Interval Notation

We can think of an interval as a subset of real numbers. For instance, the set of integers $\mathbb{Z}$ is a subset of the set of real numbers $\mathbb{R}$.

So an interval notation is simply a compact way of representing subsets of real numbers using two numbers (left and right endpoints), the comma symbol, parentheses ( ), brackets [ ], and infinity symbols (positive or negative).

If you want to see how much you already know about interval notation, you can click on the link below to do some practice problems before moving on to the lesson.

Interval Notation Practice Problems with Answers

Here’s a quick example of an interval notation showing its parts:

Geometrically, it can be thought of as a line segment which is a part (i.e. portion) of the real number line.

An interval notation may look exactly like a point so don’t mix things up! Here is an example.

What does the interval $\large{(a,b)}$ represent?

• It is not a point. $\large{a}$ does not stand for the x-coordinate and $\large{b}$ does not stand for the y-coordinate.
• However, it represents all real numbers between $\large{a}$ and $\large{b}$.

Let’s go over the different cases of interval notations.

Cases of Interval Notations

Case 1: $\Large{\left( {{\color{red}a},\infty } \right)}$

• The left endpoint is $\large{a}$ and the use of the parenthesis ( ) implies that $\large{a}$ is not included (excluded).
• The right endpoint is not a real number (positive infinity $+\infty$ symbol), it suggests that the interval is unbounded on the right which means that it goes on indefinitely.

In words:

All real numbers that are greater than $\large{a}$.

As a set builder notation:

$\large{\{ {x|x > a}\}}$

As an inequality:

$\large{x > a}$

As a graph in number line:

We use a hollow dot (unshaded) for $\color{red}\large{a}$ to indicate that it not included in the interval.

Case 2: $\Large{\left[ {{\color{red}a},\infty } \right)}$

• The left endpoint is $\large{a}$ and the use of the square bracket [ ] implies that $\large{a}$ is included.
• The right endpoint is positive infinity ($+\infty$) which implies that the interval goes on indefinitely to the right.

In words:

All real numbers that are greater than or equal to $\large{a}$.

As a set builder notation:

$\large{\{ {x|x \ge a}\}}$

As an inequality:

$\large{x \ge a}$

As a graph in number line:

We use a solid dot (shaded) for $\color{red}\large{a}$ to indicate that it included in the interval.

Case 3: $\Large{\left( { – \infty ,{\color{blue}b}} \right)}$

• The left endpoint is not a number rather the negative infinity symbol ($\large{- \infty}$). It means that the interval is unbounded on the left.
• The right endpoint is $\large{b}$ and the use of the parenthesis ( ) implies that $\large{b}$ is not included (excluded).

In words:

All real numbers that are less than $\large{b}$.

As a set builder notation:

$\large{\{ {x|x < b}\}}$

As an inequality:

$\large{x < b}$

As a graph in number line:

We use a hollow dot (unshaded) for $\color{blue}\large{b}$ to indicate that it not included in the interval.

Case 4: $\Large{\left( { – \infty ,{\color{blue}b}} \right]}$

• The left endpoint is negative infinity ($\large{- \infty}$) thus it goes on to the left indefinitely.
• The right endpoint is $\large{b}$ and the use of the square bracket [ ] implies that $\large{b}$ is included in the interval.

In words:

All real numbers that are less than or equal to $\large{b}$.

As a set builder notation:

$\large{\{ {x|x \le b}\}}$

As an inequality:

$\large{x \le b}$

As a graph in number line:

We use a solid dot (shaded) for $\color{blue}\large{b}$ to indicate that it not included in the interval.

Case 5: $\Large{\left( {{\color{red}a},{\color{blue}b}} \right)}$

• $\large{a}$ is the left endpoint, and the parenthesis ( ) indicates that $\large{a}$ is not included.
• $\large{b}$ is the right endpoint, and the parenthesis ( ) indicates that $\large{b}$ is not included (excluded).
• This type of interval is known as open interval.

In words:

All real numbers between $\large{a}$ and $\large{b}$, but not including both $\large{a}$ and $\large{b}$.

As a set builder notation:

$\large{\{ {x|a < x < b}\}}$

As an inequality:

$\large{a < x < b}$

As a graph in number line:

Both $\color{red}\large{a}$ and $\color{blue}\large{b}$ are not included in the interval so we use hollow dots.

Case 6: $\Large{\left[ {{\color{red}a},{\color{blue}b}} \right)}$

• $\large{a}$ is the left endpoint, and the square bracket [ ] indicates that $\large{a}$ is included.
• $\large{b}$ is the right endpoint, and the parenthesis ( ) indicates that $\large{b}$ is not included (excluded).
• This type of internal is known as half-open or half-closed interval.

In words:

All real numbers between $\large{a}$ and $\large{b}$, including $\large{a}$ but not including $\large{b}$.

As a set builder notation:

$\large{\{ {x|a \le x < b}\}}$

As an inequality:

$\large{a \le x < b}$

As a graph in number line:

We use a solid dot for $\color{red}\large{a}$ because it included while a hollow dot for $\color{blue}\large{b}$ because it is not included.

Case 7: $\Large{\left( {{\color{red}a},{\color{blue}b}} \right]}$

• $\large{a}$ is the left endpoint, and the parenthesis ( ) indicates that $\large{a}$ is not included (excluded).
• $\large{b}$ is the right endpoint, and the square bracket [ ] indicates that $\large{b}$ is included.
• This type of interval is known as half-open or half-closed interval.

In words:

All real numbers between $\large{a}$ and $\large{b}$, not including (excluding) $\large{a}$ but including $\large{b}$.

As a set builder notation:

$\large{\{ {x|a < x \le b}\}}$

As an inequality:

$\large{a < x \le b}$

As a graph in number line:

We use a hollow dot for $\color{red}\large{a}$ because it not included while a solid dot for $\color{blue}\large{b}$ because it is included.

Case 8: $\Large{\left[ {{\color{red}a},{\color{blue}b}} \right]}$

• $\large{a}$ is the left endpoint, and the parenthesis [ ] indicates that $\large{a}$ is included in the interval.
• $\large{b}$ is the right endpoint, and the square bracket [ ] indicates that $\large{b}$ is included in the interval.
• This type of interval is known as closed interval.

In words:

All real numbers between $\large{a}$ and $\large{b}$, including both $\large{a}$ and $\large{b}$.

As a set builder notation:

$\large{\{ {x|a \le x \le b}\}}$

As an inequality:

$\large{a \le x \le b}$

As a graph in number line:

Both $\color{red}\large{a}$ and $\color{blue}\large{b}$ are included in the interval so we use solid dots.

Case 9: $\Large{\left( { {\color{red}- \infty} ,{\color{blue}\infty} } \right)}$

• The negative infinity symbol ($\color{red}{ – \infty }$) at the left endpoint denotes that the interval is unbounded to the LEFT.
• The positive infinity symbol ($\color{blue}{ + \infty }$) at the right endpoint denotes that the interval is unbounded to the RIGHT.

In words:

All real numbers

As a set builder notation:

As a graph in number line:

You may also be interested in these related math lessons or tutorials:

Interval Notation Practice Problems with Answers