July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a essential principle that students are required learn because it becomes more essential as you grow to more complex arithmetic.

If you see advances arithmetics, such as integral and differential calculus, in front of you, then knowing the interval notation can save you time in understanding these theories.

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

What Is Interval Notation?

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

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

Fundamental problems you encounter primarily composed of single positive or negative numbers, so it can be challenging to see the utility of the interval notation from such straightforward utilization.

However, intervals are usually employed to denote domains and ranges of functions in more complex math. Expressing these intervals can increasingly become complicated as the functions become more tricky.

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

  • x is higher than negative four but less than two

Up till now we understand, this inequality notation can be expressed 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.

As we can see, interval notation is a method of writing intervals concisely and elegantly, using set rules that make writing and understanding intervals on the number line easier.

The following sections will tell us more about the rules of expressing a subset in a set of all real numbers with interval notation.

Types of Intervals

Many types of intervals place the base for denoting the interval notation. These kinds of interval are important to get to know because they underpin the entire notation process.


Open intervals are applied when the expression does not contain the endpoints of the interval. The last notation is a fine example of this.

The inequality notation {x | -4 < x < 2} describes x as being more than negative four but less than two, which means that it does not contain either of the two numbers referred to. 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 negative four and two, those two values are excluded.

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


A closed interval is the opposite of the last type of interval. Where the open interval does not include the values mentioned, a closed interval does. In text form, a closed interval is written as any value “greater 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 implies that the interval consist of those two boundary values: -4 and 2.

On the number line, a shaded circle is employed to describe an included open value.


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

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

In an inequality notation, this would be expressed 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 denotes the value which are not included 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 excludes the endpoints on the real number line, while a closed interval does. A half-open interval consist of one value on the line but excludes the other value.

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

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

  • ( ): The parentheses are used when the interval is open, or when the two endpoints on the number line are not included in the subset.

  • [ ]: The square brackets are employed when the interval is closed, or when the two points on the number line are not excluded in the subset of real numbers.

  • ( ]: Both the parenthesis and the square bracket are utilized 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 called a left open interval.

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

Number Line Representations for the Different Interval Types

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

The table below will show all the different types of intervals as they are represented 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 know 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

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

This sample question is a simple conversion; just use the equivalent symbols when writing 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 should have a at least 3 teams. Represent this equation in interval notation.

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

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

Plus, since no maximum number was stated with concern to the number of teams a school can send to the debate competition, this number should be positive to infinity.

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

These types of intervals, where there is 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 do a diet program constraining their regular calorie intake. For the diet to be successful, they should have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you write this range in interval notation?

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

The problem suggest that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, denoted with the inequality 1800 ≤ x ≤ 2000.

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

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

Interval Notation FAQs

How To Graph an Interval Notation?

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

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

How To Transform Inequality to Interval Notation?

An interval notation is just a different technique of expressing an inequality or a combination of real numbers.

If x is greater than or less a value (not equal to), then the number should be expressed 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 employed.

How Do You Rule Out Numbers in Interval Notation?

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

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