2-7 Solving Absolute-Value Inequalities
Objectives
Solve compound inequalities in one variable
involving absolute-value expressions.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Find all numbers whose absolute value is less than 5.
Absolute value inequality:
Compound inequality:
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
To Solve Absolute-Value Inequalities
1. Perform inverse operations to isolate the absolute
value bars.
2.
3. Solve the compound inequality.
4. Graph the solution set.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Additional Example 1A: Solving Absolute-Value
Inequalities Involving <
Solve the inequality and graph the solutions.
|x|– 3 < –1
|x|– 3 < –1
+3 +3
|x| < 2
Since 3 is subtracted from |x|, add 3
to both sides to undo the
subtraction.
x > –2 AND x < 2
Write as a compound inequality.
2 units
–2
–1
2 units
0
Holt McDougal Algebra 1
1
2
2-7 Solving Absolute-Value Inequalities
Additional Example 1B: Solving Absolute-Value
Inequalities Involving <
Solve the inequality and graph the solutions.
|x – 1| ≤ 2
x – 1 ≥ –2 AND x – 1 ≤ 2 Write as a compound inequality.
+1 +1
+1 +1 Solve each inequality.
AND
x ≥ –1
–3
–2
–1
0
Holt McDougal Algebra 1
1
x ≤ 3 Write as a compound inequality.
2
3
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 1a
Solve the inequality and graph the solutions.
2|x| ≤ 6
2|x| ≤ 6
2
2
|x| ≤ 3
x ≥ –3 AND x ≤ 3
3 units
–3
–2
–1
Holt McDougal Algebra 1
Since x is multiplied by 2, divide both
sides by 2 to undo the
multiplication.
Write as a compound inequality.
3 units
0
1
2
3
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 1b
Solve each inequality and graph the solutions.
|x + 3|– 4.5 ≤ 7.5
|x + 3|– 4.5 ≤ 7.5
+ 4.5 +4.5
|x + 3| ≤ 12
x + 3 ≥ –12 AND x + 3 ≤ 12
–3
–3
–3 –3
x ≥ –15 AND
x≤9
–20 –15 –10
–5
Holt McDougal Algebra 1
0
5
10
15
Since 4.5 is subtracted from
|x + 3|, add 4.5 to both
sides to undo the
subtraction.
Write as a compound
inequality.
Subtract 3 from both
sides of each inequality.
2-7 Solving Absolute-Value Inequalities
Find all numbers whose absolute value is greater than 5.
Absolute value inequality:
Compound inequality:
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Additional Example 2A: Solving Absolute-Value
Inequalities Involving >
Solve the inequality and graph the solutions.
|x| + 14 ≥ 19
|x| + 14 ≥ 19
– 14 –14
|x|
≥ 5
x ≤ –5 OR x ≥ 5
Since 14 is added to |x|, subtract 14
from both sides to undo the addition.
Write as a compound inequality.
5 units 5 units
–10 –8 –6 –4 –2
0
Holt McDougal Algebra 1
2
4
6
8 10
2-7 Solving Absolute-Value Inequalities
Additional Example 2B: Solving Absolute-Value
Inequalities Involving >
Solve the inequality and graph the solutions.
3 + |x + 2| > 5
Since 3 is added to |x + 2|,
subtract 3 from both sides to
undo the addition.
3 + |x + 2| > 5
–3
–3
|x + 2| > 2
Write as a compound inequality.
x + 2 < –2 OR x + 2 > 2
Solve each inequality.
–2 –2
–2 –2
x
< –4 OR x
> 0 Write as a compound inequality.
–10 –8 –6 –4 –2
0
Holt McDougal Algebra 1
2
4
6
8 10
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 2a
Solve each inequality and graph the solutions.
|x| + 10 ≥ 12
|x| + 10 ≥ 12
– 10 –10
|x|
≥
Since 10 is added to |x|, subtract 10
from both sides to undo the
addition.
2
x ≤ –2 OR x ≥ 2 Write as a compound inequality.
2 units 2 units
–5 –4 –3 –2 –1
0
Holt McDougal Algebra 1
1
2
3
4
5
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 2b
Solve the inequality and graph the solutions.
