Download Solving Absolute-Value Inequalities

2-7
Solving Absolute-Value
Inequalities
Connection: Connecting Absolute-Value
and Compound Inequalities
Essential question: How does solving absolute-value inequalities relate to solving
compound inequalities?
Standards for
Mathematical Content
• How would the solution and graph have been
different if the inequality had been |x| + 2 ≤ 5?
The solution would have been all real numbers
that are less than or equal to 3 units from 0. The
graph would have been the same except that the
circles would have been solid.
A-REI.2.3 Solve linear inequalities…in one
variable…
Vocabulary
absolute value
2
Math Background
EXplore
To solve an absolute-value inequality, properties
of inequality can be used to isolate the absolutevalue expression on one side of the inequality.
Then the inequality can be rewritten and solved
as a compound inequality that does not involve
absolute value. When an absolute-value inequality
involves < or ≤, the compound inequality uses
AND. When an absolute-value inequality involves > or ≥, the compound inequality uses OR.
Questioning Strategies
• Why is 3 added to each side of the inequality? To
INTR O D U C E
Essential Question
How does solving absolute-value inequalities relate
to solving compound inequalities?
isolate the absolute-value expression on one side.
• How can the graph be used to write a compound
inequality? The graph shows the solutions are
either less than -5 or greater than 5. This is the
solution to the inequality x < -5 OR x > 5.
CLOSE
You can solve an absolute-value inequality by
isolating the absolute-value expression, writing a
description of the solution, graphing the solution
on a number line, and writing a compound
inequality that describes the graph. You can also
solve an absolute-value inequality by isolating the
absolute-value expression, writing a compound
inequality, and solving the compound inequality.
When the absolute-value inequality involves < or
≤, the compound inequality involves AND. When
the absolute-value inequality involves > or ≥, the
compound inequality involves OR.
TEAC H
1
explore
Questioning Strategies
• Why is 2 subtracted from each side of the
inequality? To isolate the absolute-value
Summarize
Have students write a journal entry in which they
describe how to solve an absolute-value inequality
that involves < or ≤ and one that involves > or ≥.
Ask them to explain how they know whether the
solution of each is an AND or an OR compound
inequality.
expression on one side.
• How can the graph be used to write a compound
inequality? The graph shows that the solutions
are both greater than -3 and less than 3. This is
the solution to the inequality x > -3 AND x < 3.
Chapter 2 103
Lesson 7
© Houghton Mifflin Harcourt Publishing Company
Review absolute value with students, reminding
them that the absolute value of a number is its
distance from 0. Demonstrate by sketching the
solution of |x| = 4 on a number line. By looking
at the number line, ask students to name some
solutions of |x| > 4 and some solutions of |x| < 4.
Name
Class
Notes
2-7
Date
Solving Absolute-Value Inequalities
Connection: Connecting Absolute-Value and
Compound Inequalities
Essential question: How does solving absolute-value inequalities relate to
solving compound inequalities?
The absolute value of a number is its distance from 0 on a number
line. To solve an absolute-value inequality, you rewrite the inequality
as a compound inequality.
A-REI.2.3
1
EXPLORE
Solving Absolute-Value Inequalities with <
Solve the inequality ⎪x⎥ + 2 < 5.
A
Use the Subtraction Property of Inequality to isolate the absolute-value expression.
⎪x⎥ + 2 < 5
⎪x⎥ + 2 -
2
<5-
⎪x⎥ <
2
3
B
Write a description of the solution of the inequality.
C
Draw the graph of the solution on the number line.
less
All real numbers that are
© Houghton Mifflin Harcourt Publishing Company
-7 -6 -5 -4 -3 -2 -1
D
0
1
2
3
than
3
4
5
units from 0
6
Use the graph to rewrite ⎪x⎥ < 3 as a compound inequality.
AND
x > -3
3 , or -3 < x <
x<
3
REFLECT
1a. How did you decide whether to use AND or OR in the compound inequality?
The graph is a segment with open circles, so the solutions are between these two
values. This is the graph of a compound inequality involving AND.
1b. Solve ⎪x + 2⎥ ≤ 5 by rewriting it as a compound inequality. Show your work.
⎪x + 2⎥ ≤ 5
x + 2 ≥ -5 AND x + 2 ≤ 5
x + 2 - 2 ≥ - 5 - 2 AND x + 2 - 2 ≤ 5 - 2 (Subtraction Property of Inequality)
x ≥ - 7 AND x ≤ 3
Chapter 2
Lesson 7
103
A-REI.2.3
2
EXPLORE
Solving Absolute-Value Inequalities with >
Solve the inequality ⎪x⎥ - 3 > 2.
A
Use the Addition Property of Inequality to isolate the absolute value expression.
3
>2+
⎪ x⎥ >
B
3
5
Write a description of the solution of the inequality.
All real numbers
greater
than
5
units from 0.
C
Draw the graph of the solution on the number line.
D
Use the graph to rewrite ⎪x⎥ > 5 as a compound inequality.
-7 -6 -5 -4 -3 -2 -1
x < -5
OR
0
1
x>
2
3
4
5
6
5
REFLECT
2a. How did you decide whether to use AND or OR in the compound inequality?
The graph is two rays pointing in opposite directions, so the solution is not
between these two values. This is the graph of an OR compound inequality.
2b. Solve ⎪x - 3⎥ ≥ 2 by rewriting it as a compound inequality. Show your work.
⎪x - 3⎥ ≥ 2
x - 3 ≤ -2 OR x - 3 ≥ 2
x - 3 + 3 ≤ -2 + 3 OR x - 3 + 3 ≥ 2 + 3 (Addition Property of Inequality)
x ≤ 1 OR x ≥ 5
2c. Write a generalization for absolute-value inequalities that relate the inequality
symbol to the type of compound inequality that represents a solution.
When the absolute-value inequality has a < or ≤ symbol, the solution will be a
compound inequality involving AND. When the absolute-value inequality has a >
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
⎪x⎥ - 3 > 2
⎪x⎥ - 3 +
or ≥ symbol, the solution will be a compound inequality involving OR.
Chapter 2
Chapter 2
104
Lesson 7
104
Lesson 7
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. |x - 2| ≤ 0.05; 1.95 ≤ x ≤ 2.05
2. |x - 134| ≤ 8; 126 ≤ x ≤ 142
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
3. |x - 50| > 11
4. |x - 15.3| ≤ 0.4
Answers
5. B
Additional Practice
6. G
7. A
1. x ≥ -5 AND x ≤ 5
2. x > -3 AND x < 1
3. x ≥ 3 AND x ≤ 9
4. x > -7 AND x < 1
5. x < -3 OR x > 3
6. x < 2 OR x > 10
© Houghton Mifflin Harcourt Publishing Company
7. x ≤ -9 OR x ≥ -1
8. x ≤ 0.5 OR x ≥ 3.5
9. |x -350| ≤ 35; 315 ≤ x ≤ 385
10. |x - 88| ≤ 7.5; 80.5 ≤ x ≤ 95.5
Chapter 2
105
Lesson 7
Name
Class
Notes
2-7
Date
Additional Practice
© Houghton Mifflin Harcourt Publishing Company
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Chapter 2
Lesson 7
105
Problem Solving
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Chapter 2
Chapter 2
106
Lesson 7
106
Lesson 7