July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a crucial concept that pupils need to learn because it becomes more important as you grow to more difficult arithmetic.

If you see more complex math, something like integral and differential calculus, on your horizon, then knowing the interval notation can save you hours in understanding these ideas.

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

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

Though, intervals are usually employed to denote domains and ranges of functions in higher mathematics. Expressing these intervals can increasingly become complicated as the functions become further tricky.

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

x is greater than negative four but less than two

So far we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. Though, it can also be written 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 concisely and elegantly, using set rules that make writing and understanding intervals on the number line simpler.

In the following section we will discuss about 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 kinds of interval are necessary to get to know because they underpin the complete notation process.

Open

Open intervals are applied when the expression does not include 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 more than -4 but less than 2, which means that it does not include neither 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 represent that in a given set of real numbers, such as the interval between -4 and 2, those two values are not included.

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 contain 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 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 can be written as {x | -4 < x < 2}.

In an interval notation, this is written with brackets, or [-4, 2]. This implies that the interval includes 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 previous types of intervals. Of the two points on the line, one is included, and the other isn’t.

Using the prior 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 couldn’t possibly be equal to the value two.

In an inequality notation, this would be written 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 denotes the value which are not included 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 contain 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 examples above, there are different symbols for these types under the interval notation.

These symbols build the actual interval notation you create 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 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 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 among 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 Various Interval Types

Aside from being written with symbols, the different interval types can also be represented in the number line using both shaded and open circles, relying 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’ve understood 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 problem is a straightforward conversion; simply utilize 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 expressed as (-6, 9].

Example 2

For a school to participate in a debate competition, they need minimum of three teams. Express 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 means that three is a closed value.

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

Thus, the interval notation should be written as [3, ∞).

These types of intervals, when one side of the interval that stretches to either positive or negative infinity, are called unbounded intervals.

Example 3

A friend wants to undertake a diet program limiting their regular calorie intake. For the diet to be successful, they should have minimum of 1800 calories every day, but no more than 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, written with the inequality 1800 ≤ x ≤ 2000.

Thus, the interval notation is written 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 FAQs

How To 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 filled circle, and an open integral is denoted with an unfilled circle. This way, you can quickly check the 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 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 value should be expressed with parentheses () in the notation.

If x is greater 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 above to see how these symbols are used.

How To Exclude Numbers in Interval Notation?

Values excluded 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 value is excluded from the combination.

Grade Potential Can Guide You Get a Grip on Arithmetics

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

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