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 should learn due to the fact that it becomes more essential as you progress to more complex arithmetic.

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

This article will discuss 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 method to express a subset of all real numbers through 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.)

Basic difficulties you encounter mainly consists of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such simple applications.

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

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

  • x is greater than negative four but less than two

As we know, this inequality notation can be written 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 segregated by a comma.

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

The following sections will tell us more about 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 interval types are important to get to know because they underpin the entire notation process.


Open intervals are applied when the expression does not include the endpoints of the interval. The previous notation is a good example of this.

The inequality notation {x | -4 < x < 2} express x as being more than -4 but less than 2, which means that it excludes neither 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 implies 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 exclude 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 would be expressed 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 represent an included open value.


A half-open interval is a combination of prior types of intervals. Of the two points on the line, one is included, and the other isn’t.

Using the last 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 two.” This states that x could be the value -4 but cannot possibly be equal to the value two.

In an inequality notation, this would be expressed 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 excluded 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 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 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 excluded from 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 not included in the set, and the right endpoint is not excluded. 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 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 Various Interval Types

Aside from being written with symbols, the various interval types can also be described in the number line utilizing 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 prepared 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 easy conversion; simply 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 three teams. Represent this equation in interval notation.

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

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

Additionally, because no upper limit was stated with concern to 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 written 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 do a diet program constraining their regular calorie intake. For the diet to be a success, they must have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you write this range in interval notation?

In this question, the number 1800 is the minimum while the value 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, 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 also known as a bounded interval.

Interval Notation Frequently Asked Questions

How To Graph an Interval Notation?

An interval notation is simply a way of representing inequalities on the number line.

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

How To Convert Inequality to Interval Notation?

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

If x is greater than or less a value (not equal to), then the number should be stated with parentheses () in the notation.

If x is higher than or equal to, or lower than or equal to, then the interval is denoted with closed brackets [ ] in the notation. See the examples of interval notation prior to see how these symbols are employed.

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 means that the number is ruled out from the set.

Grade Potential Can Assist You Get a Grip on Arithmetics

Writing interval notations can get complicated fast. There are multiple difficult 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 are required to review them with the expert help and study materials that the professional tutors of Grade Potential provide.

Unlock your arithmetics skills with Grade Potential. Connect with us now!