Solve absolute value inequalities and express the solution as an interval, a union, all real numbers, or no solution.
Last updated: June 2026 | By Patchworkr Team
An absolute value inequality describes a range of numbers based on distance from a target value. Instead of looking for one exact answer, you are usually looking for all values that fall inside or outside a boundary.
That is why the solution is often written as an interval or as a union of two intervals.
For example, |x - 3| < 7 means x must be less than 7 units away from 3.
The symbols |x| mean the absolute value of x.
Absolute value measures distance from zero, so when you see an expression like |x - 3|, it represents the distance between x and 3.
In an inequality, the absolute value expression is the part that gets isolated before you solve for the solution set.
The first step is to isolate the absolute value expression.
If the isolated right side is negative, the answer depends on the inequality sign. Because absolute value is never negative, |x| < -3 has no solution, but |x| > -3 and |x| >= -3 are true for every real number.
|x - a| < b
Values inside the boundary. The solution is an interval between two endpoints.
|x - a| <= b
Values inside the boundary, including the endpoints.
|x - a| > b
Values outside the boundary. The solution splits into two intervals.
|x - a| >= b
Values outside the boundary, including the endpoints.
For |x - 3| < 7, you are looking for numbers less than 7 units away from 3.
The solution is the interval between -4 and 10, because every value in that range is within 7 units of 3.
A "less than" inequality asks for values inside a boundary, so the answer is usually one continuous interval.
A "greater than" inequality asks for values outside a boundary, so the answer is usually two separate intervals joined by or.
For example, |x - 3| > 7 means x is more than 7 units away from 3, so the solution splits into the values left of -4 and the values right of 10.
When do I use an interval?
Use an interval for less-than inequalities that describe values inside a boundary.
When do I use a union?
Use a union for greater-than inequalities that describe values outside a boundary.
Can the answer be all real numbers?
Yes, if the inequality is always true for every real number.
Can the answer be no solution?
Yes, if the absolute value would need to be less than a negative number.
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