July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental principle that learners need to understand because it becomes more important as you grow to more difficult math.

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

This article will talk about what interval notation is, what are its uses, and how you can interpret it.

What Is Interval Notation?

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

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

Fundamental problems you encounter primarily consists of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such effortless applications.

Though, intervals are usually used to denote domains and ranges of functions in more complex arithmetics. 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 4 but less than two

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

So far we understand, interval notation is a way to write intervals concisely and elegantly, using set rules that help writing and understanding intervals on the number line easier.

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

Types of Intervals

Several types of intervals lay the foundation for denoting the interval notation. These interval types are essential to get to know due to the fact they underpin the entire notation process.


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

The inequality notation {x | -4 < x < 2} express x as being greater than negative four but less than two, meaning that it does not include neither of the two numbers mentioned. As such, this is an open interval expressed with parentheses or a round bracket, such as the following.

(-4, 2)

This implies that in a given set of real numbers, such as the interval between negative four and two, 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 text form, a closed interval is expressed 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 negative four and less than or equal to two.”

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

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

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


A half-open interval is a blend of previous 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 couldn’t 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 denoted 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

In brief, there are different types of interval notations; open, closed, and half-open. An open interval doesn’t include 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 last example, there are various symbols for these types under the interval notation.

These symbols build the actual interval notation you create 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 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 excluded in the set, and the right endpoint is not excluded. Also known as 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 included in the set, while the right endpoint is not included. This is also called a right-open interval.

Number Line Representations for the Various Interval Types

Apart from being denoted with symbols, the different interval types can also be represented in the number line utilizing both shaded and open circles, relying 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 prepared 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 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 written as (-6, 9].

Example 2

For a school to join in a debate competition, they require minimum of three teams. Represent this equation in interval notation.

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

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

Furthermore, because no upper limit was stated with concern to the number of maximum 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, 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 undertake a diet program constraining their daily calorie intake. For the diet to be a success, they should have minimum of 1800 calories every day, but no more than 2000. How do you write this range in interval notation?

In this question, the number 1800 is the minimum while the value 2000 is the maximum value.

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

Therefore, the interval notation is written 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 Frequently Asked Questions

How To Graph an Interval Notation?

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

There are rules of expressing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is denoted 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 Do You Change Inequality to Interval Notation?

An interval notation is basically a diverse way of expressing an inequality or a set of real numbers.

If x is higher than or lower than 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 denoted with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are employed.

How To Rule Out Numbers in Interval Notation?

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

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