Interval Notation Calculator — Inequality to Interval Converter
Select whether each endpoint is included (closed bracket) or excluded (open bracket), enter the bounds, and instantly get the interval in standard notation, inequality form, and set-builder form — with an animated number-line diagram.
Lower bound type
Upper bound type
How does this calculator work?
Interval notation uses square brackets for included endpoints and round brackets for excluded ones: [a, b] means a ≤ x ≤ b; (a, b) means a < x < b; [a, b) and (a, b] are half-open. Infinity always takes a round bracket. Choose your bound types and values above to get the notation instantly.
Formula
How this is calculated
Interval notation is a compact way to describe a continuous set of real numbers between two bounds. A square bracket [ or ] means the endpoint is included (closed); a round bracket ( or ) means it is excluded (open). The notation pairs a left bracket with the lower bound, a comma, the upper bound, and a right bracket — for example [2, 8] means all real numbers from 2 to 8 inclusive, while (2, 8) excludes the endpoints.
When a bound is infinite, the bracket on that side is always round because infinity is not a reachable value: (-∞, 5] means all numbers up to and including 5, and [3, +∞) means all numbers from 3 upwards. This calculator constructs the notation from your choices and also shows the equivalent inequality (e.g. 2 ≤ x ≤ 8) and set-builder form ({ x ∈ ℝ : 2 ≤ x ≤ 8 }).
Interval notation is used throughout calculus (domain and range), real analysis (open and closed sets), and statistics (confidence intervals). The key rule: brackets and inequalities must agree — a closed bracket always pairs with ≤ or ≥, an open bracket with < or >.
Frequently asked questions
Use a square bracket [ or ] when the endpoint is part of the set (the value is included). Use a round bracket ( or ) when the endpoint is not included. Infinity always uses round brackets because it is not an actual number.
[2, 8] is the closed interval: it includes 2 and 8 and every number between them. (2, 8) is the open interval: it includes all numbers strictly between 2 and 8 but not the endpoints themselves.
A single point is written [5, 5] (both brackets closed, same value). However, a single point has zero length and is rarely expressed this way — more commonly it is just written as x = 5.
Also known as
TG we-Calculate Editorial Team. (2026). Interval Notation Calculator — Inequality to Interval Converter [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/interval-notation-calculator
TG we-Calculate Editorial Team. "Interval Notation Calculator — Inequality to Interval Converter." TG we-Calculate. 2026. https://we-calculate.com/calculator/interval-notation-calculator.
TG we-Calculate Editorial Team, "Interval Notation Calculator — Inequality to Interval Converter," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/interval-notation-calculator
@misc{wecalculate_interval_notation_calculator, title = {Interval Notation Calculator — Inequality to Interval Converter}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/interval-notation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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