July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental concept that learners are required grasp due to the fact that it becomes more critical as you grow to more difficult arithmetic.

If you see more complex mathematics, something like differential calculus and integral, in front of you, then knowing the interval notation can save you hours in understanding these concepts.

This article will talk about what interval notation is, what it’s used for, and how you can interpret it.

What Is Interval Notation?

The interval notation is simply 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 ∞ signifies infinity.)

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

Despite that, intervals are typically employed to denote domains and ranges of functions in more complex mathematics. Expressing these intervals can increasingly become difficult as the functions become progressively more complex.

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

  • x is higher than negative four but less than 2

As we understand, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. However, it can also be denoted 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 set rules that help writing and comprehending intervals on the number line less difficult.

In the following section we will discuss 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 necessary to get to know because they underpin the entire notation process.


Open intervals are applied when the expression do not comprise the endpoints of the interval. The prior 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, meaning that it does not include 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 -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 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 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 two.”

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

In an interval notation, this is expressed with brackets, or [-4, 2]. This means that the interval includes those two boundary values: -4 and 2.

On the number line, a shaded circle is utilized to represent 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 previous 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 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 included 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

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 does not include 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 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 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 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 among the two. In this instance, 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

Apart from being denoted 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 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 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 use 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 three teams. Represent this equation in interval notation.

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

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

Plus, because no maximum number was stated regarding 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, where there is one side of the interval that stretches to either positive or negative infinity, are called unbounded intervals.

Example 3

A friend wants to participate in diet program constraining their daily 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 write this range in interval notation?

In this question, the number 1800 is the lowest while the value 2000 is the highest value.

The question implies 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 restricted to a range between two values, and doesn’t stretch to either positive or negative infinity, it is called a bounded interval.

Interval Notation Frequently Asked Questions

How Do You Graph an Interval Notation?

An interval notation is fundamentally a technique of representing inequalities on the number line.

There are laws to writing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is written with an unfilled circle. This way, you can promptly check the number line if the point is excluded or included from the interval.

How Do You Change 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 higher than or lower than 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 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 writing an open interval, which states that the value is ruled out from the combination.

Grade Potential Can Assist You Get a Grip on Math

Writing interval notations can get complicated 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 master these concepts quickly, you need to revise them with the professional guidance and study materials that the professional teachers of Grade Potential provide.

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