July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental principle that pupils are required understand due to the fact that it becomes more critical as you advance to more complex arithmetic.

If you see higher math, 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 talk in-depth what interval notation is, what it’s used for, and how you can understand it.

What Is Interval Notation?

The interval notation is simply a way to express a subset of all real numbers across 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 encounter mainly composed of single positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such straightforward utilization.

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

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

  • x is higher than negative four but less than two

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

So far we know, interval notation is a method of writing intervals concisely and elegantly, using fixed principles that help writing and understanding intervals on the number line simpler.

The following sections will tell us more about 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 because they underpin the entire notation process.


Open intervals are used when the expression does not comprise the endpoints of the interval. The last notation is a great example of this.

The inequality notation {x | -4 < x < 2} describes x as being higher than -4 but less than 2, meaning that it excludes either 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 implies that in a given set of real numbers, such as the interval between negative four and two, those two values are not included.

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


A closed interval is the contrary 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 written 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 written with brackets, or [-4, 2]. This states that the interval consist of those two boundary values: -4 and 2.

On the number line, a shaded circle is employed 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 previous example as a guide, if the interval were half-open, it would be expressed as “x is greater than or equal to -4 and less than 2.” This implies that x could be the value negative four but cannot 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 included in the interval, and the unshaded circle denotes 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 include the endpoints on the real number line, while a closed interval does. A half-open interval consist of one value on the line but does not include the other value.

As seen in the examples above, there are various symbols for these types under the interval notation.

These symbols build the actual interval notation you develop when plotting 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 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 excluded in the set, and the right endpoint is included. 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 case, 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 Various Interval Types

Aside from being denoted with symbols, the various 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 display 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 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 easy conversion; simply 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 join in a debate competition, they require at least three teams. Represent 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 number 3 is consisted in the set, which implies that 3 is a closed value.

Plus, because no maximum number was mentioned with concern to the number of maximum teams a school can send to the debate competition, this value should be positive to infinity.

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

These types of intervals, where there is 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 daily calorie intake. For the diet to be successful, they must have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you describe this range in interval notation?

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

The question suggest that both 1800 and 2000 are included in the range, so the equation is a close interval, denoted with the inequality 1800 ≤ x ≤ 2000.

Thus, the interval notation is written 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 also known as a bounded interval.

Interval Notation FAQs

How Do You 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 filled circle, and an open integral is denoted with an unfilled circle. This way, you can quickly see on a number line if the point is included or excluded from the interval.

How To Transform Inequality to Interval Notation?

An interval notation is basically a different way of expressing an inequality or a combination 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 expressed with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are used.

How To Exclude Numbers in Interval Notation?

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

Grade Potential Could Help You Get a Grip on Arithmetics

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