two cases must you consider? 1–3. See margin.
Write an inequality for each graph. 4.
4 321 0 1 2 3 4 5 6
5. 1
2
3
4
4 3 2 1
5
x3
0
17.
x 2
54 321 0 1 2 3 4 5
18.
Solve each inequality. Check your solution. 6. 2 x 12 {xx 10}
7. 5t 6 4t 3 {tt 9}
8. 8 4t {tt 2}
9. 0.2x 6 {xx 30}
321 0 1 2 3 4 5 6 7
19. 21 0 1 2 3 4 5 6 7 8
t 10. 1 {tt 4}
11. m 10 {mm 25}
12. 3r 1 16 {rr 5}
13. 7x 12 30 {xx 6}
14. 2(h 3) 6 {hh 6}
15. 8(1 2z) 25 z {zz 1)
4
2 5
20. 8 7654321 0 1 2
21. 54 321 0 1 2 3 4 5
22.
8 6 4 2
Solve each inequality. Graph the solution. 16–21. See margin for graphs. 16. x 1 2 and x 1 6
17. 4 3j 2 7 {j2 j 3}
18. 2n 5 15 or 2n 5 3
19. 6c 24 or c 0.25 1.3 {cc 4}
20. x 3 4 {xx 1 or x 7}
21. 4b 16 {b4 b 4}
16. {x3 x 5}
864
22–23. See margin.
22. y 5x 6
O
2 4 6 8x
4 6 8
18. {nn 5 or n 1}
Graph each inequality.
y
23. 4x 2y 6
24. Car Rental Justine is renting a car that costs $32 a day with free unlimited mileage. Since she is under the age of 25, it costs her an additional $10 per day. Justine does not want to pay any more than $200 on car rental costs. For what number of days can she rent a car? 4 or fewer days
Assessment and Evaluation Masters, pp. 227–228 DATE
PERIOD
Chapter 12 Test, Form 2B
Write an inequality to describe each number. Use x as the variable. 1. a number greater than 3
1.
x3
2. a number less than 5
2.
x 5
For Questions 3–5, graph each inequality on a number line.
Chapter 12 Test 545
www.algconcepts.com/chapter_test
NAME
12
25. Manufacturing Ball bearings are used to connect moving parts and minimize friction. Ball bearings for an automobile will work properly only if their diameter is within 0.01 inch, inclusive, of 5 inches. Write and solve an inequality to represent the range of acceptable diameters for these ball bearings. m 5 0.01; {m4.99 m 5.01}
3. x 2
3.
4. y 1
4.
5. d 2.5
5.
6. Write an inequality for the graph at the right.
–3 –2 –1 0
1
2
3
4
5
6
6.
Solve each inequality.
23. Write an inequality in slopeintercept form that describes the shaded 1 region. y x 2 2
y
y
x O
O
x
–4 –3 –2 –1 0 1 2 3 4 5
{x  x 4}
7. x 3 5
7.
{x  x 8}
8.
{ y  y 6}
9.
{ p  p 8}
10. 10 y 4
10.
{ y  y 14}
11. 3.1 p 1.4
11.
{ p  p 4.5}
1 2 12. y 3 3
12.
y  y 13
13. 6h 36
13.
{h h 6}
d 14. 4 2
14.
{d  d 8}
15. 2y 26
15.
{ y  y 13}
16. 8t 32
16.
{t  t 4}
x 17. 18 3
17.
{x  x 54}
18. v 12 4
18.
{v  v 48}
19. 3h 6 33
19.
{h h 9}
1
©
–4 –3 –2 –1 0 1 2 3 4 5
8. y 4 2 9. p 2 6
Answer
Chapter Test Bonus Question
–4 –3 –2 –1 0 1 2 3 4 5
Glencoe/McGrawHill 20. 2b 12 ©
13
Glencoe/McGrawHill
{b Applications b 12.5} Geometry: Concepts and 20.

81 227
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CHAPTER
Preparing for Standardized Tests
12
Pages 546–547 are part of a complete test preparation course that is described in detail on page T9 of the Teacher’s Handbook. The test items on these pages were written in the same style as those in state proficiency tests and standardized tests like ACT and SAT. These questions were aligned and verified by The Princeton Review, the nation’s leader in test preparation.
Preparing for Standardized Tests
Angle, Triangle, and Quadrilateral Problems You are already aware of some geometry concepts that you need to know for standardized tests. Be sure that you also know the meanings and definitions of the following terms. right equilateral reflection
obtuse isosceles translation
acute similar ( ) rotation
triangle congruent ( ) dilation
quadrilateral collinear
State Test Example
ACT Example
The triangles below are similar. Find the length of side KL. J
Diagnosis and Prescription Each of the 10 test questions on page 547 is crossreferenced to the chapter where that SAT or ACT skill is covered. If students miss a particular type of problem, you can have them study that skill. (See chart at the bottom of page 547.)
