July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental concept that students are required grasp because it becomes more essential as you grow to higher arithmetic.

If you see advances mathematics, such as differential calculus and integral, in front of you, then being knowledgeable of interval notation can save you time in understanding these concepts.

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 through the number line.

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

Fundamental difficulties you face primarily consists of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such straightforward utilization.

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

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

  • x is greater 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. Though, it can also be written with interval notation (-4, 2), signified by values a and b segregated by a comma.

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

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

Various types of intervals lay the foundation for writing the interval notation. These kinds of interval are essential to get to know because they underpin the complete notation process.


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

The inequality notation {x | -4 < x < 2} describes x as being greater than -4 but less than 2, meaning that it does not contain 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 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 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 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 would be expressed as {x | -4 < x < 2}.

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

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


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

Using the last 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 two.” This implies that x could be the value negative four but cannot 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 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 which are not included 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 excludes 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 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 included 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 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 included 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

Aside from being written with symbols, the different interval types can also be described 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 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 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 problem is a straightforward 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 require minimum of three teams. Represent this equation in interval notation.

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

Since the number of teams needed is “three and above,” the number 3 is included on the set, which implies that 3 is a closed value.

Furthermore, 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.

Thus, 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 regular calorie intake. For the diet to be a success, they must have minimum of 1800 calories every day, but maximum intake restricted to 2000. How do you express this range in interval notation?

In this word problem, the number 1800 is the minimum while the number 2000 is the highest value.

The question 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 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 also known as a bounded interval.

Interval Notation FAQs

How To Graph an Interval Notation?

An interval notation is simply a technique 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 quickly check the number line if the point is included or excluded from the interval.

How To Change Inequality to Interval Notation?

An interval notation is just a different way of expressing 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 expressed with parentheses () in the notation.

If x is higher 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 check how these symbols are used.

How To Exclude Numbers in Interval Notation?

Values excluded from the interval can be denoted with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which states that the number is ruled out from the set.

Grade Potential Can Help You Get a Grip on Mathematics

Writing interval notations can get complex fast. There are multiple nuanced topics in this concentration, such as those working on the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and more.

If you want to conquer these concepts quickly, you are required to revise them with the professional help and study materials that the professional teachers of Grade Potential delivers.

Unlock your math skills with Grade Potential. Connect with us now!