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 are required learn because it becomes more critical as you grow to more complex mathematics.

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

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 along 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 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 straightforward utilization.

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

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

  • x is greater than negative 4 but less than two

As we understand, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. Though, it can also be expressed with interval notation (-4, 2), denoted by values a and b segregated by a comma.

As we can see, interval notation is a method of writing intervals concisely and elegantly, using predetermined principles that help writing and understanding intervals on the number line simpler.

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

Types of Intervals

Many types of intervals lay the foundation for writing the interval notation. These kinds of interval are important to get to know due to the fact they underpin the complete notation process.


Open intervals are applied when the expression do not include 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 more 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 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 -4 and 2, those two values are excluded.

On the number line, an unshaded circle denotes an open value.


A closed interval is the contrary 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 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 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 utilized to represent 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 means 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 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 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 last example, there are numerous symbols for these types under the interval notation.

These symbols build the actual interval notation you create when plotting 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 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 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 case, the left endpoint is included 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 denoted with symbols, the various interval types can also be represented in the number line utilizing 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 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 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 written as (-6, 9].

Example 2

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

In this word problem, let x stand for 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.

Additionally, since no upper limit was stated with concern to 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 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 limiting their regular calorie intake. For the diet to be successful, they should have at least 1800 calories regularly, but no more than 2000. How do you describe this range in interval notation?

In this question, the number 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, written with the inequality 1800 ≤ x ≤ 2000.

Thus, the interval notation is described 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 basically a technique of describing 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 expressed with an unshaded circle. This way, you can quickly check the number line if the point is included or excluded from the interval.

How To Change Inequality to Interval Notation?

An interval notation is just a different way of expressing an inequality or a combination of real numbers.

If x is greater than or less a value (not equal to), then the number should be expressed 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 above to see how these symbols are utilized.

How Do You Rule Out Numbers in Interval Notation?

Numbers excluded 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 value is excluded from the set.

Grade Potential Could Guide You Get a Grip on Arithmetics

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

If you want to master these ideas quickly, you are required to revise them with the expert help and study materials that the professional tutors of Grade Potential delivers.

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