Download Inequalities

Inequalities
Objectives To find and represent all the values that make
an
a inequality in one variable true; and to represent real-world
situations with inequalities.
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eToolkit
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Practice
EM Facts
Workshop
Game™
Teaching the Lesson
Key Concepts and Skills
• Translate between word and
number sentences. [Patterns, Functions, and Algebra Goal 1]
• Identify relation symbols in
number sentences. [Patterns, Functions, and Algebra Goal 2]
• Determine whether inequalities are true or
false. [Patterns, Functions, and Algebra Goal 2]
• Recognize that most inequalities have
an infinite number of solutions. [Patterns, Functions, and Algebra Goal 2]
Key Activities
Students extend their work with equations
to finding solution sets of inequalities. They
practice solving inequalities by playing Solution
Search. Students write and graph inequalities
that represent real-world situations.
Family
Letters
Assessment
Management
Common
Core State
Standards
Ongoing Learning & Practice
Dividing Decimals by Decimals:
Part 1
Math Journal 2, pp. 242 and 243
Students solve division problems with
decimal divisors by renaming the
dividend and divisor.
Math Boxes 6 12
Math Journal 2, p. 245
straightedge
Students practice and maintain skills
through Math Box problems.
Study Link 6 12
Curriculum
Focal Points
Interactive
Teacher’s
Lesson Guide
Differentiation Options
READINESS
Reviewing Relation Symbols
and Inequalities
Math Masters, p. 211
Students review relation symbols and
properties of number sentences.
ENRICHMENT
Graphing Compound Inequalities
Math Masters, p. 212
Students graph compound inequalities
and write inequalities to describe graphs.
Math Masters, p. 210
Students practice and maintain skills
through Study Link activities.
ENRICHMENT
Playing Solution Search
with Student-Created Cards
index cards
Students play Solution Search with
self-created cards.
Ongoing Assessment:
Informing Instruction See page 599.
Ongoing Assessment:
Recognizing Student Achievement
Use journal page 244. [Patterns, Functions, and Algebra Goal 2]
Key Vocabulary
inequality solution set
Materials
Math Journal 2, pp. 244–244B
Student Reference Book, pp. 244 and 332
Study Link 6 11
per group: Math Masters, p. 473, complete
deck of number cards (the Everything Math
Deck, if available) scissors
Advance Preparation
Plan to spend 2 days on this lesson. For Part 1, make a copy of Math Masters, page 473 for every three or four students.
Have students cut apart the game cards.
Teacher’s Reference Manual, Grades 4–6 pp. 77–79
596
Unit 6
Number Systems and Algebra Concepts
Mathematical Practices
SMP1, SMP2, SMP4, SMP6
Getting Started
Content Standards
6.NS.3, 6.EE.5, 6.EE.6, 6.EE.8
Mental Math and Reflexes
Math Message
Tell whether each inequality is true or false.
Students estimate products and quotients by showing
thumbs-up for an answer greater than 1 and thumbsdown for an answer less than 1.
1. 5 ∗ 4 ≠ 20 false
2. 5 ∗ 4 ≤ 20 true
54
3. _
9 > 7 false
4. 17 - 6 ≥ 9 true
Work with a partner to write an inequality that is neither true
nor false.
Suggestions:
1 ∗ 0.99 thumbs-down
1
1÷_
2 thumbs-up
9
1
_
_
∗
thumbs-down
8
Sample answer: x + 3 < 6
4
3
1
_
_
2 ÷ 8 thumbs-up
0.3 ÷ 4 thumbs-down
0.5 ÷ 0.25 thumbs-up
Study Link 6 11 Follow-Up
Go over the answers with the class. Ask volunteers
to share solution strategies.
If time permits, compare the dividend and quotient when the
divisor is less than 1 and when it is greater than 1.
1 Teaching the Lesson
▶ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
Algebraic Thinking Problems 1–4 of the Math Message review
the relation symbols discussed in Lesson 6-7. Briefly go over the
answers. Then ask students to share the inequalities they wrote.
Some may think the task is impossible. Others may extend
the concept of open sentences to include inequalities and use
a variable.
For example, the inequality x + 3 < 6 is neither true nor false
until a value is assigned to x. If x = 2, then the inequality is true;
if x = 3, then the inequality is false.