In the figure below, ON is congruent to L N , mLON 30, and mLMN 40. What is mNLM? O
A 6m
3m
12 m
B 6
N xm
K
C 8
L 40˚
D 9
Hint The measures of corresponding sides of similar polygons are proportional. Solution Find the corresponding sides of the two triangles. Side AB of ABC corresponds to side JK of JKL. Side BC of ABC corresponds to side KL of JKL. Using these two pairs of corresponding sides, write a proportion.
L
A 40
B 80
C 90
12
NAME
DATE
Substitute side measures.
1
2. Give the coordinates of the points graphed at the right. (Lesson 2–1)
Side KL measures 8 meters. The answer is C.
4x 1
1.
3. Simplify 34 (17). (Lesson 2–6)
3.
4. Solve k (6) 13. (Lesson 3–6)
4.
5. Simplify 9.5 5. (Lesson 4–3)
5.
4, 0, 1, 3 2 7 1.9
6.
9
6. Solve
Divide each side by 3.
Chapter 12 Cumulative Review
1. Simplify 12x 3 8x 4. (Lesson 1–4)
2 x 3
L
3x 24
PERIOD
–5 –4 –3 –2 –1 0
1
2
3
4
6. (Lesson 4–4)
2.
y y3 7. Solve . (Lesson 5–1) 6 24
7.
1
8. A single die labeled 1 through 6 is rolled. Find the odds of rolling a number greater than 4. (Lesson 5–6)
8.
1:2
9. Graph y 2x 2. (Lesson 6–3)
9.
The answer is B.
546 Chapter 12 Inequalities
y
x
O
24
10.
11. For what value of k are the graphs of y kx 5 and 1 y x 1 perpendicular? (Lesson 7–7) 2
2
11.
12. A triangle has sides of 7, 8, and 12 inches. Determine whether the triangle is a right triangle. (Lesson 8–7)
15. Factor 2x2 7x 5. (Lesson 10–4)
15.
no 2x 15 4x2(2x y) (2x 5)(x 1)
16. Use the Quadratic Formula to solve x2 x 6 0. (Lesson 11–6)
16.
2, 3
17. Graph y 2x 1. (Lesson 11–7)
17.
12.
13. Simplify (x 3)(x 5). (Lesson 9–4)
13.
14. Factor 8x3 4x2y. (Lesson 10–2)
14.
y
x
O
18. The minimum telephone service costs the Smiths $24.95 per month. Express the monthly cost as an inequality using the variable x. (Lesson 12–1)
18.
19. Solve x 8 2. (Lesson 12–6)
19.
20. Graph y x 2. (Lesson 12–7)
20.
x 24.95 10 x 6 y O
x
y = –x – 2
©
Glencoe/McGrawHill
546 Chapter 12
232
Resource Manager
x2
Algebra: Concepts and Applications
Reproducible Masters • Assessment and Evaluation, pp. 232–234
2
40˚
M
LN Since O N , ONL is isosceles and the base angles are equal. So, m1 30. Since the sum of the angle measures in any triangle is 180, find the sum of the angle measures in OML. 180 30 40 (30 m2) 180 100 m2 80 m2 Subtract 100 from each side.
y = –2x + 2
10. Suppose y varies inversely as x and y 12 when x 4. Find y when x 2. (Lesson 6–6)
30˚
N
3x 4(6) Cross multiply. x8
D 120
Solution Label the unknown angles as 1 and 2. Find m2. O
Assessment and Evaluation Masters, p. 232
M
Hint Examine all of the triangles in the figure.
AB BC JK KL 3 4 6 x
30˚
6m
B 4m C
A 5
If there is no figure or diagram, draw one yourself.
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Chapter 12 Preparing for Standardized Tests After you work each problem, record your answer on the answer sheet provided or on a sheet of paper.
(5)(4)6
6. Simplify the expression . C 3 A 120 B 40 C 40
D 120
A bubblein answer sheet for these practice problems is available on page v of the Assessment and Evaluation Masters.
Multiple Choice 1. The rectangles are similar. Find the value of x. A A 3 B 7 C 14 D 15 2. ABC is isosceles. What is mC? B A 45 B 50 C 80 D 100
21 in. 7 in.
x in.
1 in.
