# Interval Notation Calculator

URL: https://speedycalculator.com/calculators/math/interval-notation/
Last updated: 2026-10-30

> The interval notation calculator translates between interval notation, inequalities and set-builder notation for one-sided and two-sided intervals.

## Inputs

- Conversion type
- Interval type
- Inequality
- Interval
- a
- Left bracket
- Right bracket
- b

## Results

- Result
- Set-builder form
- Equivalent form

## Formula

```
[a, b] = {x | a ≤ x ≤ b}   (a, b) = {x | a < x < b}   [a, ∞) = {x | x ≥ a}
```

## Explanation

Intervals describe continuous ranges of numbers: the domain of √x is [0, ∞), the temperatures at which water is liquid at sea level are (0 °C, 100 °C). Being fluent in the three notations makes algebra and calculus far easier to read.

### How to use the Interval Notation Calculator

- **Conversion type**
- **Interval type**
- **Inequality**
- **Interval**
- **a**
- **Left bracket**
- **Right bracket**
- **b**

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

### How it works

Pick the direction of the conversion and the kind of interval. The calculator shows the interval, the equivalent inequality and the set-builder description.

### Tips

- Rewrite the equation in standard form before reading off the coefficients.
- Check your answer by substituting it back into the original equation.
- Keep exact forms (fractions and square roots) until the final step to avoid rounding errors.
- For inequalities, flip the sign whenever you multiply or divide by a negative number.

### Common mistakes to avoid

- Dropping a negative sign when copying a coefficient.
- Forgetting the ± when taking a square root.
- Dividing by an expression that could be zero.

## Worked example

Inputs — Conversion type: Interval → inequality; Interval type: Two-sided (bounded); a: 0; Left bracket: ( open; Right bracket: ] closed; b: 100.

Results — **Result = 0 < x ≤ 100**; Set-builder form = (0, 100] = {x | 0 < x ≤ 100}; Equivalent form = (0, 100].

## Key terms

- **Set-builder notation**: {x | condition}, the set of x satisfying a condition.
- **Ray**: An interval that extends to infinity in one direction.

## FAQ

**What does a half-open interval look like?**
One bracket of each kind, such as [2, 7), which includes 2 but not 7.

**How do I write the union of two intervals?**
With ∪, for example (−∞, 1) ∪ (3, ∞).

**Can an interval contain a single number?**
[a, a] = {a} is a degenerate closed interval.

**What is the empty set?**
∅ – an interval such as (3, 3) with nothing in it.

**What formula does the Interval Notation Calculator use?**
[a, b] = {x | a ≤ x ≤ b}   (a, b) = {x | a < x < b}   [a, ∞) = {x | x ≥ a}.


## 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)