Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental principle that learners are required learn because it becomes more important as you advance to more difficult arithmetic.
If you see more complex mathematics, such as integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you time in understanding these theories.
This article will talk about what interval notation is, what are its uses, and how you can decipher it.
What Is Interval Notation?
The interval notation is simply a way to express a subset of all real numbers along the number line.
An interval refers to the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Basic difficulties you encounter primarily consists of single positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such simple utilization.
Though, intervals are usually used to denote domains and ranges of functions in higher mathematics. Expressing these intervals can progressively become difficult as the functions become further tricky.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative 4 but less than two
So far we know, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. However, it can also be denoted with interval notation (-4, 2), denoted 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 predetermined rules that make writing and comprehending intervals on the number line easier.
The following sections will tell us more about the principles of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Many types of intervals place the base for denoting the interval notation. These interval types are necessary to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are applied when the expression do not include the endpoints of the interval. The prior notation is a good example of this.
The inequality notation {x | -4 < x < 2} express x as being greater than negative four but less than two, meaning 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 last type of interval. Where the open interval does not contain 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 states that the interval consist of those two boundary values: -4 and 2.
On the number line, a shaded circle is used to describe 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 last example as a guide, if the interval were half-open, it would read as “x is greater than or equal to negative four and less than 2.” This means that x could be the value -4 but couldn’t possibly be equal to the value two.
In an inequality notation, this would be denoted 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 present in the interval, and the unshaded circle indicates the value excluded 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 contain 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 subjected to interval notation.
These symbols build the actual interval notation you develop when plotting 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 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 employed 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 instance, 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 Different Interval Types
Aside from being denoted with symbols, the various interval types can also be described in the number line employing 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.
Practice Examples for Interval Notation
Now that you know everything you are required to know about writing things in interval notations, you’re ready 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 simple conversion; simply utilize 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 expressed as (-6, 9].
Example 2
For a school to join 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 required is “three and above,” the number 3 is consisted in the set, which means that 3 is a closed value.
Furthermore, because no upper limit was stated 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 constraining their regular calorie intake. For the diet to be a success, they should have minimum of 1800 calories every day, but maximum intake restricted to 2000. How do you write this range in interval notation?
In this word problem, the value 1800 is the minimum 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, written with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is written as [1800, 2000].
When the subset of real numbers is restricted to a variation 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 basically a way 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 shaded circle, and an open integral is expressed with an unshaded circle. This way, you can quickly see on a number line if the point is included or excluded from the interval.
How To Change Inequality to Interval Notation?
An interval notation is basically a different way of describing an inequality or a set 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 less than or equal to, then the interval is written with closed brackets [ ] in the notation. See the examples of interval notation prior to check how these symbols are utilized.
How To Exclude Numbers in Interval Notation?
Values ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which states that the value is ruled out from the set.
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