Interval Notation Calculator — Convert Inequalities to Interval Notation Instantly

Free interval notation calculator converts inequalities, compound inequalities, absolute value expressions, and function domains into proper interval notation with number line graphs and step-by-step breakdowns.

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Interval Notation Calculator

Select a mode, enter your values, and convert to interval notation with a number line graph.

Select a mode and enter values, then click Calculate Interval Notation to see the result.

Interval Notation Explained

Interval notation is a concise mathematical notation that represents a set of real numbers using endpoints and brackets. It is widely used in algebra, calculus, and domain analysis to describe ranges of values efficiently.

Open Interval: (a, b) = {x | a < x < b}
Closed Interval: [a, b] = {x | a ≤ x ≤ b}
Half-Open: (a, b] = {x | a < x ≤ b}  |  [a, b) = {x | a ≤ x < b}
Unbounded: [a, ∞) = {x | x ≥ a}  |  (-∞, a) = {x | x < a}

Key Notation Rules

  • Parentheses ( ) — Indicate an open endpoint that is not included in the set.
  • Brackets [ ] — Indicate a closed endpoint that is included in the set.
  • Infinity ∞ — Always paired with a parenthesis, since infinity is not a specific number.
  • Union ∪ — Combines two or more disjoint intervals into one solution set.
  • Empty set ∅ — Represents no solution (e.g., when an absolute value inequality has no valid answers).

How to Convert an Inequality to Interval Notation

Converting any inequality to interval notation follows a simple logical process. Use these steps for accurate conversion every time:

  1. Identify the inequality type — Determine if it is simple (x > a), compound (a < x < b), absolute value (|x − h| < k), or domain-based.
  2. Find the boundary values — Solve for the critical numbers that define the interval endpoints.
  3. Determine endpoint inclusion — Strict inequalities (>, <) use parentheses; inclusive inequalities (≥, ≤) use brackets.
  4. Write the interval — Place the smaller number on the left, the larger on the right, with the appropriate bracket or parenthesis at each end.
  5. Handle unions for disconnected sets — If the solution has gaps, use ∪ to join the separate intervals.

For absolute value inequalities: |x − h| < k becomes (h − k, h + k), while |x − h| > k becomes (−∞, h − k) ∪ (h + k, ∞).

Interval Notation Calculator Examples

Example 1: Simple Inequality to Interval Notation

Convert x > 5 to interval notation.

x > 5 → (5, ∞)
Set Builder: {x | x > 5}

Example 2: Compound Inequality

Convert 2 ≤ x < 7 to interval notation.

2 ≤ x < 7 → [2, 7)
Set Builder: {x | 2 ≤ x < 7}

Example 3: Absolute Value Inequality

Convert |x − 4| < 3 to interval notation.

|x − 4| < 3 → 1 < x < 7 → (1, 7)
Set Builder: {x | 1 < x < 7}

Example 4: Domain in Interval Notation

Find the domain of √(x − 2) in interval notation.

x − 2 ≥ 0 → x ≥ 2 → [2, ∞)
Set Builder: {x | x ≥ 2}

Real-World Interval Notation Applications

  • Domain Analysis: Expressing the domain of functions like square roots, logarithms, and rational expressions in calculus and precalculus.
  • Range Descriptions: Describing the output range of trigonometric, exponential, and other transcendental functions.
  • Statistics & Probability: Defining confidence intervals, probability ranges, and data distribution boundaries.
  • Computer Science: Specifying valid input ranges, array index bounds, and algorithm precondition constraints.
  • Engineering Tolerance: Expressing acceptable measurement ranges and manufacturing specification limits.
  • Economics: Defining price floors, ceilings, and feasible production quantity ranges.
  • Music Theory: Describing musical interval ranges between notes using semitone measurements.

