Interval Notation Calculator

Convert between interval notation, inequality notation, and set-builder notation with visual number line.

Select Input Type & Enter Values

Result

Interval Notation
[2, 5)
bounded interval
Interval Notation[2, 5)
Inequality Notation2 ≤ x < 5
Set-Builder Notation{x | 2 ≤ x < 5}
Interval TypeHalf-open (left-closed, right-open)
Contains Endpoints?Left: yes, Right: no

Step-by-Step Conversion

[2, 5) means 2 ≤ x < 5

Understanding Interval Notation

Interval notation is a concise way to represent a set of real numbers between two endpoints. It uses brackets and parentheses to indicate whether endpoints are included or excluded from the set. This notation is widely used in calculus, algebra, and mathematical analysis.

Types of Intervals

Closed Interval [a, b]

Both endpoints are included. Contains all x where a ≤ x ≤ b.

[a, b] = {x | a ≤ x ≤ b}

Open Interval (a, b)

Both endpoints are excluded. Contains all x where a < x < b.

(a, b) = {x | a < x < b}

Half-Open [a, b)

Left endpoint included, right excluded. Also called left-closed, right-open.

[a, b) = {x | a ≤ x < b}

Half-Open (a, b]

Left endpoint excluded, right included. Also called left-open, right-closed.

(a, b] = {x | a < x ≤ b}

Unbounded (-inf, b]

No lower bound. Contains all x where x ≤ b. Always open at infinity.

(-inf, b] = {x | x ≤ b}

Unbounded [a, inf)

No upper bound. Contains all x where x ≥ a. Always open at infinity.

[a, inf) = {x | x ≥ a}

Notation Comparison

There are three common ways to express the same set of numbers. Interval notation is the most compact, inequality notation is the most intuitive, and set-builder notation is the most formal.

Key Rules

  • Square brackets [ ] mean the endpoint IS included (closed).
  • Parentheses ( ) mean the endpoint is NOT included (open).
  • Infinity is always paired with a parenthesis, never a bracket, because infinity is not a real number.
  • The left value must always be less than the right value.
  • A single point can be written as [a, a] which equals {a}.

Practical Applications

  • Expressing domains and ranges of functions in calculus.
  • Describing solution sets of inequalities in algebra.
  • Specifying confidence intervals in statistics.
  • Defining valid input ranges in computer science.
  • Representing time intervals in scheduling problems.