Absolute Value Inequality Solver

The Absolute Value Inequality Solver finds every x that satisfies an inequality of the form |ax + b| <, ≤, >, or ≥ c. Enter the coefficients and pick a comparison, and it returns the solution in interval notation, a status flag, and a step-by-step derivation showing how the boundaries were found.

It's built for algebra students checking homework, teachers preparing worked examples, and anyone who needs an exact interval without manually splitting the inequality into cases.

How to use it

1. Enter a — the coefficient of x inside the absolute value bars.
2. Enter b — the constant added to ax, also inside the bars.
3. Pick the comparison — <, ≤, >, or ≥.
4. Enter c — the value the absolute value is compared against.
5. Read the Solution in interval notation, or the Step-by-Step Derivation for the full working.

Results update automatically as you type. For example, entering a = 2, b = -3, ≤, c = 5 returns [-1, 4] — solved from |2x - 3| ≤ 5.

How is an absolute value inequality solved?

An absolute value inequality splits into two boundary equations, ax + b = c and ax + b = -c, whose roots mark the edges of the solution:

• Root 1: ax + b = c → x = (c − b) / a
• Root 2: ax + b = −c → x = (−c − b) / a

Once the two roots are sorted into a lower and upper bound, the shape of the answer follows a simple rule regardless of whether a is positive or negative:

• < or ≤ ("less than"): the solution is the interval between the boundaries — e.g. [-1, 4].
• > or ≥ ("greater than"): the solution is everything outside the boundaries — two rays, e.g. (-∞, -4) ∪ (2, ∞).

Strict comparisons (<, >) use open brackets/parentheses; non-strict comparisons (≤, ≥) use closed brackets.

Why does it sometimes say "no solution" or "all real numbers"?

An absolute value is never negative, so when c is negative, comparing it to a negative number is either impossible for every x (|x| < -1 has no solution) or true for every x (|x| > -1 is always true) — no boundaries need solving at all.

What if a is 0?

When a = 0, the x term disappears and the expression becomes a constant, |b|. The calculator just checks whether that constant satisfies the comparison — the answer is either all real numbers (if it does) or no solution (if it doesn't), since x never appears in the outcome either way.

What if c is 0?

When c = 0, both boundary equations collapse to the same root, x = -b/a. That turns the usual interval into a single point or its complement:

• ≤ collapses to the single point {-b/a} (only that one x makes the expression exactly 0).
• < has no solution (nothing is strictly less than 0 when the interval has shrunk to a point).
• ≥ covers all real numbers (the two closed rays meet and cover the whole line).
• > covers every real number except -b/a (the expression just needs to be nonzero).

For the equation version of this problem, see the Absolute Value Equation Solver. For a plain linear inequality without the absolute value bars, see the Linear Equation Solver.

Privacy

This calculator runs entirely in your browser. Your coefficients are never uploaded, logged, or stored — the computation happens locally on your device.