# Absolute Value Equation Calculator

URL: https://speedycalculator.com/calculators/math/absolute-value-equation/
Last updated: 2026-10-30

> The absolute value equation calculator solves |ax + b| = c, giving both solutions (or explaining why there is one or none) with the two cases written out.

## Inputs

- a (coefficient of x)
- b (constant)
- c (right-hand side)

## Results

- Solution
- Split into two equations
- Boundary points

## Formula

```
|ax + b| = c  ⇔  ax + b = c  or  ax + b = −c   (c ≥ 0)
```

## Explanation

An absolute value measures distance from zero, so |ax + b| = c asks which x values put ax + b exactly c units from zero. There are usually two answers, one on each side, which is why absolute value equations split into two ordinary linear equations.

### How to use the Absolute Value Equation Calculator

- **a (coefficient of x)**
- **b (constant)**
- **c (right-hand side)**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

If c is negative there is no solution, because a distance cannot be negative. If c = 0 there is a single solution x = −b/a. Otherwise the calculator solves ax + b = c and ax + b = −c separately.

## Worked example

Inputs — a (coefficient of x): 2; b (constant): -1; c (right-hand side): 5.

Results — **Solution = x = −2 or x = 3**; Split into two equations = |2x − 1| = 5 → 2x − 1 = 5 or 2x − 1 = −5; Boundary points = Boundary points: x = −2 and x = 3.

## Key terms

- **Absolute value**: The distance of a number from zero, |x|.

## FAQ

**Why are there two solutions?**
Two numbers have the same distance from zero – for example 5 and −5 – so ax + b can equal either c or −c.

**When is there no solution?**
When c is negative. |anything| is never negative.

**Can an absolute value equation have one solution?**
Yes, when c = 0: the only solution is x = −b/a.

**How do I check the answers?**
Substitute each x into |ax + b| and confirm you get c.


## Sources

- [OpenStax – College Algebra 2e](https://openstax.org/details/books/college-algebra-2e)
- [OpenStax – Intermediate Algebra 2e](https://openstax.org/details/books/intermediate-algebra-2e)