Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental concept that learners are required understand because it becomes more important as you grow to higher arithmetic.
If you see higher arithmetics, such as integral and differential calculus, on your horizon, then being knowledgeable of interval notation can save you time in understanding these theories.
This article will talk in-depth what interval notation is, what are its uses, and how you can understand it.
What Is Interval Notation?
The interval notation is merely a method 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 ∞ means infinity.)
Basic problems you encounter essentially consists of single positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such effortless utilization.
Despite that, intervals are usually employed to denote domains and ranges of functions in higher mathematics. Expressing these intervals can progressively become complicated as the functions become more complex.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative four but less than 2
As we understand, this inequality notation can be denoted 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 segregated by a comma.
As we can see, interval notation is a method of writing intervals elegantly and concisely, using predetermined rules that make writing and comprehending intervals on the number line simpler.
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 place the base for writing the interval notation. These interval types are important to get to know because they underpin the entire notation process.
Open
Open intervals are applied when the expression does not include the endpoints of the interval. The prior notation is a fine example of this.
The inequality notation {x | -4 < x < 2} express x as being greater than -4 but less than 2, which means that it does not contain neither 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 implies that in a given set of real numbers, such as the interval between -4 and 2, those 2 values are not included.
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 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 previous 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 would be expressed as {x | -4 < x < 2}.
In an interval notation, this is written with brackets, or [-4, 2]. This means that the interval consist of those two boundary values: -4 and 2.
On the number line, a shaded circle is used 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 last example as a guide, 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 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 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 which are not included 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 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 does not include 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 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 not excluded 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 instance, the left endpoint is not excluded in the set, while the right endpoint is not included. This is also called a right-open interval.
Number Line Representations for the Various Interval Types
Apart from being denoted with symbols, the different interval types can also be described 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.
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 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 join in a debate competition, they need at least three teams. Express this equation in interval notation.
In this word question, let x stand for the minimum number of teams.
Because the number of teams required is “three and above,” the value 3 is included on the set, which means that 3 is a closed value.
Furthermore, because no maximum number was mentioned with concern to 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 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 participate in 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 minimum 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, denoted with the inequality 1800 ≤ x ≤ 2000.
Therefore, 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 To Graph an Interval Notation?
An interval notation is basically a way of describing inequalities on the number line.
There are laws to writing an interval notation to the number line: a closed interval is denoted with a shaded circle, and an open integral is expressed 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 Change Inequality to Interval Notation?
An interval notation is basically a diverse way of describing an inequality or a set of real numbers.
If x is higher than or lower than a value (not equal to), then the number 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 expressed with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are utilized.
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 writing an open interval, which states that the value is excluded from the combination.
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