Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential concept that learners need to learn due to the fact that it becomes more critical as you advance to higher arithmetic.
If you see higher math, something like differential calculus and integral, in front of you, then knowing the 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 understand it.
What Is Interval Notation?
The interval notation is merely a way 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 ∞ means infinity.)
Fundamental problems you encounter essentially consists of single positive or negative numbers, so it can be challenging to see the utility of the interval notation from such straightforward applications.
However, intervals are generally employed to denote domains and ranges of functions in advanced math. Expressing these intervals can increasingly 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 two
As we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be denoted with interval notation (-4, 2), denoted by values a and b separated by a comma.
So far we understand, interval notation is a method of writing intervals concisely and elegantly, using set rules that make writing and understanding intervals on the number line less difficult.
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
Several types of intervals lay the foundation for denoting the interval notation. These kinds of interval are essential to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are applied when the expression do not comprise the endpoints of the interval. The previous notation is a good 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 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 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 contrary 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 written 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 would be expressed as {x | -4 < x < 2}.
In an interval notation, this is stated with brackets, or [-4, 2]. This states that the interval contains 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 blend of previous 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 negative four and less than two.” This means 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 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 included in the interval, and the unshaded circle signifies 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 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 expressing points on a number line.
( ): The parentheses are used when the interval is open, or when the two endpoints on the number line are not included in the subset.
[ ]: The square brackets are utilized 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 not included in the set, and the right endpoint is included. 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 excluded. This is also known as a right-open interval.
Number Line Representations for the Various Interval Types
Aside from being written with symbols, the various interval types can also be represented in the number line utilizing both shaded and open circles, depending on the interval type.
The table below will show all the different types of intervals as they are described in the number line.
Practice Examples for Interval Notation
Now that you know 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
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 take part in a debate competition, they should have a minimum of 3 teams. Express this equation in interval notation.
In this word problem, let x be the minimum number of teams.
Since the number of teams needed is “three and above,” the value 3 is included on the set, which means that three is a closed value.
Plus, since no maximum number was mentioned regarding 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, 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 constraining their daily calorie intake. For the diet to be a success, they must have minimum of 1800 calories every day, but maximum intake restricted to 2000. How do you write this range in interval notation?
In this question, the number 1800 is the lowest while the number 2000 is the maximum value.
The question suggest that both 1800 and 2000 are inclusive 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 restricted 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 To Graph an Interval Notation?
An interval notation is fundamentally a technique of representing inequalities on the number line.
There are laws of expressing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is written with an unshaded circle. This way, you can quickly check the number line if the point is excluded or included from the interval.
How To Change Inequality to Interval Notation?
An interval notation is just a different technique of expressing an inequality or a set of real numbers.
If x is higher than or lower than a value (not equal to), then the value 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 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 stated with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which states that the value is excluded from the combination.
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