▶ Introducing Solution Sets
to Inequalities
WHOLE-CLASS
DISCUSSION
ELL
(Student Reference Book, p. 244)
Algebraic Thinking During the discussion, write the key ideas
along with examples on the board to support English language
learners. While most equations students have solved so far have
had only one solution, many inequalities have an infinite number
of solutions. To demonstrate this point, write x < 2 on the board.
Have volunteers help you generate a list of possible solutions.
Include rational and
_ irrational numbers in this list (for example,
3 , -π, -√2 , 0.3
⎯).
1.835, -1_
5
When students recognize that listing all solutions of x < 2 is not
viable, discuss the following options for representing all the values
that make an inequality true:
Lesson 6 12
597
Describe the solution set of an inequality in words. For
example: The solution set of x < 2 consists of all real numbers
less than 2.
Graph the solution set of an inequality on a number line.
Include the following steps in your discussion about graphing the
solution set of x < 2.
Step 1: On a number line, draw an open circle at 2. An open circle
shows that 2 is not a solution but numbers close to 2 are.
–4 –3 –2 –1
0
1
2
3
Step 2: Plot three solutions of x < 2 on the number line.
–4 –3 –2 –1
0
1
2
3
Step 3: Start at the open circle and draw a line that is thick
enough to cover the three plotted solutions. Draw an
arrowhead to show that there are an infinite number
of solutions. Then use one additional solution to check
your graph.
–4 –3 –2 –1
0
1
2
3
7 is included in the graph and -4_
7 is less
For example, -4_
8
8
than 2. So, the graph is correct.
If an inequality contains the relation symbol <, >, or ≠, the
endpoint of the graph is shown with an open circle, indicating that
the circled point is not part of the solution set. If an inequality
contains the relation symbol ≥ or ≤, the endpoint of the graph is
shown with a closed, or filled-in circle, indicating that the circled
point is part of the solution set.
Student Page
Date
Have students work in pairs to name three solutions of an
inequality and then graph that inequality on a number line.
Suggestions:
Time
LESSON
Inequalities
6 12
1. Name two solutions of each inequality.
10, 12
56, 60
2, 1
0, 1
a. 15 > r
c. t ≥ 56
21 > y
e. _
7
g. 6.5 > 3 ∗ d
b.
d.
f.
h.
Sample answers:
9, 12
8<m
4, 5
15 - 11 ≤ p
-1, 2
w > -3
0.4, 0.1
g < 0.5
c. y ≤ 2.6 + 4.3
1, 0.5
10, 100
y ≥ -3 Sample solutions: -3, -1, 3
Sample answers
0, -2
3, 2
2. Name two numbers that are not solutions of each inequality.
a. (7 + 3) ∗ q > 40
244
b.
1 + _
1 < t
_
2
4
d.
6 g > 12
3. Describe the solution set of each inequality.
–4 –3 –2 –1
0
1
2
3
4
5
6
7
8
9 10
m - 2 > 0 Sample solutions: 2.5, 3, 5
Example: t + 5 < 8
Solution set: All numbers less than 3
a. 8 - y > 3
Solution set
b. 4b ≥ 8
Solution set
All numbers less than 5
All numbers greater than or equal to 2
–2 –1 0
a. x < 5
1
2
3
4
5
6
7
8
9 10
1
2
3
4
5
6
7
8
9 10
1
2
3
4
5
6
7
8
9 10
0
b. 6 > b
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0
Math Journal 2, p. 244
205_246_EMCS_S_G6_MJ2_U06_576442.indd 244
598
Unit 6
3
1
2
3
4
5
6
Before assigning independent journal practice, work with students
to solve the Check Your Understanding problems at the bottom of
page 244 of the Student Reference Book.
_1
c. 1 2 ≥ h
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0
2
n ≠ 5 Sample solutions: -2, 4.9, 5.1
4. Graph the solution set of each inequality.
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
1
8/29/11 12:34 PM
Number Systems and Algebra Concepts
Ongoing Assessment: Informing Instruction
Some students may recognize that for inequalities such as x < 2 and y ≥ -3,
the direction of the relation symbol and the arrowhead on the graph are the
same. Encourage students to use more reliable methods for deciding which part
of the number line they should include in their graphs.
▶ Solving Inequalities
PARTNER
ACTIVITY
(Math Journal 2, p. 244)
PROBLEM
PRO
P
RO
R
OB
BLE
BL
LE
L
LEM
EM
SO
S
SOLVING
OL
O
LV
LV
VIIN
VIN
NG
G
Algebraic Thinking Circulate and assist as students work in pairs
to complete the problems on the journal page. Remind students to
use a solution to check each graph.
