Absolute Value Inequalities Calculator — |ax + b| < c
Solve any absolute value inequality of the form |ax + b| < c (or ≤, >, ≥) and get the solution set in interval notation plus a number-line diagram.
Inequality
|1x + 2| < 5
How does this calculator work?
|ax+b| < c splits into −c < ax+b < c — a bounded interval between x = (−c−b)/a and x = (c−b)/a. For |ax+b| > c the solution is the two outer rays. Enter a, b, c and the inequality type to get the exact solution set and a number-line diagram.
Formula
How this is calculated
An absolute value inequality |ax + b| OP c is solved by splitting it into two linear cases. For |ax + b| < c (or ≤ c), the condition requires −c < ax + b < c, which gives a bounded interval between two boundary values. For |ax + b| > c (or ≥ c), the condition requires ax + b > c or ax + b < −c, which gives two outward-pointing rays — a union of two intervals.
The two boundary x-values come from solving ax + b = c and ax + b = −c: x = (c − b)/a and x = (−c − b)/a. The sign of the coefficient a determines which boundary is lower, but taking min/max handles both cases automatically.
Special cases: if c < 0, the less-than inequality has no solution (|anything| ≥ 0 cannot be less than a negative number), and the greater-than inequality is satisfied by all real numbers. The number line marks both boundaries with open (strict) or closed (non-strict) circles to match the inequality type.
Frequently asked questions
The "less than" case constrains x to stay within distance c of the root, forming a bounded segment. The "greater than" case requires x to be far enough from the root, leaving two unbounded tails on either side.
An absolute value is always ≥ 0, so |ax + b| < c has no solution when c < 0. Conversely, |ax + b| > c is satisfied by all real numbers when c < 0, because every absolute value exceeds any negative number.
The only value that satisfies this is |ax + b| = 0, i.e., x = −b/a. Enter c = 0 with the ≤ operator: the calculator returns the single-point interval [−b/a, −b/a].
TG we-Calculate Editorial Team. (2026). Absolute Value Inequalities Calculator — |ax + b| < c [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/absolute-value-inequalities-calculator
TG we-Calculate Editorial Team. "Absolute Value Inequalities Calculator — |ax + b| < c." TG we-Calculate. 2026. https://we-calculate.com/calculator/absolute-value-inequalities-calculator.
TG we-Calculate Editorial Team, "Absolute Value Inequalities Calculator — |ax + b| < c," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/absolute-value-inequalities-calculator
@misc{wecalculate_absolute_value_inequalities_calculator, title = {Absolute Value Inequalities Calculator — |ax + b| < c}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/absolute-value-inequalities-calculator}}, year = {2026}, note = {TG we-Calculate} }
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