Absolute Value Inequalities
An absolute value inequality is an inequality that has an absolute value sign with a variable inside.
Absolute Value Inequalities (<):
The inequality | x | < 4 means that the distance between x and 0 is less than 4.
So, x > –4 AND x < 4. The solution set is 
.
When solving absolute value inequalities, there are two cases to consider.
Case 1: The expression inside the absolute value symbols is positive.
Case 2: The expression inside the absolute value symbols is negative.
The solution is the intersection of the solutions of these two cases.
In other words, for any real numbers a and b , if | a | < b, then a < b AND a > – b .
Example 1:
Solve and graph.
|x – 7| < 3
To solve this sort of inequality, we need to break it into a compound inequality.
x – 7 < 3 AND x – 7 > –3
–3 < x – 7 < 3
Add 7 to each expression.
–3 + 7 < x – 7 + 7 < 3 + 7
4 < x <10
The graph looks like this:
Absolute Value Inequalities (>):
The inequality | x | > 4 means that the distance between x and 0 is greater than 4.
So, x < –4 OR x > 4. The solution set is 
.
When solving absolute value inequalities, there are two cases to consider.
Case 1: The expression inside the absolute value symbols is positive.
Case 2: The expression inside the absolute value symbols is negative.In other words, for any real numbers a and b , if | a | > b, then a > b OR a < – b .
Example 2:
Solve and graph.
Split into two inequalities.
Subtract 2 from each sides of each inequality.
The graph looks like this:
