Interval Notation - Brackets, Parentheses, Examples

TL;DR

Interval notation is a compact way to write a set of real numbers using two symbols and two punctuation marks. A square bracket [ or ] includes the endpoint; a parenthesis ( or ) excludes it. So the inequality 1≤x<5 becomes [1,5) . Infinity is always written with a parenthesis — never a bracket — because infinity isn't a real number you can "reach."

What Is Interval Notation?

Interval notation writes a set of real numbers as a pair of endpoints inside brackets, parentheses, or a mix. It's the standard alternative to writing the same set as an inequality.

The two conventions:

• Square brackets [ or ] mean "the endpoint is included."
• Parentheses ( or ) mean "the endpoint is excluded."

So [2,7] is the set of all real numbers with 2≤x≤7 — including 2 and 7 themselves. And (2,7) is the set with 2<x<7 — excluding both endpoints.

Mixing is allowed: [2,7) includes 2 but excludes 7 — the inequality 2≤x<7.

The Four Types of Intervals

Every bounded interval falls into one of four types — defined by whether each endpoint is included or excluded.

1. Closed Interval — Both Endpoints Included

[a,b] = {x : a ≤ x ≤ b}

Both endpoints are part of the set. Example: [3,8] means 3≤x≤8 — includes 3, 8, and every real number between.

2. Open Interval — Both Endpoints Excluded

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

Neither endpoint is in the set. Example: (3,8) means 3<x<8 — excludes 3 and 8, includes everything strictly between.

3. Half-Open Interval — Left Endpoint Included

[a,b) = {x : a ≤ x < b}

Includes the left endpoint but not the right. Example: [3,8) means 3≤x<8.

4. Half-Open Interval — Right Endpoint Included

(a,b] = {x : a < x ≤ b}

Excludes the left, includes the right. Example: (3,8] means 3<x≤8.

How Does Infinity Work in Interval Notation?

Infinity (∞) and negative infinity (−∞) aren't real numbers — they're directions, not destinations. So:

Infinity is always written with a parenthesis. Never a bracket.

[3,∞) means "all real numbers with x≥3" — bounded below by 3 (included) and unbounded above. The bracket on the 3 says "3 is in the set"; the parenthesis on ∞ says "the set extends without limit."

The most common forms:

A bracket next to ∞ is always wrong. The rule has no exceptions.

How Do You Convert Inequalities to Interval Notation?

Three-step process:

1. Identify each endpoint and whether it's strict (< or >) or non-strict (≤ or ≥).
2. Strict ⇒ parenthesis; non-strict ⇒ bracket.
3. Write the smaller endpoint first, then the larger, separated by a comma.

Worked example. Convert −2≤x<5 to interval notation.

• Left endpoint: −2 with ≤ ⇒ bracket: [-2
• Right endpoint: 5 with < ⇒ parenthesis: 5)
• Combine: [-2,5)

Worked example. Convert x>7 to interval notation.

The inequality has only a lower bound. So x∈(7,∞) — parenthesis on 7 because the inequality is strict, parenthesis on ∞ always.

How Do You Combine Intervals?

Two intervals can be combined with the union symbol ∪ when the answer is "in one OR the other."

Worked example. Write "all real numbers except 5" in interval notation.

This is everything to the left of 5 or everything to the right of 5:

(−∞,5)∪(5,∞)

The two pieces are joined by ∪ (union), and 5 itself is excluded from both pieces.

Worked example. Write the set x:x≤−3 or x≥4 in interval notation.

(−∞,−3]∪[4,∞)

Brackets on −3 and 4 because both are included; parentheses on the infinities (always).

Three Worked Examples — Quick, Standard, Stretch

Quick — Inequality → Interval

Convert 1<x≤9 to interval notation.

• Left: 1 with < ⇒ parenthesis.
• Right: 9 with ≤ ⇒ bracket.
• Answer: (1,9]

Standard — Compound Inequality

Express the solution set of ∣x−2∣<5 in interval notation.

Both endpoints are strict ⇒ both parentheses: (−3,7).

Stretch — Domain of a Function

Find the domain of f(x)=1/(x−3) and write it in interval notation.

The expression under a square root must be positive. So x−3>0 gives x>3.

Domain: (3,∞) — parenthesis on 3 because 3 itself is excluded, parenthesis on ∞ always.

Why Does Interval Notation Matter? (The Real-World GROUND)

Interval notation isn't an arbitrary academic convention. It's the shorthand that makes higher mathematics readable.

• Calculus. Every domain, every range, every region of integration is described in interval notation.
• Statistics. Confidence intervals use exactly this notation.
• Computer science. When indexing arrays in many programming languages, the half-open interval [0,n) is the natural way to express "index from 0 up to but not including n."
• Engineering and physics. Specifications like "operating temperature: [-40°C,85°C]" use interval notation directly.

The notation took its modern form in the 19th century alongside the formalization of real analysis. Karl Weierstrass and Richard Dedekind needed a precise language for "set of real numbers between two values." Interval notation is what they produced.

What Are the Most Common Mistakes With Interval Notation?

Mistake 1: Putting a bracket next to ∞

Don't do this: [3,∞].

The correct way: [3,∞) . Infinity is never enclosed by a bracket.

Mistake 2: Putting the smaller endpoint on the right

Don't do this: [5,2].

The correct way: Always smaller number first: [2,5].

Mistake 3: Confusing union ∪ with intersection ∩

Don't do this: Use ∩ when the answer is "OR".

The correct way: OR in math means union (∪).

Key Takeaways

• Brackets [ include the endpoint; parentheses ( exclude it.
• Infinity is always wrapped in parentheses — a bracket on ∞ or −∞ is never correct.
• Four interval types: closed [a,b], open (a,b), half-open [a,b) or (a,b].
• Smaller endpoint first — left-to-right corresponds to small-to-large.
• Union ∪ combines intervals ( OR); intersection ∩ takes their overlap ( AND).