July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a crucial topic that students need to understand owing to the fact that it becomes more important as you advance to more difficult mathematics.

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

This article will talk in-depth 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 way to express a subset of all real numbers through the number line.

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

Basic difficulties you encounter primarily composed of one positive or negative numbers, so it can be difficult to see the utility of the interval notation from such simple utilization.

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

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

x is greater than negative four but less than two

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

So far we know, interval notation is a way to write intervals concisely and elegantly, using set principles that help writing and comprehending intervals on the number line easier.

The following sections will tell us more regarding the principles of expressing a subset in a set of all real numbers with interval notation.

Types of Intervals

Various types of intervals place the base for writing the interval notation. These interval types are important to get to know because they underpin the entire notation process.

Open

Open intervals are used when the expression does 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, meaning that it does not include either 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 means that in a given set of real numbers, such as the interval between negative four and two, those two values are not included.

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

Closed

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 “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 two.”

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

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

On the number line, a shaded circle is employed to represent an included open value.

Half-Open

A half-open interval is a blend of prior types of intervals. Of the two points on the line, one is included, and the other isn’t.

Using the previous 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 implies that x could be the value negative four but cannot possibly be equal to the value two.

In an inequality notation, this would be expressed 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 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 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 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 used 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 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 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 Different Interval Types

Aside from being written with symbols, the different interval types can also be described in the number line utilizing both shaded and open circles, relying 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	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 know everything you need 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 problem is a simple conversion; simply utilize the equivalent symbols when denoting 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 take part in a debate competition, they need at least three 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 needed is “three and above,” the number 3 is consisted in the set, which means that three is a closed value.

Furthermore, since no maximum number was referred to 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 participate in diet program limiting their daily calorie intake. For the diet to be a success, they must have at least 1800 calories every day, but no more than 2000. How do you write this range in interval notation?

In this question, the number 1800 is the minimum while the value 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, denoted with the inequality 1800 ≤ x ≤ 2000.

Therefore, the interval notation is denoted 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 also known as a bounded interval.

Interval Notation Frequently Asked Questions

How Do You Graph an Interval Notation?

An interval notation is simply a way of describing inequalities on the number line.

There are laws to writing an interval notation to the number line: a closed interval is expressed with a filled circle, and an open integral is written with an unfilled circle. This way, you can quickly check the number line if the point is excluded or included from the interval.

How Do You Change Inequality to Interval Notation?

An interval notation is just a different 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 number 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 prior to check 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 implies that you’re expressing an open interval, which means that the number is ruled out from the combination.

Grade Potential Can Help You Get a Grip on Mathematics

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

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