Ongoing Assessment:
Recognizing Student Achievement
Journal
Page 244
Problems 1 and 2
Use journal page 244, Problems 1 and 2 to assess students’ ability to
determine whether inequalities are true or false. Students are making adequate
progress if they are able to complete Problems 1 and 2. Some students may be
able to describe the solution sets in Problem 3.
[Patterns, Functions, and Algebra Goal 2]
Links to the Future
Graphing solution sets of inequalities with one variable is not a Grade 6 Goal.
The graphing activities in this lesson provide students with the skills they will
need in future algebra courses.
▶ Playing Solution Search
SMALL-GROUP
ACTIVITY
(Student Reference Book, p. 332; Math Masters, p. 473)
Algebraic Thinking Go over the directions for Solution Search.
You might want to play a few rounds of the game with three
volunteers while the rest of the class observes. Use this
opportunity to answer any questions regarding rules of play.
Then students play Solution Search in groups of three or four.
▶ Using Inequalities to Describe
WHOLE-CLASS
DISCUSSION
Real-World Situations
Present this situation to students: An amusement park’s rules state
that a guest must be at least 4 feet tall to ride a roller coaster. Ask
students to generate a list of possible heights of guests who could
1 inches;
ride the roller coaster. Sample answers: 4 feet; 5 feet 2 _
2
6 feet 1.75 inches After students realize that they cannot list all
the possible heights, ask them to describe all the possible heights
in words. All heights greater than or equal to 4 feet
Lesson 6 12
599
Student Page
Date
Time
LESSON
Next, ask students to use an inequality to represent all the possible
heights and have a volunteer graph the inequality. h ≥ 4 feet The
graph should look like the number line below.
Inequalities in Real-World Contexts
6 12
Each situation below can be described by an inequality. Follow the steps to identify an
appropriate inequality and draw a graph that fits the situation.
A city ordinance says that fences can have a maximum height of 6 feet.
1.
Sample answers: 3 feet; 5 feet 9 inches;
4 and a half feet
a.
List several allowable fence heights.
b.
Describe in words the set of all allowable fence heights.
Sample answer: All positive heights that are 6 feet or less
c.
–4
–3
–2
–1
0
1
2
3
4
5
6
Use an inequality or two inequalities to describe the set of allowable heights.
Sample answer: h ≤ 6 and h > 0
d.
–2
e.
Ask: Do all the solutions to this inequality make sense in this
situation? No. For example, a person cannot be 20 feet tall.
Explain to students that it is important to always keep the context
in mind when using inequalities to represent real-world situations.
In this example, it is not possible to say exactly what the maximum
height of a person is, so the best we can do is leave the graph as it
is. However, it is important to acknowledge that the inequality is
not a perfect model. Tell students that in other situations, it is
sometimes possible to draw a graph that better represents the
real-world situation.
Graph the set of all allowable heights on the number line below.
–1
0
1
2
3
4
5
6
7
8
Do the values represented on the graph make sense in the situation? Explain your answer.
Sample answer: Yes. A height could be any positive number,
but negative heights or a height of zero do not make sense. It is
unlikely anyone would have a fence that is 0.1 feet high, but since
we don’t know the minimum, it makes sense to end the graph at 0.
A CD player can hold up to 8 CDs.
2.
Sample answers: 8 CDs,
5 CDs, 0 CDs
a.
List several possible numbers of CDs in the player.
b.
Describe in words the set of possible numbers of CDs in the player.
c.
Sample answer: Any whole number between and including 0 and 8
CDs
Use an inequality or two inequalities to describe the possible numbers of CDs.
Sample answer: n ≥ 0 and n ≤ 8
d.
Graph the set of all possible numbers of CDs on the number line below.
e.
Do the values represented on the graph make sense in the situation? Explain your answer.
0
1
2
3
4
5
6
7
8
9
10
Sample answer: Yes. You can’t have a negative number of CDs or
part of a CD, so I only showed dots on 0 and positive whole numbers.
Outline the process that students just followed to describe a
real-world situation with an inequality.
Math Journal 2, p. 244A
205_246_EMCS_S_G6_MJ2_U06_576442.indd 244A
3/24/11 8:27 AM
List several possible values that fit the situation.
Describe the set of all possible values in words.
Represent all the possible values with an inequality
or inequalities.
