July 22, 2022

Interval Notation - Definition, Examples, Types of Intervals

Interval notation is a crucial principle that learners are required learn because it becomes more critical as you advance to more difficult arithmetic.

If you see advances arithmetics, something like integral and differential calculus, on your horizon, then being knowledgeable of interval notation can save you time in understanding these concepts.

This article will talk in-depth what interval notation is, what it’s used for, and how you can decipher it.

What Is Interval Notation?

The interval notation is simply a method 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 problems you face primarily composed of single positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such simple utilization.

However, intervals are generally used to denote domains and ranges of functions in higher arithmetics. Expressing these intervals can progressively become difficult as the functions become progressively more tricky.

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

x is higher than negative 4 but less than 2

Up till now we know, this inequality notation can be written 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 separated by a comma.

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

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

Various types of intervals lay the foundation for denoting the interval notation. These interval types 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 contain 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 higher than negative four but less than two, which means that it does not include neither of the two numbers referred to. 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 negative four and two, 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 exclude the values mentioned, a closed interval does. In text form, a closed interval is expressed as any value “greater 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 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 means that the interval includes those two boundary values: -4 and 2.

On the number line, a shaded circle is used 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 last example as a guide, if the interval were half-open, it would be expressed as “x is greater than or equal to negative four and less than 2.” This states that x could be the value -4 but couldn’t 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 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

In brief, there are different types of interval notations; open, closed, and half-open. An open interval excludes the endpoints on the real number line, while a closed interval does. A half-open interval consist of one value on the line but excludes the other value.

As seen in the examples above, there are various symbols for these types under the interval notation.

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

( ): The parentheses are utilized 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 not excluded 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 known as 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 not excluded 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 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 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 question is a straightforward conversion; just 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 written as (-6, 9].

Example 2

For a school to join in a debate competition, they should have a 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 three is a closed value.

Additionally, since no upper limit was stated with concern to the number of 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, when one side of the interval that stretches to either positive or negative infinity, are called unbounded intervals.

Example 3

A friend wants to participate in diet program limiting their daily calorie intake. For the diet to be successful, 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 question, the value 1800 is the lowest while the value 2000 is the maximum value.

The problem suggest that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, expressed with the inequality 1800 ≤ x ≤ 2000.

Thus, the interval notation is described 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 FAQs

How Do You Graph an Interval Notation?

An interval notation is fundamentally a way 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 shaded circle, and an open integral is denoted with an unfilled circle. This way, you can quickly check the number line if the point is included or excluded from the interval.

How Do You Transform Inequality to Interval Notation?

An interval notation is just a different technique of describing an inequality or a set of real numbers.

If x is greater than or less a value (not equal to), then the number should be written 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 see how these symbols are employed.

How Do You Rule Out Numbers in Interval Notation?

Numbers ruled out 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 ruled out from the combination.

Grade Potential Could Guide You Get a Grip on Arithmetics

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