# Absolute Value Calculator

Calculate absolute values, find distances between numbers, and solve absolute value equations. Free online absolute value calculator with step-by-step.

## What this calculates

Calculate the absolute value of any number, find the distance between two points on the number line, or solve absolute value equations of the form |x - a| = b. Get instant results with clear explanations.

## Inputs

- **Calculation Mode** — options: Absolute value of a number, Distance between two numbers, Solve |x - a| = b — Choose the type of absolute value calculation.
- **Number** — The number to find the absolute value of.
- **Value A** — The first number or center value in |x - a| = b.
- **Value B** — The second number or distance value.

## Outputs

- **Result** — The absolute value or distance.
- **Explanation** — formatted as text — Step-by-step explanation.
- **Solutions** — formatted as text — The solution(s) for absolute value equations.

## Details

The absolute value of a number represents its distance from zero on the number line, regardless of direction. It is always non-negative. The absolute value of -5 is 5, and the absolute value of 5 is also 5.

Mathematical Definition

|x| = x if x >= 0, and |x| = -x if x < 0. In other words, the absolute value strips away the sign.

Distance Between Numbers

The distance between two numbers a and b on the number line is |a - b|. For example, the distance between -3 and 5 is |-3 - 5| = |-8| = 8.

Absolute Value Equations

Equations like |x - a| = b have two solutions (when b > 0): x = a + b and x = a - b. If b = 0, there is one solution (x = a). If b < 0, there is no solution because absolute value is never negative.

## Frequently Asked Questions

**Q: What is absolute value?**

A: Absolute value is the distance of a number from zero on the number line, always expressed as a non-negative value. It is denoted by vertical bars: |x|. For example, |-7| = 7 and |7| = 7. Think of it as the magnitude of a number without regard to its sign.

**Q: Can absolute value ever be negative?**

A: No, absolute value is always zero or positive. By definition, it measures distance, which cannot be negative. The equation |x| = -3 has no solution because no number has a negative distance from zero. This is an important principle in solving absolute value equations.

**Q: How do I solve an absolute value equation?**

A: For |expression| = b where b > 0, split into two cases: expression = b and expression = -b. Solve each case separately. For example, |x - 3| = 5 gives x - 3 = 5 (x = 8) and x - 3 = -5 (x = -2). Always check that b is non-negative first.

**Q: What is the difference between absolute value and modulus?**

A: For real numbers, absolute value and modulus mean the same thing. Both refer to the non-negative magnitude. In programming, the modulo operation (%) is different, computing the remainder of division. In complex numbers, the modulus refers to the distance from the origin in the complex plane.

**Q: How is absolute value used in real life?**

A: Absolute value is used in calculating error margins (how far a measurement is from the true value), computing distances regardless of direction, financial calculations (magnitude of profit or loss), and signal processing. It also underpins the concept of norms in linear algebra and vector spaces.

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Source: https://vastcalc.com/calculators/math/absolute-value
Category: Math
Last updated: 2026-04-21