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Chapter 9
Section 1
Objective 2
Solve compound inequalities with
the word and.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 2
Solve compound inequalities with the word and.
Solving a Compound Inequality with and
Step 1 Solve each inequality individually.
Step 2 Since the inequalities are joined with and, the solution set
of the compound inequality will include all numbers that
satisfy both inequalities in Step 1 (the intersection of the
solution sets).
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 3
CLASSROOM
EXAMPLE 2
Solving a Compound Inequality with and
Solve the compound inequality, and graph the solution set.
x + 3 < 1 and x – 4 > –12
Solution:
Step 1 Solve each inequality individually.
x+3<1
x+3–3 <1–3
x < –2
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
and
x – 4 > –12
x – 4 + 4 > –12 + 4
x > –8
Slide 9.1- 4
CLASSROOM
EXAMPLE 2
Solving a Compound Inequality with and (cont’d)
Step 2 Because the inequalities are joined with the word and, the
solution set will include all numbers that satisfy both
inequalities.
x < –2
)
x > –8
(
The solution set is (–8, –2).
(
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
)
Slide 9.1- 5
CLASSROOM
EXAMPLE 3
Solving a Compound Inequality with and
Solve and graph.
2x ≤ 4x + 8
3 x ≥ −9
and
Solution:
Step 1 Solve each inequality individually.
2x ≤ 4x + 8
and
3 x ≥ −9
x ≥ −3
−2 x ≤ 8
x ≥ −4
Remember
to reverse
the
inequality
symbol.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 6
CLASSROOM
EXAMPLE 3
Solving a Compound Inequality with and (cont’d)
Step 2
x ≥ −4
[
x ≥ −3
[
The overlap of the graphs consists of the numbers that are greater
than or equal to – 4 and are also greater than or equal to – 3.
The solution set is [–3,∞).
[
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 7
CLASSROOM
EXAMPLE 4
Solving a Compound Inequality with and
Solve and graph.
x + 2 > 3 and 2x + 1 < –3
Solution:
Solve each inequality individually.
x+2>3
and
2x + 1 < –3
2x < –4
x>1
x < –2
x>1
(
x < –2
)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 8
CLASSROOM
EXAMPLE 4
Solving a Compound Inequality with and (cont’d)
There is no number that is both greater than 1 and less than –2, so the
given compound inequality has no solution.
The solution set is
∅.
∅
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 9
Objective 4
Solve compound inequalities with
the word or.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 10
Solve compound inequalities with the word or.
Solving a Compound Inequality with or
Step 1 Solve each inequality individually.
Step 2 Since the inequalities are joined with or, the solution set of
the compound inequality includes all numbers that satisfy
either one of the two inequalities in Step 1 (the union of the
solution sets).
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 11
CLASSROOM
EXAMPLE 6
Solving a Compound Inequality with or
Solve and graph the solution set.
x − 1 > 2 or 3 x + 5 < 2 x + 6
Solution:
Step 1 Solve each inequality individually.
x −1 > 2
3x + 5 < 2 x + 6
x <1
or
x>3
x>3
(
x<1
)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 12
CLASSROOM
EXAMPLE 6
Step 2
Solving a Compound Inequality with or (cont’d)
The graph of the solution set consists of all numbers greater than 3 or
less than1.
The solution set is
( −∞,1)  ( 3, ∞ ) .
)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
(
Slide 9.1- 13
CLASSROOM
EXAMPLE 7
Solving a Compound Inequality with or
Solve and graph.
3 x − 2 ≤ 13 or x + 5 ≤ 7
Solution:
Solve each inequality individually.
3 x − 2 ≤ 13
3 x ≤ 15
x≤5
or
x+5≤ 7
or
x≤2
x≤5
]
x≤2
]
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 14
CLASSROOM
EXAMPLE 7
Solving a Compound Inequality with or (cont’d)
The solution set is all numbers that are either less than or equal to 5
or less than or equal to 2. All real numbers less than or equal to 5 are
included.
The solution set is
(−∞,5].
]
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 15
CLASSROOM
EXAMPLE 8
Solving a Compound Inequality with or
Solve and graph.
3 x − 2 ≤ 13 or x + 5 ≥ 7
Solution:
Solve each inequality individually.
3 x − 2 ≤ 13
3 x ≤ 15
x≤5
x≤5
or
x+5≥ 7
or
x≥2
]
[
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
x≥2
Slide 9.1- 16
CLASSROOM
EXAMPLE 8
Solving a Compound Inequality with or (cont’d)
The solution set is all numbers that are either less than or equal to 5
or greater than or equal to 2. All real numbers are included.
The solution set is
(−∞, −∞).
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.1- 17
Chapter 9
Section 2
Objective 1
Use the distance definition of
absolute value.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.2- 2
Use the distance definition of absolute value.
The absolute value of a number x, written |x|, is the distance from x
to 0 on the number line.
For example, the solutions of |x| = 5 are 5 and −5, as shown below.
We need to understand the concept of absolute value in order to
solve equations or inequalities involving absolute values. We solve
them by solving the appropriate compound equation or inequality.
Distance is
5, so |−5| = 5.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Distance is
5, so |5| = 5.
Slide 9.2- 3
Use the distance definition of absolute value.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.2- 4
Objective 2
Solve equations of the form
|ax + b| = k, for k > 0.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.2- 5
Use the distance definition of absolute value.
Remember that because absolute value refers to distance from
the origin, an absolute value equation will have two parts.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.2- 6
CLASSROOM
EXAMPLE 1
Solving an Absolute Value Equation
Solve |3x – 4| = 11.
Solution:
3x – 4 = −11
3x – 4 + 4 = −11 + 4
3x = −7
7
x= −
3
or
3x – 4 = 11
3x – 4 + 4 = 11 + 4
3x = 15
x=5
7 and 5 into the original absolute value
3
7 
equation to verify that the solution set is 
−
 ,5 .
 3 
Check by substituting −
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.2- 7
Objective 3
Solve inequalities of the form
|ax + b| < k and of the form
|ax + b| > k, for k > 0.
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Slide 9.2- 8
CLASSROOM
EXAMPLE 2
Solving an Absolute Value Inequality with >
Solve |3x – 4| ≥ 11.
Solution:
3x – 4 ≤ −11
3x – 4 + 4 ≤ −11 + 4
3x ≤ −7
3x – 4 ≥ 11
3x – 4 + 4 ≥ 11 + 4
3x ≥ 15
x≥5
or
7
x≤−
3
Check the solution. The solution set is
The graph consists of two intervals.
-5
-4
-3
]
-2
-1
0
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
1
2
3
7

