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 grasp because it becomes more important as you progress to higher math.

If you see advances math, such as integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you time in understanding these ideas.

This article will talk in-depth 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 method 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 ∞ means infinity.)

Fundamental problems you encounter essentially composed of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such effortless utilization.

However, intervals are usually used to denote domains and ranges of functions in more complex math. Expressing these intervals can increasingly become difficult as the functions become further tricky.

Let’s take a straightforward 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. Despite that, it can also be denoted with interval notation (-4, 2), signified 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 predetermined principles that make writing and understanding intervals on the number line less difficult.

In the following section we will discuss 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 kinds of interval are important 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 prior notation is a fine example of this.

The inequality notation {x | -4 < x < 2} describes x as being higher than -4 but less than 2, which means that it excludes 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 implies that in a given set of real numbers, such as the interval between negative four and two, those two values are excluded.

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 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 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 can be written as {x | -4 < x < 2}.

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

On the number line, a shaded circle is used 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 for assistance, if the interval were half-open, it would read as “x is greater than or equal to -4 and less than 2.” This states that x could be the value negative four but couldn’t possibly be equal to the value two.

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 last example, there are various 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 utilized 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 not included 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 between the two. In this instance, the left endpoint is not excluded in the set, while the right endpoint is excluded. This is also known as a right-open interval.

Number Line Representations for the Various Interval Types

Aside from being denoted with symbols, the various interval types can also be described in the number line using 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’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 problem is a straightforward conversion; simply 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 expressed as (-6, 9].

Example 2

For a school to take part in a debate competition, they should have a minimum of three teams. Express this equation in interval notation.

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

Because the number of teams required is “three and above,” the number 3 is consisted in the set, which means that 3 is a closed value.

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

Thus, 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 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 successful, they must have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you describe this range in interval notation?

In this word problem, the number 1800 is the lowest while the number 2000 is the maximum value.

The problem suggest that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, denoted with the inequality 1800 ≤ x ≤ 2000.

Thus, the interval notation is written 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 Frequently Asked Questions

How Do You 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 written with a shaded circle, and an open integral is expressed with an unshaded circle. This way, you can promptly check the 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 way of describing 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 expressed with parentheses () in the notation.

If x is higher than or equal to, or lower 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 Rule Out Numbers in Interval Notation?

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

Grade Potential Could Assist You Get a Grip on Mathematics

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