Interval Notation - Brackets, Parentheses, Examples

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 $x$ with $2 \le x \le 7$ — 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 $2 \le x < 7$.

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

$$[a, b] = {x : a \le x \le b}$$

Both endpoints are part of the set. Example: $[3, 8]$ means $3 \le x \le 8$ — 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

$$[a, b) = {x : a \le x < b}$$

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

4. Half-Open Interval — Right Endpoint Included

$$(a, b] = {x : a < x \le b}$$

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

How Does Infinity Work in Interval Notation?

Infinity ($\infty$) and negative infinity ($-\infty$) aren't real numbers — they're directions, not destinations. So:

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

$$[3, \infty) \quad \text{not} \quad [3, \infty]$$

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

The most common forms:

A bracket next to $\infty$ 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 ($\le$ or $\ge$).

2. Strict $\Rightarrow$ parenthesis; non-strict $\Rightarrow$ bracket.

3. Write the smaller endpoint first, then the larger, separated by a comma.

Worked example. Convert $-2 \le x < 5$ to interval notation.

• Left endpoint: $-2$ with $\le$ $\Rightarrow$ bracket: [-2

• Right endpoint: $5$ with $<$ $\Rightarrow$ parenthesis: 5)

• Combine: $[-2, 5)$

Worked example. Convert $x > 7$ to interval notation.

The inequality has only a lower bound. So $x \in (7, \infty)$ — parenthesis on 7 because the inequality is strict, parenthesis on $\infty$ always.

How Do You Combine Intervals?

Two intervals can be combined with the union symbol $\cup$ 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:

$$(-\infty, 5) \cup (5, \infty)$$

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

Worked example. Write the set ${x : x \le -3 \text{ or } x \ge 4}$ in interval notation.

$$(-\infty, -3] \cup [4, \infty)$$

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

Three Worked Examples — Quick, Standard, Stretch

Quick — Inequality $\to$ Interval

Convert $1 < x \le 9$ to interval notation.

• Left: $1$ with $<$ $\Rightarrow$ parenthesis.

• Right: $9$ with $\le$ $\Rightarrow$ bracket.

• Answer: $(1, 9]$

Standard — Compound Inequality

Express the solution set of $|x - 2| < 5$ in interval notation.

The inequality $|x - 2| < 5$ unpacks as $-5 < x - 2 < 5$, i.e., $-3 < x < 7$.

Both endpoints are strict $\Rightarrow$ both parentheses: $(-3, 7)$.

Stretch — Domain of a Function

Find the domain of $f(x) = \dfrac{1}{\sqrt{x - 3}}$ and write it in interval notation.

The expression under a square root must be positive (zero would make the denominator zero, which is undefined). So $x - 3 > 0$, giving $x > 3$.

Domain: $(3, \infty)$ — parenthesis on $3$ because $3$ itself is excluded, parenthesis on $\infty$ always.

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

"Mathematics is the science of patterns." — adapted from Keith Devlin.

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. $\int_0^\pi \sin x , dx$ assumes you understand $[0, \pi]$ as a closed interval. The shorthand is everywhere.

• Statistics. Confidence intervals — "the 95% confidence interval is $[1.23, 1.47]$" — use exactly this notation. So do prediction intervals, tolerance intervals, and credible intervals in Bayesian statistics.

• 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 $n$." Python's range(0, n) is exactly this.

• Engineering and physics. Specifications like "operating temperature: $[-40°\text{C}, 85°\text{C}]$" use interval notation directly. Tolerance ranges, frequency bands, voltage ranges — all intervals.

The notation took its modern form in the 19th century alongside the formalisation of real analysis. Karl Weierstrass and Richard Dedekind — building rigorous foundations for calculus — needed a precise language for "set of real numbers between two values." Interval notation is what they (and their successors) produced.

A Worked Example

Find the domain of $f(x) = \dfrac{1}{x - 4}$ and write it in interval notation.

The intuitive (wrong) approach. A student in a hurry writes $(-\infty, 4) \cup [4, \infty)$ — thinking they're "marking 4 as excluded" by closing the bracket on the left piece.

Why it fails. The right piece $[4, \infty)$ includes 4 in the domain — but 4 is exactly the value that makes the denominator zero, so 4 must be excluded. Both brackets at 4 should be parentheses.

The correct method. The denominator $x - 4$ is zero when $x = 4$, so 4 is excluded. Everything else is fine.

$$\text{Domain} = (-\infty, 4) \cup (4, \infty)$$

Check. At $x = 4$: $f(4) = 1/(4-4) = 1/0$, undefined. ✓ At $x = 3.99$: $f(3.99) = 1/(-0.01) = -100$, defined. At $x = 4.01$: $f(4.01) = 1/0.01 = 100$, defined. The values approach 4 from both sides but never reach it.

At Bhanzu, our trainers walk through this wrong-path-first sequence intentionally — the bracket-vs-parenthesis confusion at an excluded point is the most common interval-notation slip, and the only way the rule sticks is to feel the sign flip once.

What Are the Most Common Mistakes With Interval Notation?

Mistake 1: Putting a bracket next to $\infty$

Where it slips in: Writing $[3, \infty]$ or $(-\infty, 5]$ with brackets on the infinity side.

Don't do this: $[3, \infty]$.

The correct way: $[3, \infty)$. Infinity is never enclosed by a bracket — it's not a real number you can include.

Mistake 2: Putting the smaller endpoint on the right

Where it slips in: Writing $[5, 2]$ instead of $[2, 5]$.

Don't do this: $[5, 2]$ — this is meaningless in interval notation.

The correct way: Always smaller number first: $[2, 5]$. The convention is left-to-right corresponds to small-to-large on the number line.

Mistake 3: Confusing union $\cup$ with intersection $\cap$

Where it slips in: A problem asks for "all $x$ with $x < 2$ or $x > 7$" and the student writes $(-\infty, 2) \cap (7, \infty)$ instead of $\cup$.

Don't do this: Use $\cap$ when the answer is "OR".

The correct way: OR in math means union ($\cup$). AND means intersection ($\cap$). The two operators look similar but mean opposite things — $\cup$ for combine, $\cap$ for overlap only. The memorizer who pattern-matches "U for union" is right to anchor it visually.

Key Takeaways

• Brackets [ ] include the endpoint; parentheses ( ) exclude it.

• Infinity is always wrapped in parentheses — a bracket on $\infty$ or $-\infty$ 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 $\cup$ combines intervals (OR); intersection $\cap$ takes their overlap (AND).

A Practical Next Step

Try these three before moving on to inequalities and function domains.

1. Write the inequality $-3 \le x < 6$ in interval notation.

2. Write the domain of $f(x) = \sqrt{x - 2}$ in interval notation.

3. Write "all real numbers less than 0 or greater than 5" in interval notation.

If problem 2 felt tricky, $\sqrt{x - 2}$ requires $x - 2 \ge 0$, so $x \ge 2$, giving domain $[2, \infty)$. Want a Bhanzu trainer to walk through more interval-notation problems? Book a free demo class — online globally.