Interval Notation in Math: Definition & Examples

Definitionnoun phrase

What is interval notation in math?

Interval notation is a short way to write a set of numbers between two endpoints, using brackets and parentheses. A bracket, [ or ], means the endpoint is included; a parenthesis, ( or ), means it is not. So −2 ≤ x < 3 is written [−2, 3), and x > 4 is written (4, ∞).

In Spanish: notación de intervalos
Also called: interval form, bracket notation
Grade level: Algebra 1

Algebra 1Algebra 2PrecalculusCCSS HSA-REI.B.3CCSS HSF-IF.A.1SATACTACCUPLACER

Interval Notation in a picture

Read the number line from left to right and copy what you see: a filled dot becomes a bracket, an open dot or an arrow to infinity becomes a parenthesis.

Formal definition and notation

For real numbers a < b, the intervals are [a, b] = {x ∣ a ≤ x ≤ b}, (a, b) = {x ∣ a < x < b}, and the half-open [a, b) and (a, b]. Unbounded intervals use ∞ or −∞, which are not numbers and always take a parenthesis: [a, ∞) = {x ∣ x ≥ a} and (−∞, ∞) is all real numbers. A set made of separate intervals is written with the union symbol ∪.

Symbol	Read it as	Meaning
[a, b]	closed interval from a to b	a ≤ x ≤ b: both endpoints included
(a, b)	open interval from a to b	a < x < b: neither endpoint included
[a, b)	a included, b not included	a ≤ x < b, a half-open interval
(a, ∞)	a to infinity	x > a: every number greater than a
∪	union	joins two intervals: (−∞, 1] ∪ (4, ∞)

Key facts

The bracket or parenthesis matches the dot on a number line: filled dot means bracket, open dot means parenthesis.
∞ and −∞ always get parentheses, because infinity is not a number you can reach or include.
(3, 7) can look like an ordered pair. Context tells you which: on a number line it is the interval between 3 and 7.
All real numbers: (−∞, ∞). A single number such as 5 is not an interval; write it as the set {5}.
Domain and range are usually written in interval notation: the domain of y = √x − 2 is [2, ∞).

Inequality to interval notation

a ≤ x < b → [a, b) x > a → (a, ∞)

[, ]: use with ≤ or ≥
(, ): use with <, >, and always with ∞

Always write the smaller number on the left, exactly as the numbers appear on a number line from left to right.

Try it: inequality to interval notation

Interactive

Drag the endpoints, switch each end open or closed, or send an end to infinity. The readout gives the inequality and the matching interval notation, including rays such as [2, ∞).

This interactive needs JavaScript. The worked examples below show the same ideas step by step.

Interval Notation examples, step by step

Example 1: A ray in interval notation

Warm-up

Write x ≥ −5 in interval notation.

The numbers start at −5 and go on forever to the right.
−5 is included (≥), so use a bracket: [−5.
There is no right end, so write ∞ with a parenthesis.

Answer: [−5, ∞).

Example 2: A half-open interval

Core

Write −1 < x ≤ 6 in interval notation, and say whether 6 and −1 are in the set.

Left end −1 with <: not included, parenthesis.
Right end 6 with ≤: included, bracket.
Write the smaller number first: (−1, 6].

Answer: (−1, 6]: 6 is in the set and −1 is not.

Example 3: Solve, then write the interval

Test-style

Solve 3 − 2x > 7 and write the solution in interval notation.

Subtract 3: −2x > 4.
Divide by −2 and reverse the sign: x < −2.
Every number less than −2, with −2 itself left out.
Interval: (−∞, −2).

Answer: (−∞, −2).

Common mistakes

✗ Mistakex ≥ 2 written as [2, ∞].

✓ Instead[2, ∞).

Infinity is not a number that can be included, so it always gets a parenthesis.

✗ Mistakex < 3 written as (3, −∞).

✓ Instead(−∞, 3).

Interval notation lists the smaller end first, the way a number line reads from left to right.

✗ Mistake−4 ≤ x < 1 written as (−4, 1].

✓ Instead[−4, 1).

Match each end separately: −4 is included (≤), so bracket; 1 is not (<), so parenthesis.

Not to be confused with

Interval Notationvs.Compound inequality

A compound inequality is the statement, such as −2 ≤ x < 3. Interval notation is one way to write its solution set: [−2, 3).

Interval Notationvs.Ordered pair

(2, 5) as an ordered pair is one point in the plane. (2, 5) as an interval is every number between 2 and 5 on a number line.

Interval Notationvs.Set-builder notation

Set-builder notation describes the rule, {x ∣ x > 4}. Interval notation shows the endpoints, (4, ∞). Both name the same set.

Quick check: 3 questions

Question 1 of 3

Which interval matches x ≤ 7?

Show answer and explanation

Answer: A. All numbers up to and including 7: (−∞, 7]. (−∞, 7) leaves out 7, [7, ∞) goes the wrong way, and −∞ can never take a bracket.

Question 2 of 3

Which number is in the interval (−3, 2]?

Show answer and explanation

Answer: B. The bracket at 2 means 2 is included. The parenthesis at −3 means −3 is not, and 3 and −4 lie outside the interval.

Question 3 of 3

Write x < −1 or x ≥ 4 in interval notation.

Show answer and explanation

Answer: D. Two pieces joined by union: (−∞, −1) leaves out −1, and [4, ∞) includes 4. The second choice swaps the brackets, and the other two describe the numbers between −1 and 4 instead.

Interval Notation: frequently asked questions

When do you use a bracket and when a parenthesis?

Use a bracket, [ or ], when the endpoint is part of the set: the inequality has ≤ or ≥, and the number line shows a filled dot. Use a parenthesis, ( or ), when the endpoint is not part of the set: < or >, or an open dot. Infinity always gets a parenthesis.

Why does infinity always get a parenthesis?

A bracket means “this endpoint is included”, and infinity is not a real number that could be included in the set. It only says the interval keeps going without end. So x ≥ 2 is [2, ∞): 2 is included, but there is no last number on the right.

How do you write all real numbers in interval notation?

Write (−∞, ∞). It means every real number, with no endpoint on either side, so both ends get parentheses. You will see it as the domain of polynomials such as y = x² − 3x, and as the solution of inequalities that are always true, such as x + 1 > x.

Practice and tools

Inequality CalculatorSolve an inequality and see its interval notation Domain and Range CalculatorDomain and range written in interval notation Compound Inequality CalculatorTurn and/or inequalities into intervals and unions

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-REI.B.3, CCSS HSF-IF.A.1

Cite this entry

Effortless Math Team. “Interval Notation.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/interval-notation/

Flashcard

TermInterval Notation[−2, 3)

DefinitionBrackets include an endpoint, parentheses leave it out, and infinity always uses a parenthesis: −2 ≤ x < 3 is [−2, 3).

Study the full deck

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Interval Notation

What is interval notation in math?

Interval Notation in a picture

Formal definition and notation

Key facts

Inequality to interval notation

Try it: inequality to interval notation

Interval Notation examples, step by step

Example 1: A ray in interval notation

Example 2: A half-open interval

Example 3: Solve, then write the interval

Common mistakes

Not to be confused with

Quick check: 3 questions

Interval Notation: frequently asked questions

When do you use a bracket and when a parenthesis?

Why does infinity always get a parenthesis?

How do you write all real numbers in interval notation?

Related terms

Practice and tools

Sources

Flashcard

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