Solve inequalities of the form a|x + b| <, ≤, >, or ≥ c and see the solution as both a compound inequality and interval notation.
Last updated: August 2026 | By Summa Calculator
An expression such as |x - 3| is the distance from x to 3. Absolute value inequalities compare that distance with a boundary.
“Less than” means points inside the boundary around the center. “Greater than” means points outside it. That geometric idea explains why less-than answers are normally one interval while greater-than answers normally split into two pieces.
For a positive bound d:
Strict inequalities use open endpoints. Inclusive inequalities use closed endpoints when the endpoint is finite.
This solver accepts negative values of a. When you divide an inequality by a negative number, the inequality direction must reverse.
Missing that sign reversal is one of the easiest ways to get an absolute value inequality wrong.
Every point between -4 and 10 is less than 7 units from the center 3. The endpoints are excluded because the inequality is strict.
Once the absolute value is isolated, its left side is always at least zero. That creates several important edge cases:
|u| < 0 has no solution, while |u| ≤ 0 requires u = 0.
|u| > 0 allows every value except u = 0, while |u| ≥ 0 is true for every real value.
For a negative bound, less-than forms have no solution and greater-than forms are true for every real number.
What does a union mean?
It joins two separate parts of the solution set, which commonly happens when values must lie outside an absolute-value boundary.
What do parentheses and brackets mean?
Parentheses exclude a finite endpoint; brackets include it. Infinity always uses a parenthesis.
Can the answer be one point?
Yes. After isolation, |u| ≤ 0 is true only when u = 0, so the solution can collapse to a single x-value.
What does this calculator not solve?
It handles a|x + b| compared with c. It does not parse arbitrary algebraic inequalities, multiple absolute-value terms, or nonlinear expressions inside the bars.
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