July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental topic that students are required understand owing to the fact that it becomes more essential as you progress to higher arithmetic.

If you see advances math, such as differential calculus and integral, in front of you, then knowing the interval notation can save you hours 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 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 ∞ denotes infinity.)

Fundamental difficulties you encounter primarily composed of one positive or negative numbers, so it can be difficult to see the utility of the interval notation from such effortless utilization.

Despite that, intervals are typically used to denote domains and ranges of functions in higher arithmetics. Expressing these intervals can increasingly become difficult as the functions become further complex.

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

  • x is greater than negative 4 but less than two

As we know, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. Though, it can also be written with interval notation (-4, 2), signified by values a and b segregated by a comma.

As we can see, interval notation is a way to write intervals elegantly and concisely, using predetermined rules that make 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 lay the foundation 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 comprise the endpoints of the interval. The last notation is a great example of this.

The inequality notation {x | -4 < x < 2} express x as being greater than -4 but less than 2, meaning that it excludes either 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 represent 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 previous 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 “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 two.”

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

In an interval notation, this is written with brackets, or [-4, 2]. This implies that the interval consist of those two boundary values: -4 and 2.

On the number line, a shaded circle is employed to denote 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 be expressed as “x is greater than or equal to -4 and less than 2.” This states that x could be the value -4 but couldn’t possibly be equal to the value 2.

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

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 consist of one value on the line but excludes the other value.

As seen in the examples above, there are different symbols for these types under the interval notation.

These symbols build the actual interval notation you create when plotting points on a number line.

  • ( ): The parentheses are used 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 not excluded 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 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 known as a right-open interval.

Number Line Representations for the Different Interval Types

Aside from being denoted with symbols, the various interval types can also be represented in the number line employing both shaded and open circles, depending on the interval type.

The table below will display 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 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 expressed as (-6, 9].

Example 2

For a school to join in a debate competition, they should have a at least three 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 number 3 is included on the set, which means that three is a closed value.

Furthermore, since no maximum number was mentioned 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, 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 do a diet program constraining their daily calorie intake. For the diet to be successful, they should have minimum of 1800 calories regularly, but no more than 2000. How do you express this range in interval notation?

In this word problem, the value 1800 is the minimum while the value 2000 is the highest 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 described 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 FAQs

How To Graph an Interval Notation?

An interval notation is simply 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 written with a filled circle, and an open integral is denoted with an unshaded circle. This way, you can quickly see on a number line if the point is excluded or included from the interval.

How To Change Inequality to Interval Notation?

An interval notation is just a diverse technique of expressing 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 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 above to check how these symbols are utilized.

How Do You Exclude Numbers in Interval Notation?

Numbers excluded from the interval can be stated with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which states that the value is excluded from the combination.

Grade Potential Can Assist You Get a Grip on Arithmetics

Writing interval notations can get complex fast. There are more nuanced 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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