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 learn because it becomes more critical as you progress to more difficult math.

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

This article will discuss 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 method to express a subset of all real numbers across the number line.

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

Fundamental difficulties you encounter primarily composed of single positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such straightforward applications.

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

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

  • x is higher than negative 4 but less than two

Up till now 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), denoted by values a and b separated by a comma.

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

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 interval types are necessary to get to know due to the fact they underpin the complete notation process.


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

The inequality notation {x | -4 < x < 2} express x as being higher than -4 but less than 2, which means that it does not contain neither of the two numbers referred to. 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 -4 and 2, those two values are excluded.

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


A closed interval is the contrary of the previous type of interval. Where the open interval does not include the values mentioned, a closed interval does. In text 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 two.”

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 implies that the interval contains 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 blend 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 states that x could be the value -4 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 signifies the value which are not included 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 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 excludes the other value.

As seen in the prior example, there are different 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 used 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 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 within 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 Different Interval Types

Aside from being written with symbols, the different 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 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 know 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 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 expressed as (-6, 9].

Example 2

For a school to participate in a debate competition, they need at least three teams. Represent this equation in interval notation.

In this word problem, let x be 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 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.

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 regular calorie intake. For the diet to be a success, they should have minimum of 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 number 2000 is the highest value.

The problem implies 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 described 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 also known as a bounded interval.

Interval Notation FAQs

How Do You Graph an Interval Notation?

An interval notation is simply a way of describing 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 promptly see on a number line if the point is excluded or included from the interval.

How Do You Convert Inequality to Interval Notation?

An interval notation is basically a different way of describing 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 higher 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 above to check how these symbols are used.

How To Rule Out Numbers in Interval Notation?

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

Grade Potential Could Help You Get a Grip on Math

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