STEMHub.tools
Equations · Inequalities

Absolute-Value Equations & Inequalities Calculator

Solve equations and inequalities of the form |ax+b| ◇ c by interpreting absolute value as distance on the real line.

Formula shown Runs locally Reviewed Aug 11, 2026
Follow us on Google
Equation guide

Distance-based solution of a linear expression inside an absolute value

Equality

|u|=c ⇔ u=±c

Inside

|u|<c ⇔ −c<u<c

Outside

|u|>c ⇔ u<−c or u>c
Equations · Absolute value

Solve a linear relation with modulus

Inputs and displayed decimal results use at most three decimal places. Exact internal values are retained during calculation.

|ax + b| = c

Enter a nonzero coefficient a and choose an equality or inequality.

Using the model

Symbols, assumptions and limitations

Interpretation notes

The coefficient a must be non-zero. The calculator handles equality and strict or inclusive inequalities, including zero and negative right-hand sides, and expresses solutions as points, intervals, exterior regions, all reals, or the empty set.

Worked examples

See the method in practice

01

Two-point equation

|2x−4|=6 gives x=−1 or x=5 because both points are distance 3 from the center x=2.

02

Inside interval

|2x−4|<6 gives −1<x<5, the points whose transformed distance is below the threshold.

Questions

Frequently asked

Why can an absolute-value equation have two solutions?

Positive and negative expressions can have the same positive magnitude.

What if c is negative?

Absolute value cannot be negative, so equality and below-threshold cases have no solution while above-threshold cases may include every real number.

Why must a be nonzero?

If a is zero, the relation is constant and no longer a linear absolute-value relation in x.

Sources & review

Equations and examples are checked against the references below. Results are educational and should be independently verified for safety-critical work.

OpenStax College Algebra — Absolute Value Equations

Written by the STEM Hub editorial team · Reviewed August 11, 2026 · Review methodology