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 should understand because it becomes more essential as you progress to higher math.

If you see higher math, something like integral and differential calculus, on your horizon, then being knowledgeable of interval notation can save you hours in understanding these ideas.

This article will discuss what interval notation is, what it’s used for, and how you can interpret 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 difficulties you face mainly consists of one positive or negative numbers, so it can be difficult to see the utility of the interval notation from such effortless applications.

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

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

  • x is higher 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. However, it can also be denoted with interval notation (-4, 2), denoted by values a and b separated by a comma.

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

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

Types of Intervals

Several types of intervals place the base for denoting the interval notation. These interval types are essential to get to know due to the fact they underpin the entire notation process.


Open intervals are used when the expression do not include the endpoints of the interval. The prior notation is a great example of this.

The inequality notation {x | -4 < x < 2} express x as being greater than negative four but less than two, meaning that it does not include neither of the two numbers referred to. As such, this is an open interval denoted 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 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 contain 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 -4 and less than or equal to 2.”

In an inequality notation, this can 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 denote 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 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 means that x could be the value negative four but cannot possibly be equal to the value two.

In an inequality notation, this would be written 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 denotes 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 excludes 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 last example, there are various 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 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 not included 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 excluded. This is also known as a right-open interval.

Number Line Representations for the Various Interval Types

Aside from being denoted with symbols, the different 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 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 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 question is a straightforward 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 written as (-6, 9].

Example 2

For a school to take part in a debate competition, they should have a minimum of three teams. Express this equation in interval notation.

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

Because the number of teams needed is “three and above,” the value 3 is consisted in the set, which means 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 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 do a diet program constraining their regular calorie intake. For the diet to be successful, they should have at least 1800 calories every day, but maximum intake restricted to 2000. How do you write this range in interval notation?

In this word problem, the number 1800 is the minimum 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, written with the inequality 1800 ≤ x ≤ 2000.

Therefore, the interval notation is written as [1800, 2000].

When the subset of real numbers is confined to a range between two values, and doesn’t stretch to either positive or negative infinity, it is called a bounded interval.

Interval Notation Frequently Asked Questions

How To Graph an Interval Notation?

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

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

How Do You Change Inequality to Interval Notation?

An interval notation is basically a different technique of describing an inequality or a combination of real numbers.

If x is higher than or lower than a value (not equal to), then the value should be expressed 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 denoted with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which states that the number is excluded from the set.

Grade Potential Can Assist You Get a Grip on Math

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

If you desire to conquer these concepts fast, you are required to review them with the professional help and study materials that the professional instructors of Grade Potential provide.

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