July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental topic that students are required learn owing to the fact that it becomes more essential as you progress to higher mathematics.

If you see higher math, such as integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you time in understanding these concepts.

This article will discuss what interval notation is, what it’s used for, 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 values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)

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

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

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

x is higher than negative 4 but less than 2

Up till now we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. Though, it can also be expressed with interval notation (-4, 2), signified by values a and b segregated 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 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 denoting the interval notation. These kinds of interval are important to get to know due to the fact they underpin the entire notation process.

Open

Open intervals are used when the expression does not comprise 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 greater than negative four but less than two, meaning 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 two values are excluded.

On the number line, an unshaded circle denotes an open value.

Closed

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 “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 two.”

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

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

On the number line, a shaded circle is utilized to denote an included open value.

Half-Open

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 last example for assistance, if the interval were half-open, it would be expressed as “x is greater than or equal to -4 and less than two.” This states that x could be the value -4 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 indicates 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 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 under the interval notation.

These symbols build the actual interval notation you create 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 used 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 instance, the left endpoint is included in the set, while the right endpoint is excluded. This is also known as a right-open interval.

Number Line Representations for the Different Interval Types

Aside from being written with symbols, the different interval types can also be described in the number line employing both shaded and open circles, depending 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	Inequality	Interval Type
(a, b)	{x \| a < x < b}	Open
[a, b]	{x \| a ≤ x ≤ b}	Closed
[a, ∞)	{x \| x ≥ a}	Half-open
(a, ∞)	{x \| x > a}	Half-open
(-∞, a)	{x \| x < a}	Half-open
(-∞, a]	{x \| x ≤ a}	Half-open

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

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

This sample problem is a easy conversion; simply 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 join in a debate competition, they should have a at least 3 teams. Express this equation in interval notation.

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

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

Additionally, because no upper limit was mentioned with concern to the number of maximum teams a school can send to the debate competition, this value should be positive to infinity.

Therefore, the interval notation should be written 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 constraining their regular 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 question, the value 1800 is the minimum while the number 2000 is the maximum value.

The question 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.

Thus, the interval notation is described 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 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 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 shaded circle, and an open integral is expressed with an unfilled 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 expressing an inequality or a combination of real numbers.

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

How Do You Exclude Numbers in Interval Notation?

Numbers excluded 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 Can Guide You Get a Grip on Math

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