Interval Notation Explained: Brackets, Infinity and Unions

Q: What is the difference between ( and [ in interval notation?

A square bracket [ includes the endpoint; a round bracket ( excludes it. So [2, 5) includes 2 but not 5.

Q: Why does infinity always get a round bracket?

Infinity is not a number that can be reached or included, so an interval can never contain it. The round bracket shows the range goes on without end.

How to read and write intervals like $[a, b)$ and $(-\infty, 3)$ , how they match inequalities and number-line pictures, how to solve compound inequalities, and how to join pieces with a union.

3 min read Supports 3 lessons

Calculus constantly describes where things happen: where a function is defined, where it is increasing, where it is positive. Writing "all numbers between $-2$ and $5$ , including $5$ but not $-2$ " every time would be exhausting. Interval notation says it in five characters: $(-2, 5]$ .

$-2 < x \le 5$ is the interval $(-2, 5]$ .

The brackets

Brackets

a \le x \le b \;\Leftrightarrow\; [a, b] \qquad a < x < b \;\Leftrightarrow\; (a, b)

A square bracket means the endpoint is included (a filled dot). A round bracket means it is excluded (an open dot). They can be mixed: $[a, b)$ .

Tip

A square bracket has corners that "hold on" to the endpoint. A round bracket curves away from it.

Infinity

When a range goes on for ever, use $\infty$ or $-\infty$ as the endpoint. Infinity is not a number you can reach, so it always gets a round bracket.

Unbounded intervals

x \ge 3 \;\Leftrightarrow\; [3, \infty) \qquad x < 1 \;\Leftrightarrow\; (-\infty, 1)

All real numbers is $(-\infty, \infty)$ .

Watch out

Never write $[3, \infty]$ . You cannot include infinity, because there is nothing there to include.

Compound inequalities

A compound inequality like $-3 \le 2x + 1 < 7$ squeezes $x$ from both sides. Solve it by doing the same thing to all three parts at once.

$-3 \le 2x + 1 < 7$
Three parts. Whatever you do, do it to all of them.

1/4

Watch out

Dividing by a negative flips both signs, so the ends swap over: $-4 < -2x \le 6$ becomes $-3 \le x < 2$ , which is $[-3, 2)$ .

Unions: "or"

Sometimes the answer is two separate pieces, such as $x < -1$ or $x \ge 4$ . Join them with the union symbol $\cup$ .

$x < -1$ or $x \ge 4$ is $(-\infty, -1) \cup [4, \infty)$ .

(-\infty, -1) \cup [4, \infty)

Key idea

You will use intervals to state domains and ranges, to describe where a graph rises or falls, and to say how close $x$ must be to a point in the definition of a limit.

Common questions

What is the difference between ( and [ in interval notation?: A square bracket $[$ includes the endpoint; a round bracket $($ excludes it. So $[2, 5)$ includes $2$ but not $5$ .
Why does infinity always get a round bracket?: Infinity is not a number that can be reached or included, so an interval can never contain it. The round bracket shows the range goes on without end.

Practise interval notation

The Compound Inequalities & Interval Notation lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What is the difference between ( and [ in interval notation?

A square bracket $[$ includes the endpoint; a round bracket $($ excludes it. So $[2, 5)$ includes $2$ but not $5$ .

Why does infinity always get a round bracket?

Infinity is not a number that can be reached or included, so an interval can never contain it. The round bracket shows the range goes on without end.