An absolute value inequality is an inequality that has an absolute value sign with a variable inside.
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
Add 7 to both sides of the first inequality.
x < 10 AND – (x – 7) < 3
Multiply both sides of the second inequality by –1. Remember to reverse the direction of the inequality.
x < 10 AND x – 7 > –3
Add 7 to both sides of the second inequality.
x < 10 AND x > 4
We can write this more simply as:
4 < x < 10
Its graph looks like this:
Some absolute value inequalities result in an "OR" inequality, rather than an "AND" inequality. The graphs of these will be two rays in opposite directions.
Example 2:
Solve and graph.
|x + 2|
4
Split into two inequalities.
x + 2
4 OR – (x + 2)
4
Subtract 2 from both sides of the first inequality.
x
2 OR – (x + 2)
4
Multiply the second inequality by –1. Again, remember to reverse the direction.
x
2 OR x + 2
–4
Subtract 2 from both sides of the second inequality.
x
2 OR x
– 6