People Also Ask About Interval Notation

Interval notation is a mathematical notation that represents a set of real numbers using endpoints and brackets. Parentheses ( ) indicate open endpoints not included, while brackets [ ] indicate closed endpoints that are included. For example, (2, 5] means all numbers greater than 2 and up to and including 5.
To convert an inequality to interval notation, identify the bound value and the inequality direction. For x > a, use (a, ∞). For x ≥ a, use [a, ∞). For x < a, use (−∞, a). For x ≤ a, use (−∞, a]. Compound inequalities like a < x < b become (a, b), and inclusive versions use brackets as appropriate.
The ∪ symbol in interval notation represents the union of two or more intervals, meaning the solution set includes all values that belong to at least one of the intervals. For example, (−∞, 2) ∪ (5, ∞) represents all numbers less than 2 OR greater than 5, excluding 2 and 5 themselves.
To write a function's domain in interval notation, identify all x-values for which the function is defined. Exclude values that cause division by zero, negative values under even roots, or invalid logarithm arguments. For example, the domain of √(x−3) is [3, ∞), and the domain of 1/(x−2) is (−∞, 2) ∪ (2, ∞).
An open interval (a, b) excludes both endpoints a and b, meaning a < x < b. A closed interval [a, b] includes both endpoints, meaning a ≤ x ≤ b. Half-open intervals like (a, b] or [a, b) include one endpoint and exclude the other. Infinity is always written with a parenthesis.

Frequently Asked Questions About Interval Notation

Yes. Select the Absolute Value mode to convert expressions like |x − h| < k or |x − h| > k into proper interval notation. The calculator handles both less-than (single interval) and greater-than (union of two intervals) cases.
Yes. The Domain Finder mode supports common function types including square roots, rational functions, square roots in denominators, and natural logarithms. Enter the parameter value and the calculator returns the domain in interval notation.
If the absolute value inequality has no solution (for example, |x − 3| < −1), the calculator returns the empty set symbol ∅. This occurs because an absolute value can never be less than a negative number.
The number line graph visually represents the interval. Filled circles indicate closed (included) endpoints, while open circles indicate open (excluded) endpoints. A thick colored line shows the solution set region. For unions, multiple regions are shown.
Set builder notation is an alternative way to describe sets: {x | condition}. For example, {x | x > 5} reads "the set of all x such that x is greater than 5." The calculator provides both interval notation and set builder notation for each result.
Infinity (∞) is always paired with a parenthesis because it is not a specific real number that can be "reached" or "included." Writing [a, ∞) would imply infinity is an attainable endpoint, which is mathematically incorrect.

Interval Notation Glossary

Interval Notation

A notation using parentheses and brackets to describe a continuous set of real numbers between two endpoints.

Open Interval

An interval that excludes both endpoints, written with parentheses: (a, b) means a < x < b.

Closed Interval

An interval that includes both endpoints, written with brackets: [a, b] means a ≤ x ≤ b.

Half-Open Interval

An interval including one endpoint and excluding the other: (a, b] or [a, b).

Set Builder Notation

An alternative notation: {x | condition} that defines a set by describing the properties its members must satisfy.

Union (∪)

The combination of two or more disjoint intervals into a single solution set covering all values in any of the intervals.

Empty Set (∅)

Represents a set with no elements, used when an inequality has no solution.

Unbounded Interval

An interval that extends infinitely in one or both directions, using ∞ or −∞ with parentheses.

Editorial Review & Methodology

This interval notation calculator was built and reviewed by the NumbrWiz Editorial Team. Interval notation is a fundamental concept in algebra and precalculus, verified against standard mathematics curricula including Common Core standards and college-level algebra textbooks.

  • Notation verification: Cross-checked against multiple authoritative algebra, precalculus, and calculus sources.
  • Edge case testing: Tested with negative bounds, zero values, empty sets, unbounded intervals, and absolute value edge cases.
  • UX review: Designed for intuitive mode selection with clear error messaging and visual number line graphs.

Transparency note: All calculations run client-side in your browser. No data is ever collected, stored, or transmitted. Results are for educational purposes; verify critical calculations independently.

Page last reviewed: May 2026 · NumbrWiz Editorial Team