July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a essential principle that pupils are required understand because it becomes more critical as you progress to more difficult arithmetic.

If you see more complex arithmetics, something like differential calculus and integral, on your horizon, then knowing the interval notation can save you hours in understanding these theories.

This article will talk about what interval notation is, what it’s used for, and how you can understand it.

What Is Interval Notation?

The interval notation is merely a way to express a subset of all real numbers across the number line.

An interval refers to the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)

Basic problems you encounter essentially consists of one positive or negative numbers, so it can be challenging to see the utility of the interval notation from such effortless utilization.

Despite that, intervals are generally used to denote domains and ranges of functions in advanced arithmetics. Expressing these intervals can increasingly become difficult as the functions become more complex.

Let’s take a straightforward compound inequality notation as an example.

  • x is higher than negative four but less than 2

Up till now we understand, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. However, it can also be written with interval notation (-4, 2), denoted by values a and b separated by a comma.

So far we understand, interval notation is a method of writing intervals concisely and elegantly, using fixed rules that help writing and understanding intervals on the number line simpler.

In the following section we will discuss regarding the principles of expressing a subset in a set of all real numbers with interval notation.

Types of Intervals

Various types of intervals lay the foundation for writing the interval notation. These interval types are essential to get to know because they underpin the complete notation process.


Open intervals are used when the expression do not include the endpoints of the interval. The prior notation is a great example of this.

The inequality notation {x | -4 < x < 2} describes x as being higher than -4 but less than 2, which means that it excludes neither of the two numbers mentioned. As such, this is an open interval denoted with parentheses or a round bracket, such as the following.

(-4, 2)

This means that in a given set of real numbers, such as the interval between -4 and 2, those two values are not included.

On the number line, an unshaded circle denotes an open value.


A closed interval is the contrary of the last type of interval. Where the open interval does exclude the values mentioned, a closed interval does. In word form, a closed interval is written as any value “higher than or equal to” or “less than or equal to.”

For example, if the last example was a closed interval, it would read, “x is greater than or equal to -4 and less than or equal to two.”

In an inequality notation, this can be expressed as {x | -4 < x < 2}.

In an interval notation, this is expressed with brackets, or [-4, 2]. This means that the interval contains those two boundary values: -4 and 2.

On the number line, a shaded circle is used to represent an included open value.


A half-open interval is a blend of prior types of intervals. Of the two points on the line, one is included, and the other isn’t.

Using the previous example as a guide, if the interval were half-open, it would be expressed as “x is greater than or equal to -4 and less than two.” This means that x could be the value -4 but cannot possibly be equal to the value 2.

In an inequality notation, this would be denoted as {x | -4 < x < 2}.

A half-open interval notation is written with both a bracket and a parenthesis, or [-4, 2).

On the number line, the shaded circle denotes the number included in the interval, and the unshaded circle signifies the value excluded from the subset.

Symbols for Interval Notation and Types of Intervals

To recap, there are different types of interval notations; open, closed, and half-open. An open interval doesn’t contain the endpoints on the real number line, while a closed interval does. A half-open interval consist of one value on the line but does not include the other value.

As seen in the prior example, there are different symbols for these types under the interval notation.

These symbols build the actual interval notation you develop when plotting points on a number line.

  • ( ): The parentheses are utilized when the interval is open, or when the two endpoints on the number line are excluded from the subset.

  • [ ]: The square brackets are used when the interval is closed, or when the two points on the number line are included in the subset of real numbers.

  • ( ]: Both the parenthesis and the square bracket are employed when the interval is half-open, or when only the left endpoint is not included in the set, and the right endpoint is included. Also known as a left open interval.

  • [ ): This is also a half-open notation when there are both included and excluded values within the two. In this instance, the left endpoint is included in the set, while the right endpoint is not included. This is also known as a right-open interval.

Number Line Representations for the Various Interval Types

Apart from being written with symbols, the various interval types can also be represented in the number line employing both shaded and open circles, depending on the interval type.

The table below will display all the different types of intervals as they are described in the number line.

Interval Notation


Interval Type

(a, b)

{x | a < x < b}


[a, b]

{x | a ≤ x ≤ b}


[a, ∞)

{x | x ≥ a}


(a, ∞)

{x | x > a}


(-∞, a)

{x | x < a}


(-∞, a]

{x | x ≤ a}


Practice Examples for Interval Notation

Now that you’ve understood everything you are required to know about writing things in interval notations, you’re ready for a few practice problems and their accompanying solution set.

Example 1

Transform the following inequality into an interval notation: {x | -6 < x < 9}

This sample problem is a easy conversion; simply utilize the equivalent symbols when stating the inequality into an interval notation.

In this inequality, the a-value (-6) is an open interval, while the b value (9) is a closed one. Thus, it’s going to be expressed as (-6, 9].

Example 2

For a school to take part in a debate competition, they need at least three teams. Represent this equation in interval notation.

In this word question, let x stand for the minimum number of teams.

Since the number of teams required is “three and above,” the number 3 is consisted in the set, which states that three is a closed value.

Furthermore, since no maximum number was referred to regarding the number of teams a school can send to the debate competition, this value should be positive to infinity.

Thus, the interval notation should be expressed as [3, ∞).

These types of intervals, when one side of the interval that stretches to either positive or negative infinity, are also known as unbounded intervals.

Example 3

A friend wants to participate in diet program limiting their regular calorie intake. For the diet to be a success, they should have minimum of 1800 calories every day, but maximum intake restricted to 2000. How do you express this range in interval notation?

In this word problem, the number 1800 is the lowest while the number 2000 is the highest value.

The question suggest that both 1800 and 2000 are included in the range, so the equation is a close interval, written with the inequality 1800 ≤ x ≤ 2000.

Therefore, the interval notation is written as [1800, 2000].

When the subset of real numbers is confined to a range between two values, and doesn’t stretch to either positive or negative infinity, it is called a bounded interval.

Interval Notation FAQs

How Do You Graph an Interval Notation?

An interval notation is fundamentally a way of describing inequalities on the number line.

There are laws of expressing an interval notation to the number line: a closed interval is expressed with a filled circle, and an open integral is denoted with an unshaded circle. This way, you can promptly check the number line if the point is excluded or included from the interval.

How To Convert Inequality to Interval Notation?

An interval notation is just a different way of describing an inequality or a set of real numbers.

If x is greater than or less a value (not equal to), then the number should be written with parentheses () in the notation.

If x is greater than or equal to, or less than or equal to, then the interval is expressed with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are utilized.

How To Rule Out Numbers in Interval Notation?

Values ruled out from the interval can be written with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which means that the number is excluded from the combination.

Grade Potential Could Help You Get a Grip on Mathematics

Writing interval notations can get complex fast. There are multiple nuanced topics within this concentration, such as those dealing with the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and many more.

If you desire to conquer these ideas fast, you need to revise them with the expert assistance and study materials that the professional instructors of Grade Potential provide.

Unlock your mathematic skills with Grade Potential. Book a call now!