July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a crucial principle that pupils are required learn because it becomes more critical as you grow to more difficult arithmetic.

If you see advances mathematics, such as integral and differential calculus, on your horizon, then knowing the interval notation can save you time in understanding these theories.

This article will talk in-depth what interval notation is, what it’s used for, and how you can decipher it.

What Is Interval Notation?

The interval notation is merely 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 encounter primarily composed of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such straightforward applications.

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

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

  • x is greater than negative four but less than 2

Up till now we understand, this inequality notation can be written 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 segregated by a comma.

So far we know, interval notation is a way to write intervals elegantly and concisely, using predetermined rules that help writing and comprehending 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 denoting the interval notation. These kinds of interval are essential to get to know due to the fact they underpin the complete notation process.


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

The inequality notation {x | -4 < x < 2} describes x as being greater than -4 but less than 2, which means that it excludes neither of the two numbers mentioned. As such, this is an open interval denoted 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 negative four and two, those two 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 include 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 previous example was a closed interval, it would read, “x is greater than or equal to negative four 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 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 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 2.” This means that x could be the value negative four but couldn’t 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 included in the interval, and the unshaded circle signifies 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 contain 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 numerous symbols for these types subjected to interval notation.

These symbols build the actual interval notation you develop when stating 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 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 case, the left endpoint is not excluded in the set, while the right endpoint is not included. This is also called a right-open interval.

Number Line Representations for the Various Interval Types

Apart from being denoted with symbols, the various interval types can also be represented 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 know 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

Convert the following inequality into an interval notation: {x | -6 < x < 9}

This sample question is a easy conversion; just 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 take part in a debate competition, they should have a at least 3 teams. Express this equation in interval notation.

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

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

Additionally, since no upper limit was referred to 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 expressed 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 do a diet program constraining their regular calorie intake. For the diet to be successful, 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 number 1800 is the lowest while the number 2000 is the maximum value.

The problem suggest that both 1800 and 2000 are included in the range, so the equation is a close interval, expressed 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 also known as a bounded interval.

Interval Notation Frequently Asked Questions

How Do You Graph an Interval Notation?

An interval notation is simply a technique 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 shaded circle, and an open integral is denoted 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 Transform Inequality to Interval Notation?

An interval notation is basically a different 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 value should be stated with parentheses () in the notation.

If x is greater than or equal to, or less than or equal to, then the interval is denoted with closed brackets [ ] in the notation. See the examples of interval notation prior to see 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 writing an open interval, which means that the value is excluded from the set.

Grade Potential Can Guide You Get a Grip on Mathematics

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