The Absolute Value Calculator computes |x| and solves absolute value equations and inequalities.
Three modes: calculate absolute value, solve |ax+b|=c (equations), and solve |ax+b|<c or |ax+b|>c (inequalities). Includes interval visualization on a number line.
Calculator information
📋 How to use this calculator
- Choose a mode: compute the absolute value of a single number, solve an equation |ax+b|=c, or solve an inequality.
- For the number mode, enter x; the result |x| is always >= 0.
- For equations, enter a, b, c (c must be >= 0 to have a real solution).
- For inequalities, select the sign (<, <=, >, >=) and enter the coefficients.
- Review the solution visualization on the number line, presented in interval notation.
- Tip: remember |x| < c is equivalent to -c < x < c, while |x| > c is equivalent to x < -c or x > c.
🧮 Definition and properties of absolute value
|x| = x if x >= 0; |x| = -x if x < 0
- |x| = distance of x from zero on the number line
- |ax + b| = c => ax + b = c or ax + b = -c
- |ax + b| < c => -c < ax + b < c
- |ax + b| > c => ax + b < -c or ax + b > c
- Properties: |ab| = |a|*|b|; |a + b| <= |a| + |b| (triangle inequality)
Inequalities with c < 0 have no solution for |.| < c and all real numbers for |.| > c.
💡 Worked example: Solve |2x - 5| < 3
Given:- a = 2
- b = -5
- c = 3 (>= 0, valid)
- Sign = <
Steps:- Convert to compound form: -3 < 2x - 5 < 3.
- Add 5 to all parts: 2 < 2x < 8.
- Divide all parts by 2: 1 < x < 4.
- Write in interval notation: (1, 4).
- Check an example: x = 3 -> |2*3 - 5| = |1| = 1 < 3 OK.
Result: Solution set: x in (1, 4) or 1 < x < 4.
❓ Frequently asked questions
What is the geometric meaning of absolute value?
The absolute value |x| is the distance of the point x from zero on the number line, which is always non-negative. |x - a| is the distance of x from a. This property is useful for interpreting equations such as |x - 3| = 5 as 'points x at distance 5 from 3', i.e., x = -2 or x = 8.
How do I solve absolute value equations with two expressions?
For |f(x)| = |g(x)|, split into two cases: f(x) = g(x) or f(x) = -g(x). Solve each, then verify each root in the original equation to filter out extraneous solutions. This approach works for equations such as |x - 1| = |2x + 3|.
Why does |x| < -2 have no solution?
The definition of absolute value states |x| >= 0 for all real x. No real number has an absolute value smaller than a negative number. Therefore inequalities |x| < c with c <= 0 always yield the empty set.
What is the triangle inequality and what is it used for?
The triangle inequality states |a + b| <= |a| + |b| for all real (or vector) a, b. Geometrically it implies that the length of one side of a triangle does not exceed the sum of the other two sides. In analysis it is used to prove continuity, convergence, and many fundamental theorems.
Is |x| differentiable?
|x| is differentiable for all x != 0, with d|x|/dx = sign(x). At x = 0 the function is not differentiable because it has a sharp corner. In optimization and machine learning, subgradients are used to handle this singular point (for example in L1 regularization).
📚 Sources & references
Last updated: May 11, 2026