# Holt McDougal Algebra 1 2-7 Solving Absolute-Value Inequalities Solve compound inequalities in one variable involving absolute-value expressions. Objectives

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Find all numbers whose absolute value is less than 5.
Absolute value inequality:
|x|– 3 < –1
x > –2 AND x < 2
Since 3 is subtracted from |x|, add 3 to both sides to undo the subtraction.
Write as a compound inequality.
+3 +3
|x| < 2
Additional Example 1B: Solving Absolute-Value Inequalities Involving <
Solve the inequality and graph the solutions.
x – 1 ≥ –2 AND x – 1 ≤ 2
Solve each inequality.
+1 +1
+1 +1
x ≥ –1
x ≤ 3
Solve the inequality and graph the solutions.
2|x| ≤ 6
x ≥ –3 AND x ≤ 3
Since x is multiplied by 2, divide both sides by 2 to undo the multiplication.
|x| ≤ 3
2|x| ≤ 6
|x + 3|– 4.5 ≤ 7.5
Since 4.5 is subtracted from |x + 3|, add 4.5 to both sides to undo the subtraction.
Solve each inequality and graph the solutions.
x + 3 ≥ –12 AND x + 3 ≤ 12
Write as a compound inequality.
Subtract 3 from both sides of each inequality.
+ 4.5 +4.5
–20
–15
–10
–5
0
5
10
15
Find all numbers whose absolute value is greater than 5.
Absolute value inequality:
|x| + 14 ≥ 19
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.
–10
–8
–6
–4
–2
0
2
4
6
8
10
3 + |x + 2| > 5
Since 3 is added to |x + 2|, subtract 3 from both sides to undo the addition.
Write as a compound inequality. Solve each inequality.
Additional Example 2B: Solving Absolute-Value Inequalities Involving >
Write as a compound inequality.
|x + 2| > 2
–2 –2
–2 –2
–10
–8
–6
–4
–2
0
2
4
6
8
10
|x| + 10 ≥ 12
|x| + 10 ≥ 12
x ≤ –2 OR x ≥ 2
Write as a compound inequality.
Since 10 is added to |x|, subtract 10 from both sides to undo the addition.
Solve each inequality and graph the solutions.
– 10 –10
|x| ≥ 2
Solve the inequality and graph the solutions.
Write as a compound inequality.
Write as a compound inequality. Solve each inequality.
Since is added to |x + 2 |, subtract from both sides to undo the addition.
OR
Solve the inequality and graph the solutions.
–7
–6
–5
–4
–3
0
1
2
3
–2
–1
Homework:
Sec. 2-7 Practice B Wksht (1-8) & Sec. 2-7 Practice A Wksht (1-8)
Holt McDougal Algebra 1
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 absolute-value 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 ≤ 3
Solve the two inequalities.
+95 +95
+95 +95
98
100
96
94
92
90
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 absolute-value 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 ≤ 75
p – 125 ≤ 75
|p – 125| ≤ 75
Solve the two inequalities.
+125 +125
+125 +125
200
225
175
150
125
100
75
50
25
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
Solve the inequality.
Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers.
All real numbers are solutions.
|x + 4|– 5 > – 8
Solve the inequality.
Absolute-value expressions are always nonnegative. Therefore, the statement is false for all values of x.
The inequality has no solutions.
|x – 2| + 9 < 7
Solving Absolute-Value Inequalities
An absolute value represents a distance, and distance cannot be less than 0.
Remember!
Solve the inequality.
|x| – 9 ≥ –11
Absolute-value expressions are always nonnegative. Therefore, the statement is true for all real numbers.
All real numbers are solutions.
|x| – 9 ≥ –11
Solve the inequality.
Divide both sides by 4.
The inequality has no solutions.
4|x – 3.5| ≤ –8

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