# Absolute Value Inequalities Calculator

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

> The absolute value inequalities calculator solves |ax + b| < c, ≤ c, > c or ≥ c and writes the answer in interval notation, including the special cases where every number or no number works.

## Inputs

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

## Results

- Solution set (interval notation)
- Rewritten without absolute values
- Boundary points

## Formula

```
|u| < c  ⇔  −c < u < c
|u| > c  ⇔  u < −c  or  u > c
```

## Explanation

"Less than" absolute value inequalities describe numbers close to a point and give a single interval; "greater than" inequalities describe numbers far from a point and give two separate rays. Tolerances in engineering ("within ±0.2 mm") are classic examples.

### How to use the Absolute Value Inequalities Calculator

- **a (coefficient of x)**
- **b (constant)**
- **Inequality sign**
- **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

The calculator rewrites the inequality as a compound inequality, solves for x and adjusts for a negative a (which flips the inequality signs). Strict inequalities give open brackets, non-strict ones closed brackets.

## Worked example

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

Results — **Solution set (interval notation) = (−2, 3)**; Rewritten without absolute values = |2x − 1| < 5 → −5 < 2x − 1 < 5; Boundary points = Boundary points: x = −2 and x = 3.

## Key terms

- **Compound inequality**: Two inequalities joined by "and" or "or".
- **Interval notation**: Writing a set of numbers as (a, b), [a, b] and so on.

## FAQ

**What is the solution of**
x| < 3? | All x between −3 and 3: (−3, 3).

**What is the solution of**
x| > 3? | x < −3 or x > 3: (−∞, −3) ∪ (3, ∞).

**What if c is negative?**
|ax + b| < c has no solution; |ax + b| > c is true for every x.

**What do round and square brackets mean?**
A round bracket excludes the endpoint, a square bracket includes it.


## 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)