1.7B Solve Absolute Value Inequalities
Goal  Solve absolute value (equations) and inequalities.
VOCABULARY
Absolute value, Extraneous solution
ABSOLUTE VALUE INEQUALITIES
You can solve an absolute value inequality by rewriting it as a ___compound___ ___inequality___ and
then solving ___each___ ___part___.
In the inequalities below, c > 0.
Inequality
Equivalent Form
|ax + b| _<_ c
c < ax + b < c
|ax + b| _≤_ c
c ≤ ax + b ≤ c
|ax + b| _>_ c
ax + b < c or ax + b > c
|ax + b| _≥_ c
ax + b ≤ c or ax + b ≥ c
Graph of Solution
Example 1: Solve an inequality of the form |ax + b| > c
Solve |2x + 5 | > 3. Then graph the solution.
The absolute value inequality is equivalent to 2x + 5 < _3_ or 2x + 5 > _3_ .
First Inequality
Second Inequality
2x + 5 < _3_
Write inequalities.
2x + 5 > _3_
2x < _8_
Subtract _5_ from each side.
2x > _2_
2x < _4_
Divide each side by _2_ .
x > _1_
The solutions are all real numbers less than _4_ or greater than _1_.
Example 2: Solve an inequality of the form |ax + b|  c
Solve | 3x - 7 |  5 . Then graph the solution.
The absolute value inequality is equivalent to 2x + 5 < _3_ or 2x + 5 > _3_ .
First Inequality
3x - 7 ≤ _-5_
Write the inequalities.
3x ≤ _2_
Add _7_ to each side.
x ≤ _2/3_
Divide each side by _3_ .
Second Inequality
3x -7 ≥ _5_
3x ≥ _12_
x ≥ _4_
The solutions are all real numbers less than or equal to _2/3_ or greater than or equal to _4_.
You Try: Solve the inequality. Then graph the solution.
1. |x  2|  7
x  5 or x  9
Example 3: Solve an inequality of the form | ax + b | < c
Solve | 2x + 1 | < 9 . Then graph the solution.
First Inequality
Second Inequality
2x + 1 < _9_
Write inequalities.
2x + 1 > _-9_
2x < _8_
Subtract _1_ from each side.
2x > _10_
x < _4_
Divide each side by _2_ .
x > _5_
The solution is between _-5_ and _4_, inclusive. -5 ≤ x ≤ 4
Example 4: Solve an inequality of the form | ax + b |  c
Solve |x  1.5|  4.5. Then graph the solution.
First Inequality
x – 1.5 ≤ _4.5_
Write inequalities.
x ≤ _6_
Add _1.5_ to each side.
Second Inequality
x - 1.5 ≥_-4.5_
x ≥ _-3_
The solution is between _-3_ and _6_, inclusive. -3 ≤ x ≤ 6
You Try: Solve the inequality. Then graph the solution.
2. |4x  1| < 9
2 < x < 2.5
Example 5: Solve Word Problems Using Absolute Value Inequalities
A food manufacturer specifies that every family-sized box of cereal should have a net weight of 25
ounces, with a tolerance of 1.2 ounces. Write and solve an absolute value inequality that describes the
acceptable net weights for the cereal in a family-size box.
Let w = weight of the cereal box
|w  25|  1.2
First Inequality
x – 25 ≤ _1.2_
Write inequalities.
x ≤ _26.2__ Add _25_ to each side.
Second Inequality
x - 25 ≥ _-1.2_
x ≥ _23.8_
The solution is between _23.8_ and _26.2_, inclusive. 23.8 ≤ x ≤ 26.2
You Try: You have found that your new winter coat is comfortable to wear when the outdoor
temperature is between 10°F and 42°F, inclusive. Write an absolute value inequality for this
temperature range, where t represents the temperature in degrees Fahrenheit.
Given: 10 ≤ x ≤ 42
Let t = temperature
|t  26|  16