Graph the possible values on a number line.
Check by asking whether the solutions on the graph make sense
in the real-world situation.
Next, present the following situation: A restaurant allows children
under the age of 10 to eat from the kids’ menu. Guide students
through the process above.
Ask students to list several possible ages of children who can
eat from the kids’ menu. Sample answers: 3 years; 4 and a half
years; 5 years and 9 months; 2 years, 6 months, and 8 days
Student Page
Date
Time
LESSON
6 12
3.
Inequalities in Real-World Contexts
continued
Next, ask students to describe all the possible ages in words.
Sample answer: All ages less than 10 years old
The manager of an apartment building turns on the heat when the outside
temperature drops below 50°F.
a.
List several possible outside temperatures when the heat is on.
Sample answers: 45°F; 15°F; –10°F
b.
Ask students how they might describe the solution set using an
inequality. Encourage students to keep the context in mind. Ask:
Can a child have an age that is not a whole number of years?
Yes. A child can be part of a year old. Can a child have a
negative age? No. Age is positive. If students suggest a < 10,
ask them whether this accurately represents the fact that age
cannot be negative. If nobody mentions it, suggest that students
might use two inequalities to represent the possible ages: a < 10
and a > 0.
Describe in words the set of outside temperatures when the heat is on.
Sample answer: All temperatures less than 50°F
c.
Use an inequality or two inequalities to describe the possible outside
temperatures when the heat is on.
t < 50
d.
Graph the set of all possible temperatures on the number line below.
e.
Do the values represented on the graph make sense in the situation? Explain your answer.
– 30
– 20
–10
0
10
20
30
40
50
60
70
Sample answer: Not all of the values make sense. You can have negative
temperatures and parts of degrees, but the temperature will never be
4. a.
–1,000°F. Since I don’t know the minimum, this is the best graph I can draw.
Describe your own real-world situation that can be modeled by an inequality.
Answers vary.
b.
List several possible values for your situation.
Answers vary.
c.
Describe the set of possible values in words and with an inequality or two.
NOTE The restaurant situation could also be represented by the compound
Answers vary.
d.
e.
Graph the set of all the possible values on the number line below.
inequality 0 < a < 10. Acknowledge students who point this out, but do not insist
that they write this inequality. Keep the focus of this activity on writing simple
inequalities and drawing graphs that are appropriate for the situation. See the
Enrichment activity in Part 3 for work with compound inequalities.
Answers vary.
Do the values represented on the graph make sense in the situation? Explain your answer.
Answers vary.
Math Journal 2, p. 244B
205_246_EMCS_S_G6_MJ2_U06_576442.indd 244B
599A Unit 6
3/4/11 2:02 PM
Number Systems and Algebra Concepts
Student Page
Ask a volunteer to graph the inequalities a < 10 and a > 0 together
on one number line. The graph should show only the numbers that
satisfy both inequalities, as shown on the number line below.
Date
Time
LESSON
6 12
䉬
Dividing Decimals by Decimals: Part 1
When you multiply the numerator and denominator of a fraction by the same
nonzero number, you rename the fraction without changing its value.
3
5
10
10
30
50
3
5
42–44
30
50
For example: ⴱ , so 3
5
In general, you can think of a fraction as a division problem. The fraction equals
3 5, or 53
. The numerator of the fraction is the dividend; the denominator is the
divisor. As you do with a fraction, you can multiply the dividend and divisor by the
same number without changing the value of the quotient.
–1
0
1
2
3
4
5
6
7
8
9
10
11
Study the patterns in the table below.
Fraction
6.08
0.08
6.08
0.08
6.08
0.08
Ask students whether the ages represented on the graph make
sense in the real-world context. Sample answer: Yes. The graph
shows all ages between 0 and 10 years.
Present one more situation to students: You may have no more
than 6 books checked out of the library at one time. Again, guide
students through the process above to help them describe the
possible values in this situation.
List several possible numbers of books. Sample answers:
6, 3, 0, 2
10
10
100
100
ⴱ ⴱ
Division Problem
Quotient
0.086
.0
8
76
0.86
0
.8
76
86
0
8
76
What do you notice about the quotient when you multiply the dividend and the
divisor by the same number?
The quotient stays the same.
Rename each division problem so the divisor is a whole number. Then solve the
equivalent problem using partial-quotients division or another method.
