# 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:

| Notation | Meaning | Inequality |
| --- | --- | --- |
| (a,∞) | All real x with x > a | x > a |
| \[a,∞) | All real x with x ≥ a | x ≥ a |
| (−∞,b) | All real x with x < b | x < b |
| (−∞,b\] | All real x with x ≤ b | x ≤ b |
| (−∞,∞) | All real numbers | (no restriction) |

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_).