July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental concept that learners should learn owing to the fact that it becomes more critical as you progress to more difficult mathematics.

If you see higher arithmetics, something like differential calculus and integral, in front of you, then knowing the interval notation can save you time in understanding these ideas.

This article will discuss what interval notation is, what it’s used for, 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 through the number line.

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

Fundamental problems you face essentially consists of one positive or negative numbers, so it can be difficult to see the utility of the interval notation from such effortless applications.

Though, intervals are generally employed to denote domains and ranges of functions in advanced mathematics. Expressing these intervals can increasingly become difficult as the functions become more tricky.

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

  • x is higher than negative four but less than 2

As we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be written 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 set rules that make writing and understanding intervals on the number line less difficult.

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

Various types of intervals place the base for writing the interval notation. These interval types are important to get to know because they underpin the complete notation process.


Open intervals are used when the expression does not comprise 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 higher than negative four but less than two, 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 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 opposite of the last type of interval. Where the open interval does exclude 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 -4 and less than or equal to 2.”

In an inequality notation, this would be written as {x | -4 < x < 2}.

In an interval notation, this is written with brackets, or [-4, 2]. This implies 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 prior example as a guide, if the interval were half-open, it would read as “x is greater than or equal to -4 and less than two.” This means that x could be the value negative four but cannot possibly be equal to the value 2.

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 indicates 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 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 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 employed when the interval is open, or when the two endpoints on the number line are not included in the subset.

  • [ ]: The square brackets are utilized 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 used when the interval is half-open, or when only the left endpoint is not included 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 among the two. In this case, 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

Apart from being written with symbols, the different interval types can also be represented in the number line utilizing 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 need 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 use 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 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 needed is “three and above,” the value 3 is included on the set, which means that three is a closed value.

Additionally, since no maximum number was stated 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, where there is 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 daily calorie intake. For the diet to be a success, they must have at least 1800 calories every day, but maximum intake restricted to 2000. How do you express this range in interval notation?

In this question, the value 1800 is the minimum while the number 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, written with the inequality 1800 ≤ x ≤ 2000.

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

Interval Notation FAQs

How Do You Graph an Interval Notation?

An interval notation is simply 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 filled 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 Transform Inequality to Interval Notation?

An interval notation is basically a diverse way 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 number should be stated with parentheses () in the notation.

If x is higher than or equal to, or lower than or equal to, then the interval is denoted with closed brackets [ ] in the notation. See the examples of interval notation prior to check how these symbols are used.

How To Exclude Numbers in Interval Notation?

Values ruled out from the interval can be stated with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which states that the number is ruled out from the combination.

Grade Potential Can Guide You Get a Grip on Mathematics

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