What is Interval? — Mathematics

Definition 1.6 High School Mathematics · Chapter 1 — Numbers and Sets of Numbers

Let $a$ and $b$ be real numbers with $a < b$ . An interval is the set of all real numbers between two bounds. A square bracket means the bound is included, a parenthesis means it is excluded:

notation	description
$\intcc{a}{b}$	$a \leq x \leq b$ (both ends included)
$\intoo{a}{b}$	$a < x < b$ (both ends excluded)
$\intco{a}{b}$	$a \leq x < b$
$\intoc{a}{b}$	$a < x \leq b$
$\intco{a}{+\infty}$	$x \geq a$
$\intoo{-\infty}{b}$	$x < b$

The symbols $-\infty$ and $+\infty$ (“infinity”) are not numbers, only a way of saying that the interval continues forever; the bracket next to them is always a parenthesis. The whole line $\R$ is the interval $\intoo{-\infty}{+\infty}$ .

Intervals on the number line: a filled dot for an included bound, a hollow dot for an excluded one.

Examples

Example 1.13

Solve $\abs{x - 3} \leq 2$ . The solutions are the numbers at distance at most $2$ from $3$ : the interval $\intcc{1}{5}$ . Conversely, the interval $\intcc{-1}{7}$ has center $\frac{-1+7}{2} = 3$ and radius $\frac{7-(-1)}{2} = 4$ , so it is described by $\abs{x - 3} \leq 4$ .

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Examples

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