July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a crucial topic that students should understand owing to the fact that it becomes more important as you grow to higher mathematics.

If you see higher mathematics, such as differential calculus and integral, on your horizon, then knowing the interval notation can save you hours in understanding these theories.

This article will discuss what interval notation is, what are its uses, and how you can decipher it.

What Is Interval Notation?

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

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

Fundamental difficulties you face essentially composed of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such effortless utilization.

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

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

  • x is greater than negative 4 but less than two

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

So far we understand, interval notation is a way to write intervals elegantly and concisely, using predetermined principles that help 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

Many types of intervals place the base for denoting the interval notation. These interval types are important to get to know due to the fact 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 greater than negative four but less than two, which means that it does not contain either 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 -4 and 2, those 2 values are excluded.

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 exclude 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 previous 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 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 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 -4 and less than 2.” This implies that x could be the value -4 but couldn’t possibly be equal to the value 2.

In an inequality notation, this would be denoted 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

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 includes one value on the line but excludes the other value.

As seen in the examples above, there are various symbols for these types under the interval notation.

These symbols build the actual interval notation you create when expressing points on a number line.

  • ( ): The parentheses are utilized 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 included. Also called a left open interval.

  • [ ): This is also a half-open notation when there are both included and excluded values among the two. In this case, the left endpoint is not excluded in the set, while the right endpoint is not included. This is also called a right-open interval.

Number Line Representations for the Different Interval Types

Aside from being written 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 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 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

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

This sample question is a easy conversion; just utilize 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 need minimum of 3 teams. Represent this equation in interval notation.

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

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

Additionally, because no maximum number was referred to regarding the number of maximum teams a school can send to the debate competition, this value 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 also known as unbounded intervals.

Example 3

A friend wants to do a diet program limiting their daily calorie intake. For the diet to be successful, they must 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 lowest while the number 2000 is the highest value.

The problem implies that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, expressed 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 variation 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 technique of describing inequalities on the number line.

There are laws of expressing an interval notation to the number line: a closed interval is expressed with a shaded circle, and an open integral is written with an unshaded circle. This way, you can quickly 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 different 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 stated with parentheses () in the notation.

If x is greater 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 above to see how these symbols are used.

How Do You Rule Out Numbers in Interval Notation?

Numbers ruled out from the interval can be stated with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which states that the number is ruled out from the combination.

Grade Potential Could Assist You Get a Grip on Mathematics

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