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 essential as you grow to more complex math.

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

This article will talk about what interval notation is, what it’s used for, 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 refers to the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ signifies infinity.)

Fundamental problems you encounter mainly consists of one positive or negative numbers, so it can be challenging to see the utility of the interval notation from such simple applications.

Despite that, intervals are typically used to denote domains and ranges of functions in advanced math. Expressing these intervals can increasingly become complicated as the functions become more complex.

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

  • x is greater than negative 4 but less than 2

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

As we can see, interval notation is a way to write intervals concisely and elegantly, using predetermined principles that make writing and understanding intervals on the number line easier.

The following sections will tell us more about the principles 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 kinds of interval are essential to get to know due to the fact they underpin the entire notation process.


Open intervals are applied when the expression does not include the endpoints of the interval. The last notation is a good example of this.

The inequality notation {x | -4 < x < 2} express 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 means that in a given set of real numbers, such as the interval between negative four and two, those 2 values are excluded.

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 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 -4 and less than or equal to 2.”

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 states that the interval includes 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 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 be expressed as “x is greater than or equal to -4 and less than two.” This states 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 denotes the value excluded 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 include 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 last 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 utilized when the interval is half-open, or when only the left endpoint is excluded in the set, and the right endpoint is included. 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 not excluded in the set, while the right endpoint is excluded. This is also known as a right-open interval.

Number Line Representations for the Various Interval Types

Aside from being written with symbols, the various interval types can also be described in the number line utilizing both shaded and open circles, relying 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 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 easy conversion; just 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 take part in a debate competition, they need at least 3 teams. Express this equation in interval notation.

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

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

Plus, since no maximum number was mentioned regarding 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 also known as unbounded intervals.

Example 3

A friend wants to undertake a diet program limiting their daily calorie intake. For the diet to be a success, they must have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you express this range in interval notation?

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

The question implies 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 written 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 To Graph an Interval Notation?

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

There are laws of expressing an interval notation to the number line: a closed interval is written with a shaded 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 Transform Inequality to Interval Notation?

An interval notation is just a diverse 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 number 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 written with closed brackets [ ] in the notation. See the examples of interval notation prior to see how these symbols are employed.

How Do You Exclude Numbers in Interval Notation?

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

Grade Potential Can Guide You Get a Grip on Mathematics

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