Since is added to |x + 2 |, subtract
from both sides to undo the addition.
Write as a compound inequality.
Solve each inequality.
OR
Write as a compound
inequality.
x ≤ –6
Holt McDougal Algebra 1
x≥1
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 2b Continued
Solve the inequality and graph the solutions.
–7 –6 –5 –4 –3 –2 –1 0
Holt McDougal Algebra 1
1
2
3
2-7 Solving Absolute-Value Inequalities
Homework:
Sec. 2-7 Practice B Wksht (1-8) &
Sec. 2-7 Practice A Wksht (1-8)
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Additional Example 3: Application
A pediatrician recommends that a baby’s
bath water be 95°F, but it is acceptable for
the temperature to vary from this amount by
as much as 3°F. Write and solve an absolutevalue inequality to find the range of
acceptable temperatures. Graph the
solutions.
Let t represent the actual water temperature.
The difference between t and the ideal
temperature is at most 3°F.
t – 95
Holt McDougal Algebra 1
≤
3
2-7 Solving Absolute-Value Inequalities
Additional Example 3 Continued
t – 95
≤
3
|t – 95| ≤ 3
t – 95 ≥ –3 AND t – 95 ≤
3
+95 +95
+95 +95
t
≥ 92 AND t
≤ 98
90
92
94
96
98
Solve the two
inequalities.
100
The range of acceptable temperature is 92 ≤ t ≤ 98.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 3
A dry-chemical fire extinguisher should be
pressurized to 125 psi, but it is acceptable
for the pressure to differ from this value by
at most 75 psi. Write and solve an absolutevalue inequality to find the range of
acceptable pressures. Graph the solution.
Let p represent the desired pressure.
The difference between p and the ideal
pressure is at most 75 psi.
p – 125
Holt McDougal Algebra 1
≤
75
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 3 Continued
p – 125
≤
75
|p – 125| ≤ 75
p – 125 ≥ –75 AND p – 125 ≤ 75
+125 +125
+125 +125
p
≥
50 AND p
≤ 200
25
50
Solve the two
inequalities.
75 100 125 150 175 200 225
The range of pressure is 50 ≤ p ≤ 200.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
When solving an absolute-value inequality, you may
get a statement that is true for all values of the
variable. In this case, all real numbers are solutions
of the original inequality.
If you get a false statement when solving an
absolute-value inequality, the original inequality has
no solutions.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Additional Example 4A: Special Cases of AbsoluteValue Inequalities
Solve the inequality.
|x + 4|– 5 > – 8
|x + 4|– 5 > – 8
+5
+5
|x + 4|
>
–3
Add 5 to both sides.
Absolute-value expressions
are always nonnegative.
Therefore, the statement is
true for all real numbers.
All real numbers are solutions.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Additional Example 4B: Special Cases of AbsoluteValue Inequalities
Solve the inequality.
|x – 2| + 9 < 7
|x – 2| + 9 < 7
–9 –9
|x – 2|
< –2
Subtract 9 from both sides.
Absolute-value expressions
are always nonnegative.
Therefore, the statement is
false for all values of x.
The inequality has no solutions.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Remember!
An absolute value represents a distance, and
distance cannot be less than 0.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 4a
Solve the inequality.
|x| – 9 ≥ –11
|x| – 9 ≥ –11
+9 ≥ +9
|x|
≥ –2
Add 9 to both sides.
Absolute-value expressions
are always nonnegative.
Therefore, the statement is
true for all real numbers.
All real numbers are solutions.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Check It Out! Example 4b
Solve the inequality.
4|x – 3.5| ≤ –8
4|x – 3.5| ≤ –8
4
4
|x – 3.5| ≤ –2
Divide both sides by 4.
Absolute-value expressions
are always nonnegative.
Therefore, the statement is
false for all values of x.
The inequality has no solutions.
Holt McDougal Algebra 1
2-7 Solving Absolute-Value Inequalities
Homework:
Sec. 2-7 Practice C & Problem
Solving Worksheets
Holt McDougal Algebra 1