Inequality to Interval Notation Calculator

Enter an inequality using the inequality type selector and define your value (a) — or set up a compound inequality with two bounds — and this Inequality to Interval Notation Calculator converts it to proper interval notation (e.g. (3, ∞) or [-2, 5)). You also get the set-builder notation and a plain-English description of the solution set.

Choose the type of inequality you want to convert.

Enter the lower bound (or single bound for simple inequalities).

Enter the upper bound (only needed for compound/double inequalities).

Results

Interval Notation

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Inequality Written Out

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Set-Builder Notation

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Plain-English Description

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Interval Type

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Results Table

Frequently Asked Questions

What is interval notation in math?

Interval notation is a compact way to describe a set of real numbers between two endpoints. Square brackets [ ] indicate that an endpoint is included (closed), while parentheses ( ) indicate it is excluded (open). Infinity is always paired with a parenthesis because it is not a real number and can never be included.

How do you convert an inequality to interval notation?

Replace the variable's bounds with the endpoint values and decide on brackets vs. parentheses based on the inequality sign. Use ( or ) for strict inequalities (< or >) and [ or ] for non-strict inequalities (≤ or ≥). For example, x ≥ 3 becomes [3, ∞) and x < 5 becomes (-∞, 5).

What are the main types of intervals?

There are four main types: open intervals (a, b) where neither endpoint is included; closed intervals [a, b] where both endpoints are included; half-open (or half-closed) intervals like [a, b) or (a, b]; and unbounded intervals that extend to positive or negative infinity, such as [a, ∞) or (-∞, b).

How do you handle compound inequalities in interval notation?

Compound inequalities use either 'and' (intersection) or 'or' (union). An 'and' compound like 2 < x < 8 is written as a single interval (2, 8). An 'or' compound like x < 2 or x > 8 is written as a union: (-∞, 2) ∪ (8, ∞). The union symbol ∪ separates the two separate solution sets.

What is the interval notation for -1 ≤ x ≤ 1?

The interval notation is [-1, 1]. Since both endpoints are included (≤ on both sides), square brackets are used on both ends. This is called a closed interval.

What is set-builder notation and how does it differ from interval notation?

Set-builder notation describes a set by specifying a property its elements must satisfy, written as {x | condition} — for example, {x | x > 3}. Interval notation conveys the same idea more compactly as (3, ∞). Both are equivalent ways to describe the same set of numbers; interval notation is often preferred for its brevity.

Can infinity be written with a bracket in interval notation?

No. Infinity (∞) and negative infinity (-∞) are concepts, not real numbers, so they can never be reached or included in a set. They are always written with a parenthesis: for example (5, ∞) or (-∞, 3]. Using a bracket next to infinity is considered a mathematical error.

What does an open circle vs. a closed circle mean on a number line?

An open circle on a number line means that endpoint is not included in the solution — corresponding to a strict inequality (< or >) and a parenthesis in interval notation. A closed (filled) circle means the endpoint is included — corresponding to a non-strict inequality (≤ or ≥) and a square bracket in interval notation.

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