July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a crucial concept that pupils need to understand because it becomes more critical as you grow to more difficult mathematics.

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

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

What Is Interval Notation?

The interval notation is merely a way 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 problems you encounter primarily composed of one positive or negative numbers, so it can be challenging to see the utility of the interval notation from such straightforward utilization.

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

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

  • x is higher than negative four but less than two

Up till now we understand, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. Though, it can also be written with interval notation (-4, 2), signified by values a and b separated by a comma.

So far we understand, interval notation is a method of writing intervals elegantly and concisely, using fixed principles that make writing and understanding intervals on the number line easier.

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


Open intervals are used when the expression do not contain the endpoints of the interval. The prior 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, meaning that it does not contain 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 -4 and 2, those 2 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 contain the values mentioned, a closed interval does. In text 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 negative four and less than or equal to 2.”

In an inequality notation, this would be written as {x | -4 < x < 2}.

In an interval notation, this is written 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 previous types of intervals. Of the two points on the line, one is included, and the other isn’t.

Using the last 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 two.” This implies that x could be the value negative four but cannot possibly be equal to the value 2.

In an inequality notation, this would be expressed as {x | -4 < x < 2}.

A half-open interval notation is denoted 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 denotes 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 consist of one value on the line but excludes the other value.

As seen in the prior example, there are different symbols for these types under the interval notation.

These symbols build the actual interval notation you develop 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 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 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 among 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

Apart from being denoted with symbols, the different interval types can also be represented in the number line utilizing both shaded and open circles, relying on the interval type.

The table below will show 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 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 simple conversion; just 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 expressed as (-6, 9].

Example 2

For a school to participate in a debate competition, they should have a at least 3 teams. Represent this equation in interval notation.

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

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

Additionally, since no maximum number was mentioned regarding the number of 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, when 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 limiting their regular calorie intake. For the diet to be successful, they should have minimum of 1800 calories regularly, but maximum intake restricted to 2000. How do you express this range in interval notation?

In this question, the number 1800 is the lowest while the value 2000 is the highest 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 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 called a bounded interval.

Interval Notation Frequently Asked Questions

How To Graph an Interval Notation?

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

There are rules to writing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is expressed with an unfilled circle. This way, you can quickly 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 describing an inequality or a combination of real numbers.

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

How Do You Exclude Numbers in Interval Notation?

Numbers ruled out from the interval can be written with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which states that the value is excluded from the combination.

Grade Potential Can Guide You Get a Grip on Math

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

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