July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental principle that learners need to learn due to the fact that it becomes more critical as you grow to more complex arithmetic.

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

This article will talk in-depth what interval notation is, what are its uses, and how you can understand it.

What Is Interval Notation?

The interval notation is simply a method to express a subset of all real numbers along the number line.

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

Basic difficulties you face primarily consists of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such straightforward utilization.

Despite that, intervals are usually 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 tricky.

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

  • x is greater than negative four but less than two

As we know, 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 separated by a comma.

So far we know, interval notation is a method of writing intervals elegantly and concisely, using set rules 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

Several types of intervals lay the foundation for writing the interval notation. These interval types are essential to get to know due to the fact they underpin the entire notation process.


Open intervals are applied when the expression do not comprise the endpoints of the interval. The previous notation is a great example of this.

The inequality notation {x | -4 < x < 2} describes x as being greater than -4 but less than 2, which means that it does not include either 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 -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 opposite of the previous type of interval. Where the open interval does not include the values mentioned, a closed interval does. In text form, a closed interval is expressed as any value “greater 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 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 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 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 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 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 included in the interval, and the unshaded circle indicates the value excluded from the subset.

Symbols for Interval Notation and Types of Intervals

To recap, 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 includes one value on the line but does not include the other value.

As seen in the examples above, 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 utilized when the interval is open, or when the two endpoints on the number line are excluded from the subset.

  • [ ]: The square brackets are utilized 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 employed 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 between the two. In this case, the left endpoint is included 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 represented 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 described 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

Transform the following inequality into an interval notation: {x | -6 < x < 9}

This sample question is a easy 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 written as (-6, 9].

Example 2

For a school to take part in a debate competition, they require at least 3 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 three is a closed value.

Furthermore, since no maximum number was mentioned regarding the number of maximum teams a school can send to the debate competition, this number should be positive to infinity.

Therefore, the interval notation should be written as [3, ∞).

These types of intervals, when 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 limiting 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 describe this range in interval notation?

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

The question implies 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 restricted 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 basically a way of representing inequalities on the number line.

There are laws to writing an interval notation to the number line: a closed interval is expressed with a filled circle, and an open integral is written with an unfilled circle. This way, you can promptly see on a 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 diverse technique of describing an inequality or a set of real numbers.

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

How Do You Exclude Numbers in Interval Notation?

Numbers ruled out from the interval can be denoted 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 Guide You Get a Grip on Math

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