Interval notation is used to represent the set of all real numbers between two endpoints. The two endpoints are listed within a set of brackets.
Square brackets \([ \: ]\), or just "brackets," are used to indicate closed intervals (intervals that include both endpoints). Rounded brackets \(( \: )\), or "parentheses," are used for open intervals (intervals that do not include either endpoint). An interval can also be half-closed or half-open, if it contains one of the endpoints but not the other. This would be indicated with one square bracket and one rounded bracket.
For sets of real numbers with no upper boundary, the upper endpoint is written as \(\infty\) (infinity) with a rounded bracket in interval notation. For sets of real numbers with no lower boundary, the lower endpoint is written as \(-\infty\) (negative infinity).
Here are some examples of intervals illustrated as inequalities, number lines, and interval notation. Remember that on a number line, a solid dot means that the number is part of the interval, while an open dot means that it is not.
| Inequality |
Number Line |
Interval Notation |
| \(0 \leq x \leq 4\) |
 |
\(x \in [0, 4]\) |
| \(-3 \lt x \lt 3\) |
 |
\(x \in (-3, 3)\) |
| \(-4 \leq x \lt 1\) |
 |
\(x \in [-4, 1)\) |
| \(-1 \lt x \leq 2\) |
 |
\(x \in (-1, 2]\) |
| \(x \lt 0\) |
 |
\(x \in (-\infty, 0)\) |
| \(x \leq -3\) |
 |
\(x \in (-\infty, -3]\) |
| \(x \gt 1\) |
 |
\(x \in (1, \infty)\) |
| \(x \ge -2\) |
 |
\(x \in [-2, \infty)\) |
| \(x \in \mathbb{R}\) (\(x\) is any real number) |
 |
\(x \in (-\infty, \infty)\) |
Notice that \(-\infty\) or \(\infty\) are always listed with rounded brackets, since these are not defined endpoints. Also note that the symbol \(\in\) means "belongs to", so the notation \(x \in [0, 4]\)means that \(x\) belongs to (or is part of) the interval \( [0, 4]\).
Sometimes we need to describe a set of numbers that consists of two or more distinct intervals.
In this situation, we can use interval notation in combination with the \(\cup\) symbol (the symbol for the union of two sets).
Here are some examples:
| Inequality |
Number Line |
Interval Notation |
| \(x \lt 0\) or \(x \geq 2\) |
 |
\(x \in (-\infty, 0)\cup [2, \infty)\) |
| \(x \in \mathbb{R}, x \neq -1\) (i.e., \(x\) is any real number except \(-1\)) |
 |
\(x \in (-\infty, -1)\cup(-1, \infty)\) |
Interval notation can be useful when solving inequalities. It can be used to express the solutions to inequalities (which often consist of a range or set of numbers) and will appear occasionally throughout the rest of this lesson.