# Interval Notation

**Interval notation** is a way to describe continuous sets of real numbers by the numbers that bound them. **Intervals**, when written, look somewhat like ordered pairs. However, they are not meant to denote a specific point. Rather, they are meant to be a shorthand way to write an inequality or system of inequalities.

## Writing Interval Notation

Intervals are written with rectangular brackets or parentheses, and two numbers delimited with a comma. The number on the left denotes the least element or lower bound. The number on the right denotes the greatest element or upper bound.

The rectangular bracket symbols, \([\ ],\) are used to describe sets with a "less than or equal to" or a "greater than or equal to" element, respectively. They correspond to the \(\ge\) and \(\le\) symbols:

\[\begin{array}{lc} \text{Inequality:} & 3 \le x \le 9 \\ \text{Interval:} & [3,9]. \end{array}\]

In this case, \(x\) could equal \(3\) or \(9\).

The parentheses symbols, \( (\ ), \) are used to describe sets with a lower bound or upper bound, respectively. They correspond to the \(>\) and \(<\) symbols:

\[\begin{array}{lc} \text{Inequality:} & -1 < x < 4 \\ \text{Interval:} & (-1,4). \end{array}\]

In this case, \(x\) does not equal \(-1\) or \(4\).

The different types of brackets can be used in the same interval:

\[\begin{array}{lc} \text{Inequality:} & -3 \le x < 5 \\ \text{Interval:} & [-3,5) \end{array}\]

If an interval has no lower bound or upper bound, then the \(-\infty\) or \(\infty\) symbols are used. These symbols are always used with a parentheses bracket, because infinity is not a number that can be included in a set:

\[\begin{array}{lc} \text{Inequality:} & x \le 7 \\ \text{Interval:} & (-\infty,7] \end{array}\]

\[\begin{array}{lc} \text{Inequality:} & x >2 \\ \text{Interval:} & (2,\infty). \end{array}\]

Intersections and unions of intervals can be written with the \(\cap\) or \(\cup\) symbols:

\[\begin{array}{lc} \text{Inequality:} & x \le -4\ \cup\ 0 < x < 6 \\ \text{Interval:} & (-\infty,-4]\ \cup\ (0,6) \end{array}\]

\[\begin{array}{lc} \text{Inequality:} & x \ne 1\\ \text{Intervals:} & (-\infty,1)\ \cup\ (1,\infty). \end{array}\]

**Cite as:**Interval Notation.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/interval-notation/