Solve Absolute Value Inequalities
Learning Outcomes
- Express solutions to inequalities containing absolute value
- Identify solutions for absolute inequalities where there are no solutions
Solve Inequalities Containing Absolute Value
Let us apply what you know about solving equations that contain absolute value and what you know about inequalities to solve inequalities that contain absolute value. Let us start with a simple inequality.This inequality is read, “the absolute value of x is less than or equal to .” If you are asked to solve for x, you want to find out what values of x are units or less away from on a number line. You could start by thinking about the number line and what values of x would satisfy this equation. and are both four units away from , so they are solutions. and are also solutions because each of these values is less than units away from . So are and , and , and so on—there are an infinite number of values for x that will satisfy this inequality. The graph of this inequality will have two closed circles, at and . The distance between these two values on the number line is colored in blue because all of these values satisfy the inequality.


Writing Solutions to Absolute Value Inequalities
For any positive value of a and x, a single variable, or any algebraic expression:Absolute Value Inequality | Equivalent Inequality | Interval Notation |
or | ||
or |
Example
Solve for x.Answer: Since this is a “greater than” inequality, the solution can be rewritten according to the “greater than” rule.
Solve each inequality.
Check the solutions in the original equation to be sure they work. Check the end point of the first related equation, and the end point of the second related equation, 1.
Try , a value less than , and 5, a value greater than 1, to check the inequality.
Both solutions check! Inequality notation: Interval notation: Graph:

Example
Solve for y.Answer: Begin isolating the absolute value by adding 9 to both sides of the inequality.
Divide both sides by 3 to isolate the absolute value.
Write the absolute value inequality using the “less than” rule. Subtract 6 from each part of the inequality.
Divide by to isolate the variable.
Answer
Inequality notation: Interval notation: Graph:
Identify Cases of Inequalities Containing Absolute Value That Have No Solutions
As with equations, there may be instances where there is no solution to an inequality.Example
Solve for x.Answer: Isolate the absolute value by subtracting from both sides of the inequality.
The absolute value of a quantity can never be a negative number, so there is no solution to the inequality. There is no solution