Example:
Equivalent Problem
0.0050
.0
1
5
Quotient Equivalent Problem
Equivalent Problem
Sample answer:
42
,0
5
0
1. 0.0042
.0
5
Describe the possible numbers of books in words. All whole
numbers between and including 0 and 6. You cannot check out
a part of a book, nor can you check out a negative number of
books.
Quotient
3
Sample answer:
37
0
.8
2. 0.37
.0
8
512.5
23.6
Quotient Math Journal 2, p. 242
Describe the solution set with an inequality or inequalities.
b ≤ 6; b ≥ 0
Graph the inequalities on a number line. Ask students how
they would indicate on the graph that only whole numbers are
included. If no one suggests it, show students how to draw dots
at each possible value, as shown below.
–2
–1
0
1
2
3
4
5
6
7
8
Check that the values represented on the graph make sense in
the real-world context. Sample answer: Yes. It shows 0, 1, 2, 3,
4, 5, or 6 books could be checked out from the library.
Student Page
Date
Time
LESSON
6 12
䉬
Dividing Decimals by Decimals
continued
Rename each division problem so the divisor is a whole number. Then solve the
equivalent problem using partial-quotients division or another method.
▶ Graphing Inequalities in
Equivalent Problem
PARTNER
ACTIVITY
3.
0.142
9
4
Sample answer:
9
,4
0
0
142
42–44
Equivalent Problem
4.
0.0136
.2
4
Sample answer:
,2
4
0
136
Real-World Contexts
(Math Journal 2, pp. 244A and 244B)
Circulate and assist as students work in pairs to complete the
journal pages. Encourage partnerships to discuss how the graph
will look before starting to draw each graph.
Quotient 2,100
Quotient Equivalent Problem
5.
0.463
3
.5
8
Quotient Sample answer:
,3
5
8
463
73
480
Equivalent Problem
6.
1.671
3
.3
6
Quotient Sample answer:
,3
3
6
1671
8
Math Journal 2, p. 243
Lesson 6 12 599B
Student Page
Date
Time
LESSON
6 12
Find the solution to each equation.
1.
2 Ongoing Learning & Practice
Math Boxes
䉬
a.
y 6 28 40
y
b.
45 3n 45
n
c.
7p 19 2p 55 p d.
7 4w 10w
2.
Solve the pan-balance problem.
62
30
4
6
One
1
2
w
2
18
▶ Dividing Decimals by
weighs as much
212
as
marbles.
Decimals: Part 1
243
250
(Math Journal 2, pp. 242 and 243)
Rebecca’s Walks
40
Students rename decimal-number divisors as whole-number
divisors while maintaining the value of the quotient. They solve
equivalent problems using partial-quotients division or method
of their choice. Students complete Part 2 in Lesson 8-2.
30
Distance (mi)
Complete the table. Then graph the
data and connect the points.
Time (hr) Distance (mi)
1
(h)
(32 º h)
Rebecca walks at an average
1
1
312
speed of 3 miles per hour.
2
2
7
Rule:
1712
5
Distance 7
24.5
1
3 mph ⴱ number of hours
2
35
10
3.
INDEPENDENT
ACTIVITY
20
10
0
0 1 2 3 4 5 6 7 8 9 10
Time (hr)
The spinner at the right is
divided into 5 equal parts.
4.
Suppose you spin
this spinner 75 times.
About how many times would
you expect the spinner
to land on a prime number?
45
5.
2
1
5
3
4
Serena keeps 4 stuffed animals lined up
on a shelf over her bed. How many
different arrangements of the stuffed
animals are possible? (Hint: Label the
animals A, B, C, and D. Then make a list.)
24
▶ Math Boxes 6 12
INDEPENDENT
ACTIVITY
(Math Journal 2, p. 245)
arrangements
times
88
149 150
156
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 6-10. The skills in
Problems 4 and 5 preview Unit 7 content.
Math Journal 2, p. 245
▶ Study Link 6 12
INDEPENDENT
ACTIVITY
(Math Masters, p. 210)
Home Connection Students review some of the major
concepts of the unit. They name solutions and then graph
solution sets of inequalities.
3 Differentiation Options
Study Link Master
Name
Date
STUDY LINK
Review
6 12
䉬
1.
Time
READINESS
Write a number sentence for each word sentence.
242–244
251–252
Word sentence
2.
a.
15 is not equal to 3 times 7.
b.
5 more than a number is 75.
c.
13 more than 9 divided by 9 is
less than or equal to 14.
