An interval is used to represent a set of values that lies between two numbers, or a set of values that are greater than or less than a specific number. Intervals are usually represented algebraically with inequalities and graphically with a number line. This is done by marking the endpoints of the interval and drawing a line segment between them.

An interval consisting of real numbers contains an infinite amount of numbers. In the interval $1\le x\le 4,$ all numbers between $1$ and $4$ are part of the set. For example, $1,$ $2.7,$ $3.5,$ $3.61,$ and $4.$ An interval does not have to be limited between two ends. For example, the inequality $x\ge 0$ is the set of all positive real numbers greater than or equal to $0.$

Interval Notation

Interval notation is a way to denote an interval. In this notation, a square bracket, $[$ or $],$ is used when the end value is included, and a round bracket, $($ or $),$ is used when the end value is not included. When using interval notation, the left value is less than the right value.

Interval Notation	Description	Type of Interval
$(a,b)$	Both ends are not included	Open interval
$[a,b]$	Both ends are included	Closed interval
$[a,b)$ or $(a,b]$	One end is not included	Half-open interval

The following applet shows some more examples of intervals. Note that if the variable only has one limit, the other is infinity, $\infty$ or $\text{-}\infty ,$ and is not included in the interval.