July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a crucial concept that learners need to learn because it becomes more essential as you progress to more difficult math.

If you see more complex math, such as integral and differential calculus, in front of you, then knowing the interval notation can save you time 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 merely a method to express a subset of all real numbers through the number line.

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

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

Though, intervals are generally used to denote domains and ranges of functions in higher 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

So far we understand, this inequality notation can be expressed 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.

So far we understand, interval notation is a way to write intervals concisely and elegantly, using predetermined rules that make writing and comprehending intervals on the number line simpler.

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

Various types of intervals place the base for writing the interval notation. These kinds of interval are important to get to know because they underpin the entire notation process.


Open intervals are applied when the expression do not contain the endpoints of the interval. The previous 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, which means that it does not contain neither 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 means 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 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 includes 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 prior types of intervals. Of the two points on the line, one is included, and the other isn’t.

Using the prior example for assistance, if the interval were half-open, it would read as “x is greater than or equal to negative four and less than 2.” This implies that x could be the value negative four but cannot 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 denotes 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 doesn’t contain 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 examples above, there are different symbols for these types subjected to interval notation.

These symbols build the actual interval notation you create 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 included 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 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 case, 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

Apart from being denoted with symbols, the various interval types can also be described in the number line using 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 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 straightforward conversion; just use the equivalent symbols when stating 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 require minimum of 3 teams. Express this equation in interval notation.

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

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

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

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

The question 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 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 of expressing an interval notation to the number line: a closed interval is written with a shaded circle, and an open integral is expressed with an unfilled circle. This way, you can promptly check the number line if the point is included or excluded from the interval.

How Do You Transform Inequality to Interval Notation?

An interval notation is basically a diverse way of expressing an inequality or a set of real numbers.

If x is greater than or less a value (not equal to), then the value 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 expressed with closed brackets [ ] in the notation. See the examples of interval notation prior to check how these symbols are utilized.

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 writing an open interval, which means that the number is excluded from the set.

Grade Potential Could Assist You Get a Grip on Mathematics

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

If you want to conquer these concepts quickly, you need to review them with the expert guidance and study materials that the professional instructors of Grade Potential provide.

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