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 need to understand owing to the fact that it becomes more critical as you advance to higher arithmetic.

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

This article will talk in-depth what interval notation is, what are its uses, and how you can decipher it.

What Is Interval Notation?

The interval notation is merely 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.)

Basic problems you encounter primarily composed of single positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such simple applications.

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

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

  • x is higher than negative four but less than two

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

So far we understand, interval notation is a way to write intervals concisely and elegantly, using set principles that make writing and comprehending intervals on the number line less difficult.

In the following section we will discuss about the principles 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 important to get to know because they underpin the entire notation process.


Open intervals are applied when the expression do not contain the endpoints of the interval. The previous notation is a great example of this.

The inequality notation {x | -4 < x < 2} describes x as being greater 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 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 -4 and 2, those 2 values are excluded.

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


A closed interval is the contrary of the previous type of interval. Where the open interval does not contain 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 negative four and less than or equal to 2.”

In an inequality notation, this would be written 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 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 negative four and less than 2.” This means that x could be the value negative four but couldn’t possibly be equal to the value two.

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 present 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 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 excludes the other value.

As seen in the prior example, there are numerous 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 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 used 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 employed when the interval is half-open, or when only the left endpoint is not included in the set, and the right endpoint is not excluded. 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 instance, the left endpoint is included in the set, while the right endpoint is excluded. This is also known as a right-open interval.

Number Line Representations for the Different Interval Types

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

The table below will display 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 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 problem is a simple conversion; simply use the equivalent symbols when denoting 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 participate in a debate competition, they should have a minimum of 3 teams. Express this equation in interval notation.

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

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

Additionally, since no upper limit was referred to with concern to the number of maximum teams a school can send to the debate competition, this value should be positive to infinity.

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

These types of intervals, when one side of the interval that stretches to either positive or negative infinity, are called 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 must have at least 1800 calories every day, but maximum intake restricted to 2000. How do you describe this range in interval notation?

In this question, the value 1800 is the lowest while the value 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, expressed with the inequality 1800 ≤ x ≤ 2000.

Thus, the interval notation is described as [1800, 2000].

When the subset of real numbers is restricted to a variation 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 basically a technique of describing inequalities on the number line.

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

How Do You Change 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 greater than or less a value (not equal to), then the value should be stated with parentheses () in the notation.

If x is greater than or equal to, or lower 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 used.

How Do You Rule Out Numbers in Interval Notation?

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

Grade Potential Could Help You Get a Grip on Math

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