July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental principle that students should learn due to the fact that it becomes more important as you progress to higher math.

If you see more complex mathematics, such as integral and differential calculus, 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 understand it.

What Is Interval Notation?

The interval notation is merely a way to express a subset of all real numbers across 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.)

Basic problems you encounter essentially composed of single positive or negative numbers, so it can be challenging to see the utility of the interval notation from such simple utilization.

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

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

  • x is greater than negative 4 but less than 2

So far we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. However, it can also be written with interval notation (-4, 2), denoted 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 fixed principles that make writing and understanding intervals on the number line easier.

In the following section we will discuss regarding the rules 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 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 comprise the endpoints of the interval. The prior notation is a fine 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 implies that in a given set of real numbers, such as the interval between negative four and two, 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 exclude 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 would be written as {x | -4 < x < 2}.

In an interval notation, this is written with brackets, or [-4, 2]. This means that the interval includes those two boundary values: -4 and 2.

On the number line, a shaded circle is utilized to describe 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 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 denoted 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 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 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 examples above, there are numerous 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 included 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 excluded in the set, and the right endpoint is not excluded. Also called 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 Various Interval Types

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

The table below will show 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 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

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 written as (-6, 9].

Example 2

For a school to take part in a debate competition, they need minimum of 3 teams. Express this equation in interval notation.

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

Since the number of teams required is “three and above,” the value 3 is consisted in the set, which implies that 3 is a closed value.

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

In this word problem, the value 1800 is the minimum while the number 2000 is the maximum 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.

Therefore, the interval notation is denoted as [1800, 2000].

When the subset of real numbers is confined to a variation 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 fundamentally a way 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 unfilled circle. This way, you can quickly see on a number line if the point is excluded or included from the interval.

How To Transform Inequality to Interval Notation?

An interval notation is basically a diverse 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 value 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 written with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are employed.

How To Rule Out Numbers in Interval Notation?

Values excluded from the interval can be denoted 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 set.

Grade Potential Could Assist You Get a Grip on Math

Writing interval notations can get complex fast. There are many 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.

If you desire to master these concepts fast, you need to revise them with the professional help and study materials that the expert instructors of Grade Potential delivers.

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