July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a essential principle that learners are required grasp due to the fact that it becomes more important as you grow to higher arithmetic.

If you see more complex math, such as integral and differential calculus, on your horizon, then knowing the interval notation can save you time in understanding these concepts.

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 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 ∞ signifies infinity.)

Basic problems you face primarily composed of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such simple utilization.

Despite that, intervals are generally employed to denote domains and ranges of functions in more complex mathematics. 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 greater than negative 4 but less than 2

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

As we can see, interval notation is a method of writing intervals concisely and elegantly, using set principles that help writing and comprehending intervals on the number line simpler.

The following sections will tell us more about the principles of expressing a subset in a set of all real numbers with interval notation.

Types of Intervals

Various types of intervals lay the foundation for writing the interval notation. These interval types are important to get to know due to the fact they underpin the complete notation process.


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

The inequality notation {x | -4 < x < 2} express x as being more than negative four but less than two, meaning that it excludes 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 represent 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 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 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 contains those two boundary values: -4 and 2.

On the number line, a shaded circle is employed to describe an included open value.


A half-open interval is a combination of prior types of intervals. Of the two points on the line, one is included, and the other isn’t.

Using the previous example for assistance, 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 cannot 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 denoted 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 which are not included 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 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 prior example, there are various symbols for these types under the interval notation.

These symbols build the actual interval notation you create when stating 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 utilized 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 called 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 not included. This is also called a right-open interval.

Number Line Representations for the Various Interval Types

Apart from being written 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 show 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 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 denoting 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 require minimum of 3 teams. Represent this equation in interval notation.

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

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

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

Thus, 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 undertake a diet program limiting their daily calorie intake. For the diet to be a success, they must have at least 1800 calories regularly, but no more than 2000. How do you describe this range in interval notation?

In this question, the value 1800 is the minimum while the value 2000 is the maximum 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 denoted 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 Frequently Asked Questions

How To Graph an Interval Notation?

An interval notation is basically a way of describing inequalities on the number line.

There are laws to writing an interval notation to the number line: a closed interval is expressed with a shaded circle, and an open integral is expressed 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 Convert Inequality to Interval Notation?

An interval notation is just a different technique 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 stated with parentheses () in the notation.

If x is higher 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 Rule Out Numbers in Interval Notation?

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

Grade Potential Can Guide You Get a Grip on Math

Writing interval notations can get complicated fast. There are many difficult topics within this concentration, such as those dealing with the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and more.

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