B
7. The square of a number is 255 greater than twice the number. What is the number? B A 15 or 17 B 17 or 15 C 31 or 33
D 33 or 31
8. What is the value of b + c? A
80˚
A
110˚ a ˚ c˚
C
111˚ b˚ d ˚
3. What is the sum of a, b, and c? C a˚
A 179
B 180
C 181
D 221
b˚ c˚
A 180 B 240 C 270 D 360 E It cannot be determined from the information given. 4. The formula for the area of a trapezoid is
Additional Practice Additional test practice questions are available in the Assessment and Evaluation Masters, pp. 233–234.
Answers 10A. Sample answer: i) Not possible; the angles of an equilateral triangle all measure 60. ii)
10B. Sample answer: i)
Grid In 9. In the figure, what is the sum of a, b, and c? 270
ii) a˚ b˚ c˚
1 2
A h(b1 b2), where b1 and b2 are the lengths of the bases and h is the height. If the area of trapezoid QRST is 72 square centimeters, then what is the height? D R
6 cm
10. Draw a triangle that fits each description. If a drawing is not possible, explain why.
S
h
Q
A 1 cm
12 cm
B 2 cm
Extended Response
Part A See margin. i. an obtuse equilateral triangle
T
C 4 cm
D 8 cm
2
5. Find the xintercept of y x 4. A 3 A 6 B 6 C 4 D 0
www.algconcepts.com/standardized_test_practice
Assessment and Evaluation Masters, pp. 233–234
ii. an acute equilateral triangle
12
NAME
DATE
(Chapters 1–12)
Part B See margin. i. an obtuse isosceles triangle
Write the letter for the correct answer in the blank at the right of each problem.
ii. an acute isosceles triangle
Chapter 12 Preparing for Standardized Tests 547
1. Tell which sentence represents the equation n 8 12. A. Eight more than a number is twelve. B. Twelve more than a number is eight. C. Eight is twelve less than a number. D. The difference of a number and eight is twelve.
1.
2. Suppose you wanted to buy a music CD and DVD movie. The movie costs $10 more than the CD. The CD costs $15. Which of the following statements is true? A. The movie costs $25. B. The movie costs $10. C. Together, they cost $25. D. None are true.
2.
A
3. Evaluate 5 3. A. 8 B. 2
3.
C
C. 2
D. 8
4. In a football game, the team first lost 18 yards, then gained 16 yards. Which integer represents the net gain in yards? A. 34 B. 2 C. 2 D. 34
Chapter 12 Angle, Triangle, and Quadrilateral Problems Ex. 1 Ex. 2 1 2 3 4 5 6 7 8 9 10
similar triangles isosceles triangle similar rectangles isosceles triangle straight angles trapezoid area formula xintercept absolute value quadratic word problem angles angles triangles
PERIOD
Chapter 12 Standardized Test Practice
SPT SPT SAT SPT SPT SPT SPT SAT SAT SPT
SPT ACT Ch. 12 Ch. 12 Ch. 12 Ch. 1 Ch. 7 Ch. 3 Ch. 11 Ch. 12 Ch. 12 Ch. 12
A
4.
B
5. For what value in the replacement set {1, 0, 1, 2} is the equation x 4 3x true? A. 1 B. 0 C. 1 D. 2
5.
D
6. For what value of x is the equation 24 x (6) true? A. 30 B. 18 C. 18 D. 30
6.
A
7. A sandwich can be made of turkey, tuna, or ham and be put on white or wheat bread. Find the number of possible sandwiches. A. 2 B. 3 C. 5 D. 6
7.
D
8. For what value of x is 23 3x 4 true? A. 27 B. 19 C. 9
8.
C
9.
B
1 D. 2
10.
A
D. (3, 4)
11.
A
D. 9
9. A coat was on sale for $75. It originally cost $90. Which of the following represents the percent of change? A. 15% B. 16.7% C. 20% D. 120% 10. Ten cards in a deck are numbered from 1 to 10. A card is drawn and then replaced. Another card is drawn. Find the probability of drawing a 4 and then a 3. 1 A. 100
1 B. 10
1 C. 5
11. Which ordered pair is not a solution to 2x 3y 6? A. (4, 0) B. (0, 2) C. (3, 0)
12. The number of gallons of gas needed to drive on a trip varies directly as the number of miles traveled. If 20 gallons of gas are used to drive 250 miles, how many gallons are used to drive 400 miles? B © Glencoe/McGrawHill 8140 gal A. 30 gal B. 32 gal C. D.Geometry: 64 gal Concepts and Applications 12. ©
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233
Algebra: Concepts and Applications
Chapter 12
547