July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a fundamental concept that learners are required understand due to the fact that it becomes more important as you grow to more complex mathematics.

If you see advances math, something like differential calculus and integral, in front of you, then knowing the interval notation can save you hours in understanding these ideas.

This article will talk in-depth what interval notation is, what are its uses, and how you can interpret 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 means the values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ signifies infinity.)

Fundamental problems you face mainly composed of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such straightforward applications.

However, intervals are typically used to denote domains and ranges of functions in more complex arithmetics. Expressing these intervals can progressively become complicated as the functions become further complex.

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

x is higher than negative 4 but less than 2

So far we know, 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), denoted by values a and b segregated by a comma.

So far we know, interval notation is a way to write intervals elegantly and concisely, using set principles that make writing and understanding intervals on the number line less difficult.

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

Types of Intervals

Several types of intervals lay the foundation 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

Open intervals are applied when the expression does not comprise the endpoints of the interval. The last notation is a good example of this.

The inequality notation {x | -4 < x < 2} describes x as being higher than -4 but less than 2, which means that it does not contain 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 means 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 word 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 2.”

In an inequality notation, this would 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 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 prior example as a guide, 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 negative four but cannot possibly be equal to the value 2.

In an inequality notation, this would be expressed 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 indicates the value excluded from the subset.

Symbols for Interval Notation and Types of Intervals

To recap, 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 does not include the other value.

As seen in the prior example, there are different symbols for these types subjected to 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 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 used when the interval is half-open, or when only the left endpoint is not included 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

Aside from being denoted 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 display all the different types of intervals as they are described 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 are required 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 straightforward conversion; just use 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 participate in a debate competition, they require at least 3 teams. Represent this equation in interval notation.

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

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

Additionally, since no maximum number was mentioned regarding the number of teams a school can send to the debate competition, this number should be positive to infinity.

Therefore, 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 undertake a diet program constraining their regular calorie intake. For the diet to be a success, they must have at least 1800 calories every day, but maximum intake restricted to 2000. How do you describe this range in interval notation?

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

The question 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.

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

When the subset of real numbers is confined to a range 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 Do You Graph an Interval Notation?

An interval notation is basically a technique of describing 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 denoted with an unfilled circle. This way, you can promptly see on a number line if the point is excluded or included from the interval.

How Do You Transform Inequality to Interval Notation?

An interval notation is just a diverse way of describing an inequality or a combination of real numbers.

If x is higher than or lower than a value (not equal to), then the value should be expressed with parentheses () in the notation.

If x is higher than or equal to, or less than or equal to, then the interval is expressed with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are employed.

How Do You Rule Out Numbers in Interval Notation?

Numbers ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which means that the number is ruled out from the set.

Grade Potential Can Guide You Get a Grip on Arithmetics

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

If you want to conquer these ideas quickly, you need to revise them with the professional assistance and study materials that the professional teachers of Grade Potential provide.

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