July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a crucial principle that students should learn owing to the fact that it becomes more important as you advance to more difficult arithmetic.

If you see higher mathematics, something like integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you time in understanding these ideas.

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

What Is Interval Notation?

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

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

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

Though, intervals are typically used to denote domains and ranges of functions in higher arithmetics. Expressing these intervals can increasingly become difficult 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 2

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

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

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

Various types of intervals place the base for denoting the interval notation. These interval types are important to get to know due to the fact they underpin the complete notation process.


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

The inequality notation {x | -4 < x < 2} describes x as being higher than negative four but less than two, meaning that it does not include 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 represent 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 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 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 -4 and less than or equal to two.”

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

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

On the number line, a shaded circle is utilized to denote 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 previous example for assistance, if the interval were half-open, it would read 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 written 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 present 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 doesn’t contain the endpoints on the real number line, while a closed interval does. A half-open interval includes 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 stating points on a number line.

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

  • [ ]: The square brackets are utilized 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 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 between the two. In this instance, the left endpoint is not excluded 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 various interval types can also be described in the number line utilizing 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’ve understood everything you need to know about writing things in interval notations, you’re prepared 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 utilize 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 participate in a debate competition, they need minimum of three teams. Express this equation in interval notation.

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

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

Plus, since no maximum number was mentioned 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 called unbounded intervals.

Example 3

A friend wants to participate in diet program constraining their daily calorie intake. For the diet to be successful, they should have at least 1800 calories every day, but no more than 2000. How do you describe this range in interval notation?

In this question, the number 1800 is the minimum while the number 2000 is the maximum 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 described as [1800, 2000].

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

Interval Notation Frequently Asked Questions

How To Graph an Interval Notation?

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

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

How Do You Transform Inequality to Interval Notation?

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

If x is higher than or lower than a value (not equal to), then the value should be stated with parentheses () in the notation.

If x is higher than or equal to, or lower than or equal to, then the interval is denoted with closed brackets [ ] in the notation. See the examples of interval notation above to check how these symbols are utilized.

How Do You Exclude Numbers in Interval Notation?

Numbers excluded from the interval can be stated with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which means that the value is excluded from the set.

Grade Potential Could Guide You Get a Grip on Math

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