2-6
Solving Compound Inequalities
Extension: Solving Special Compound
Inequalities
Essential question: How can you solve special compound inequalities?
Standards for
Mathematical Content
Extra Example
Solve. Write the solution in set notation. Graph
the solution.
A-REI.2.3 Solve linear inequalities…in one
variable…
4x + 11 ≥ 7 AND 3x + 1 > 7
{x | x ≥ -1} ∩ {x | x > 2}, or {x | x > 2}
Prerequisites
Solve Two-Step and Multi-Step Inequalities
Solve Inequalities with Variables on Both Sides
Math Background
-4 -3 -2 -1
The symbols ∩ and ∪ are more formally known
as intersection and union. A ∩ B denotes the
intersection of sets A and B, or the members that
belong to both set A and to set B. A ∪ B denotes the
union of sets A and B, or the members that belong
to set A or set B. A mnemonic for remembering the
difference is that ∪ looks like the “U” in union, and
includes everything joined together.
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CLOSE
Essential Question
How can you solve special compound inequalities?
Solve each part of the inequality independent of
the other part. Graph each solution above the
number line. Determine what part of the solutions
should be used as the compound solution based
on whether the inequality uses AND or OR. Graph
that solution on the number line and write it as the
solution set.
INTR O D U C E
Summarize
Have students make a table of the different results
of compound inequalities and the resulting
graph(s) for each. Descriptions should also include
whether the inequality includes AND or OR.
TEAC H
1
Highlighting
the Standards
example
Questioning Strategies
• What would be the solution to part D if the
inequality used AND? {x | x ≥ 3}
The exercises provide opportunities to
address Mathematical Practices Standard 2
(Reason abstractly and quantitatively) and
Standard 6 (Attend to precision). • If you have a final graph of two rays pointing in
the same direction, how do you tell which ray
shows the solution? If the compound inequality
uses AND, it will be the shorter arrow; if the
inequality uses OR, it will be the longer arrow.
PRACTICE
Where skills are
taught
1 EXAMPLE
Chapter 2 99
Where skills are
practiced
EXS. 1–4
Lesson 6
© Houghton Mifflin Harcourt Publishing Company
Draw a simple Venn diagram on the board of two
overlapping circles. Discuss with students how
all members of the two sets can be designated
using the ∪ symbol, and how only the overlapping
members of the two sets can be designated using
the ∩ symbol.
Name
Class
Notes
2-6
Date
Solving Compound Inequalities
Extension: Solving Special Compound Inequalities
Essential question: How can you solve special compound inequalities?
Compound inequalities are two inequalities joined by AND () or
OR (
).
To solve a compound inequality:
1. Solve each inequality independently.
2. Graph the solutions above the same number line.
3. Decide which parts of the graphs represent the solution. If AND
is used, it’s the common points. If OR is used, it’s all points. Then
graph the solution on the number line.
A-REI.2.3
1
EXAMPLE
Solving Compound Inequalities
Solve. Write the solution in set notation. Graph the solution.
A
2x < 8 AND 3x + 2 > -4
x<4
3x > -6
x > -2
Solution set: x ⎢ x <
© Houghton Mifflin Harcourt Publishing Company
B
-4 -3 -2 -1
3x + 2 ≥ -1 OR 4 - x ≥ 2
3x ≥ -3
-x ≥ -2
x ≥ -1
x≤2
-4 -3 -2 -1
Solution set: x ⎢ x ≥ -1 x ⎢ x ≤
C
0
-4 -3 -2 -1
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numbers
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3 x ⎢ x ≤ -3 or the empty set or ∅
3x - 1 > 2 OR 2x + 2 ≥ 8
3x > 3
2x ≥ 6
x >1
x≥3
Solution set: x ⎢ x >
1
4 real
2 or the set of all
2x - 3 > 3 AND x + 4 ≤ 1
2x > 6
x ≤ -3
x>3
Solution set: x ⎢ x >
D
0
4 x ⎢ x < -2 or x⎢ -2 < x <
-4 -3 -2 -1
1 x ⎢ x ≥ 3 or x ⎢ x >
Chapter 2
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1
2
1 Lesson 6
99
REFLECT
1a. In Part C, why is the solution set the empty set?
The solution set is the common numbers, which would be the overlapping
The solution set is the set of all numbers from both graphs. However, {x x ≥ 3} is
a subset of {x x > 1}. So, the solution set can be written as {x x > 1},
PRACTICE
Solve. Write the solution in set notation. Graph the solution.
1. 4x + 2 > 14 AND x + 6 ≤ 4
4x > 12
2. -3x < 3 OR 2x + 3 ≥ 11
x > -1
x ≤ -2
2x ≥ 8
x≥4
x>3
Solution set: { x | x > 3} { x | x ≤ -2}
Solution set: { x | x > -1} { x | x ≥ 4}
the empty set or { x | x > -1}
-4 -3 -2 -1
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-4 -3 -2 -1
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3. 2 + x < 1 OR -5x + 1 < 16
x < -1
-5x < 15
x > -3
Solution set:
0
Chapter 2
Chapter 2
0
{ x | x < -1} { x | x > -3}
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3
4
2
3
4x > -4
3x ≥ 9
x > -1
x≥3
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5
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8
9
Solution set: { x | x > -1} { x | x ≥ 3}
{ x | x ≥ 3}
all real numbers
-4 -3 -2 -1
1
4. 4x - 3 > -7 AND 3x -2 ≥ 7
5
6
7
8
-4 -3 -2 -1
9
100
0
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8
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
segment. However, the graphs don’t overlap and that segment does not exist.
1b. In Part D, why is the solution set {x ⎢ x > 1}?
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Lesson 6
100
Lesson 6
Problem Solving
ADD I T I O NA L P R AC TI C E
AND PRO BL E M S O LV I N G
1. 68 ≤ t + 8 ≤ 77; 60 ≤ t ≤ 69
2. 380 ≤ m + 45 ≤ 410; 335 ≤ m ≤ 365
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
3. y < 1 OR y ≥ 5
Answers
4. 10 ≤ 2a ≤ 15; 5 ≤ a ≤ 7.5
Additional Practice
1. −2 < x < 4
2. x < -3 OR x ≥ 3
3. x ≤ −15 OR x ≥ -8
4. 0 ≤ x < 20
5. C
6. G
7. B
5. −7 < x < 4
6. 3 ≤ n < 7
7. -3 ≤ b ≤ 2
8. x < 0 OR x ≥ 6
9. k ≤ -4 OR k ≥ 4
© Houghton Mifflin Harcourt Publishing Company
10. s ≤ 2 OR s > 7
11. 20 ≤ h ≤ 20,000
12. 140 < w ≤ 147
Chapter 2
101
Lesson 6
Name
Class
Notes
2-6
Date
Additional Practice
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