July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental topic that learners are required learn due to the fact that it becomes more critical as you advance to higher math.

If you see higher arithmetics, something like integral and differential calculus, in front of you, then knowing the interval notation can save you time in understanding these theories.

This article will talk about what interval notation is, what are its uses, 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 across the number line.

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

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

Despite that, intervals are generally employed to denote domains and ranges of functions in more complex mathematics. Expressing these intervals can increasingly become difficult as the functions become more tricky.

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

  • x is greater than negative 4 but less than 2

As we know, this inequality notation can be denoted 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 separated by a comma.

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

In the following section we will discuss regarding the principles 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 necessary to get to know due to the fact they underpin the complete notation process.


Open intervals are used when the expression does not include the endpoints of the interval. The previous notation is a fine example of this.

The inequality notation {x | -4 < x < 2} describes x as being more than negative four but less than two, meaning that it does not include 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 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 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 would be expressed 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 combination of prior types of intervals. Of the two points on the line, one is included, and the other isn’t.

Using the previous 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 means that x could be the value negative four but cannot 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 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 doesn’t contain 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 different symbols for these types subjected to interval notation.

These symbols build the actual interval notation you develop when stating points on a number line.

  • ( ): The parentheses are employed 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 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 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 Different Interval Types

Apart 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 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 question is a simple conversion; just 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 participate in a debate competition, they need at least three teams. Express this equation in interval notation.

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

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

Additionally, because no maximum number was referred to 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, 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 do a diet program constraining their daily calorie intake. For the diet to be successful, they must have minimum of 1800 calories regularly, but no more than 2000. How do you express this range in interval notation?

In this word problem, the value 1800 is the lowest while the number 2000 is the highest value.

The problem implies that both 1800 and 2000 are included in the range, so the equation is a close interval, denoted 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 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 Do You Graph an Interval Notation?

An interval notation is fundamentally a technique of describing inequalities on the number line.

There are rules of expressing an interval notation to the number line: a closed interval is expressed 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 Change Inequality to Interval Notation?

An interval notation is just a diverse way of expressing an inequality or a set of real numbers.

If x is higher than or lower than a value (not equal to), then the number 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 written with closed brackets [ ] in the notation. See the examples of interval notation prior to see how these symbols are utilized.

How To Rule Out Numbers in Interval Notation?

Values excluded from the interval can be written with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which states that the number is excluded from the set.

Grade Potential Can Guide You Get a Grip on Mathematics

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

If you desire to conquer these concepts fast, you are required to revise them with the professional guidance and study materials that the expert teachers of Grade Potential provide.

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