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The absolute value of a number x, written |x|, measures its distance from zero on the number line — regardless of sign. By definition, |x| = x when x ≥ 0, and |x| = −x when x < 0. This means |5| = 5 and |−5| = 5; both are 5 units from zero.
This calculator evaluates |expression|, solves equations of the form |ax + b| = c, and solves inequalities such as |ax + b| < c and |ax + b| > c. Each result includes step-by-step working so you can follow the split-case logic.
Absolute value equations require splitting into two cases because |u| = c has the solutions u = c and u = −c (when c ≥ 0). For inequalities, |u| < c becomes −c < u < c (a bounded interval), while |u| > c becomes u < −c or u > c (two rays). Understanding these case splits is the key skill for all absolute value problems.
In geometry, absolute value computes the distance between two points on a number line: d = |a − b|. In statistics it appears in mean absolute deviation (MAD). In complex number theory, the absolute value extends to the modulus √(a² + b²). In programming, the abs() function implements exactly this operation. Despite its simple definition, absolute value underpins many areas of mathematics and science.
|x| = x if x ≥ 0; |x| = −x if x < 0 | |ax+b| = c → ax+b = c or ax+b = −c
Evaluate |x|, solve equations, or solve inequalities.
| Type | Form | Solution Method | Interval Notation |
|---|---|---|---|
| Equation | |ax+b| = c (c>0) | Two cases: ax+b = c or ax+b = −c | Two values |
| Equation | |ax+b| = 0 | One case: ax+b = 0 | One value |
| Equation | |ax+b| = c (c<0) | No solution | ∅ |
| Less than | |ax+b| < c | −c < ax+b < c (compound) | (left, right) |
| Less than or equal | |ax+b| ≤ c | −c ≤ ax+b ≤ c | [left, right] |
| Greater than | |ax+b| > c | ax+b > c OR ax+b < −c | (−∞, left) ∪ (right, ∞) |
| Greater than or equal | |ax+b| ≥ c | ax+b ≥ c OR ax+b ≤ −c | (−∞, left] ∪ [right, ∞) |
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.