 −∞, −   [5, ∞ ) .
3

4
[
5
6
7
8
Slide 9.2- 9
CLASSROOM
EXAMPLE 3
Solving an Absolute Value Inequality with <
Solve |3x – 4| ≤ 11.
Solution:
−11 ≤ 3x – 4 ≤ 11
−11 + 4 ≤ 3x – 4 ≤ 11+ 4
−7 ≤ 3x ≤ 15
7
− ≤ x≤5
3
Check the solution. The solution set is
 7
 − 3 ,

5 .

The graph consists of a single interval.
-5
-4
-3
[
-2
-1
0
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
1
2
3
4
]
5
6
7
8
Slide 9.2- 10
Solve inequalities of the form |ax + b| < k and of the
form |ax + b| > k, for k > 0.
When solving absolute value equations and inequalities of the types in
Examples 1, 2, and 3, remember the following:
1. The methods describe apply when the constant is alone on one side of the
equation or inequality and is positive.
2. Absolute value equations and absolute value inequalities of the form |ax + b| > k
translate into “or” compound statements.
3. Absolute value inequalities of the form |ax + b| < k translate into “and” compound
statements, which may be written as three-part inequalities.
4. An “or” statement cannot be written in three parts.
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.2- 11
Objective 4
Solve absolute value equations that
involve rewriting.
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Slide 9.2- 12
CLASSROOM
EXAMPLE 4
Solving an Absolute Value Equation That Requires Rewriting
Solve |3x + 2| + 4 = 15.
Solution:
First get the absolute value alone on one side of the equals sign.
|3x + 2| + 4 = 15
|3x + 2| + 4 – 4 = 15 – 4
|3a + 2| = 11
3x + 2 = −11
3x = −13
or
3x + 2 = 11
3x = 9
x=3
13
x= −
3
The solution set is
 13
 − 3 ,
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
3 .

Slide 9.2- 13
CLASSROOM
EXAMPLE 5
Solving Absolute Value Inequalities That Require Rewriting
Solve the inequality.
|x + 2| – 3 > 2
Solution:
|x + 2| – 3 > 2
|x + 2| > 5
x+2>5
or
x> 3
x+2<−5
x < −7
Solution set: (−∞, −7) ∪ (3, ∞)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.2- 14
CLASSROOM
EXAMPLE 5
Solving Absolute Value Inequalities That Require Rewriting (cont’d)
Solve the inequality.
|x + 2| – 3 < 2
Solution:
|x + 2| < 5
−5 < x + 2 < 5
−7 < x < 3
Solution set: (−7, 3)
Copyright © 2012, 2008, 2004 Pearson Education, Inc.
Slide 9.2- 15