July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a crucial principle that learners need to grasp because it becomes more important as you grow to more complex arithmetic.

If you see advances mathematics, something like 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 decipher it.

What Is Interval Notation?

The interval notation is merely a way to express a subset of all real numbers across the number line.

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

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

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

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

  • x is higher than negative 4 but less than 2

As we know, this inequality notation can be written 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.

As we can see, interval notation is a way to write intervals concisely and elegantly, using predetermined rules that make writing and understanding intervals on the number line easier.

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

Types of Intervals

Several 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 entire notation process.


Open intervals are used when the expression do not contain 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 greater than negative four but less than two, which means that it excludes either 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 represent that in a given set of real numbers, such as the interval between -4 and 2, those 2 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 not include the values mentioned, a closed interval does. In text form, a closed interval is written as any value “higher than or equal to” or “less than or equal to.”

For example, if the last 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 can be written as {x | -4 < x < 2}.

In an interval notation, this is expressed 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 used to describe 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 negative four and less than two.” This implies that x could be the value -4 but cannot 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 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 indicates 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 used when the interval is open, or when the two endpoints on the number line are not included in the subset.

  • [ ]: The square brackets are used 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 utilized 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 called a left open interval.

  • [ ): This is also a half-open notation when there are both included and excluded values within the two. In this instance, the left endpoint is not excluded in the set, while the right endpoint is excluded. This is also called a right-open interval.

Number Line Representations for the Different Interval Types

Aside from being written with symbols, the various interval types can also be represented in the number line employing 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’ve understood everything you need 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 straightforward 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 require minimum of 3 teams. Represent this equation in interval notation.

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

Because the number of teams required is “three and above,” the number 3 is included on the set, which implies that three is a closed value.

Plus, since no upper limit was referred to regarding 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 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 undertake a diet program constraining their regular calorie intake. For the diet to be a success, they should have minimum of 1800 calories every day, but no more than 2000. How do you describe this range in interval notation?

In this word problem, the value 1800 is the minimum while the value 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, expressed 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 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 To Graph an Interval Notation?

An interval notation is basically 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 denoted 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 included or excluded from the interval.

How Do You Convert 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 written with parentheses () in the notation.

If x is greater than or equal to, or less than or equal to, then the interval is written 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 excluded 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 excluded from the set.

Grade Potential Can Help You Get a Grip on Mathematics

Writing interval notations can get complex fast. There are multiple nuanced topics in this concentration, such as those working on the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and many more.

If you want to conquer these ideas quickly, you need to revise them with the expert assistance and study materials that the expert teachers of Grade Potential provide.

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