(
c.
(Math Masters, p. 211)
b.
(
)
16 2 5 3 12
2
46
40 8 24 º 2 0
5º68º2
8
2
To provide experience identifying relation symbols and properties
of number sentences, have students work with a partner to
complete Math Masters, page 211. This activity is preparation
for finding and graphing solution sets of inequalities.
18
b.
20 2 d.
42 (4 º 2) 3 º 2 8
Solve each equation.
a.
(4y 5)
b. 2
3x 5 5x 3
Solution
5.
)
200 4 º 5 10
x 1
y9
Solution
y 6.5
Name three solutions of the inequality. Then graph the solution set.
1
f 3, f 22, f 2
3
f 2
a.
3
2
Practice
6.
$2.52 12 Estimate
7.
455
7.6
0
8.
1209
3
,7
2
0
Estimate
Estimate
$0.25
1
800
1
0
1
2
Sample estimates given.
Quotient $0.21
1.28
Quotient
781
Quotient
Math Masters, p. 210
600
Unit 6
5–15 Min
and Inequalities
13 14
9
9
Use the order of operations to evaluate each expression.
a.
4.
▶ Reviewing Relation Symbols
Insert parentheses to make each equation true.
a.
3.
Number sentence
15 ≠ 3 º 7
x 5 75
PARTNER
ACTIVITY
Number Systems and Algebra Concepts
Teaching Master
INDEPENDENT
ACTIVITY
ENRICHMENT
▶ Graphing Compound
Name
Date
LESSON
6 12
䉬
15–30 Min
1.
Time
Reviewing Relation Symbols and Inequalities
Translate between word and number sentences.
Word sentence
Inequalities
7
a. 9
(Math Masters, p. 212)
b.
To further explore compound inequalities, students graph
the solution sets of compound inequalities and write
compound inequalities to describe graphs. For Problem 4,
remind students that all_square roots have a positive and a
negative value; that is, √9 = 3 or -3, because 3 ∗ 3 = 9 and
-3 ∗ -3 = 9. Students can use the word and when writing the
inequalities for Problems 5–7. For example, the graph in Problem 5
represents the solution set x ≥ -8 and x ≤ 5.
3.
7
9
19 is not equal to 54 divided by 3.
2
3
19 ≠ 54 3
c.
20 is greater than or equal to
5 less than 5 squared.
20 52 5
d.
The product of 4 and 19
is less than 80.
4 ⴱ 19 80
e.
62 plus a number y is
greater than 28.
f.
2.
Number sentence
2
3
is greater than .
62 y 28
2 is less than or equal to the
quotient of a number x and 17.
x
2 1
7
Indicate whether each inequality is true or false.
a.
5 º 4 20
c.
54 / 9 7
e.
29 12 false
false
true
51
3
false
true
true
18 2 º 7 6
b.
(7 3) º 6 60
d.
45 9 º 5
f.
Are the inequalities 17 6 9 and 9 17 6 equivalent? Explain.
Sample answer: Yes. Both the symbols and the sides of
the inequality have been reversed, so the inequalities are
equivalent.
SMALL-GROUP
ACTIVITY
ENRICHMENT
▶ Playing Solution Search
15–30 Min
with Student-Created Cards
Math Masters, p. 211
To extend Solution Search, students create sets of Solution Search
cards and play the game using these cards.
Teaching Master
Name
Date
LESSON
6 12
Time
Graphing Compound Inequalities
Graph all solutions of each inequality.
1.
-1 ≤ x < 8
(Hint: -1 ≤ x < 8 means x ≥ -1 and x < 8. For a number to be a solution, it
must make both number sentences true.)
10 9 8 7 6 5 4 3 2 1 0
2.
3<y+2≤7
3.
m ≠ -2
4.
x2 ≥ 9
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
9
10
Write an inequality for each graph.
5.
x ≥ - 8 and x ≤ 5 OR -8 ≤ x ≤ 5
6.
x ≥ - 7 and x < 1 OR - 7 ≤ x < 1
7.
x < 1 and x > 1 OR x ≠ 1
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
10 9 8 7 6 5 4 3 2 1 0
1
1
1
2
2
2
3
3
3
4
4
4
5
6
7
8
5
6
7
8
9
10
5
6
7
8
9
10
Math Masters, p. 212
180-216_EMCS_B_G6_MM_U06_576981.indd 212
8/16/11 3:52 PM
Lesson 6 12
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