July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a crucial topic that students are required understand because it becomes more critical as you advance to higher math.

If you see more complex mathematics, something like integral and differential calculus, in front of you, then knowing the interval notation can save you time in understanding these concepts.

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

What Is Interval Notation?

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

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

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

Though, intervals are usually used to denote domains and ranges of functions in advanced mathematics. Expressing these intervals can progressively become difficult as the functions become progressively more complex.

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

  • x is greater than negative four but less than 2

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

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

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

Types of Intervals

Many types of intervals lay the foundation for writing the interval notation. These kinds of interval are essential 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} express x as being greater than negative four but less than two, which means that it does not include 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 represent that in a given set of real numbers, such as the interval between negative four and two, those 2 values are not included.

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


A closed interval is the opposite of the previous type of interval. Where the open interval does exclude the values mentioned, a closed interval does. In word form, a closed interval is expressed as any value “greater than or equal to” or “less than or equal to.”

For example, if the previous 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 would be expressed as {x | -4 < x < 2}.

In an interval notation, this is stated 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 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 prior example for assistance, if the interval were half-open, it would be expressed as “x is greater than or equal to negative four and less than two.” This implies that x could be the value -4 but couldn’t possibly be equal to the value 2.

In an inequality notation, this would be expressed 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 signifies the value excluded from the subset.

Symbols for Interval Notation and Types of Intervals

To summarize, 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 subjected to interval notation.

These symbols build the actual interval notation you create when expressing 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 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 within the two. In this instance, 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 Different Interval Types

Apart from being written with symbols, the different 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 show 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 need 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 straightforward 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 participate in a debate competition, they should have a minimum of three teams. Express this equation in interval notation.

In this word question, 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 states that 3 is a closed value.

Additionally, because no maximum number was referred to regarding 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 written 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 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 describe this range in interval notation?

In this word problem, the value 1800 is the lowest while the value 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, 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 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 fundamentally a way of representing inequalities on the number line.

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

How Do You Convert Inequality to Interval Notation?

An interval notation is basically 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 value should be written 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 prior to see how these symbols are utilized.

How To Exclude Numbers in Interval Notation?

Values ruled out from the interval can be written with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which means that the value is ruled out from the set.

Grade Potential Can Help You Get a Grip on Arithmetics

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