Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential principle that students are required understand because it becomes more essential as you advance to more difficult math.
If you see advances arithmetics, 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 are its uses, and how you can interpret 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 refers to the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)
Basic problems you face primarily composed of single positive or negative numbers, so it can be challenging to see the utility of the interval notation from such straightforward utilization.
However, intervals are generally used to denote domains and ranges of functions in higher arithmetics. Expressing these intervals can increasingly become complicated as the functions become further complex.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative four but less than two
As we understand, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be written with interval notation (-4, 2), signified 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 set rules that make writing and comprehending intervals on the number line simpler.
The following sections will tell us more regarding 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 important to get to know due to the fact they underpin the complete notation process.
Open
Open intervals are used when the expression do not include the endpoints of the interval. The last notation is a fine example of this.
The inequality notation {x | -4 < x < 2} express x as being higher than negative four but less than two, which means that it does not contain neither 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 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 include 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 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 written as {x | -4 < x < 2}.
In an interval notation, this is stated 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 employed to represent 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 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 cannot 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 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 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 examples above, there are different symbols for these types subjected to interval notation.
These symbols build the actual interval notation you develop 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 utilized 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 used 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 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 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 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 written 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 problem, let x stand for the minimum number of teams.
Since the number of teams required 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.
Thus, 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 called unbounded intervals.
Example 3
A friend wants to undertake a diet program limiting 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 describe this range in interval notation?
In this question, the value 1800 is the lowest while the value 2000 is the highest value.
The question suggest that both 1800 and 2000 are included 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 also known as a bounded interval.
Interval Notation FAQs
How To Graph an Interval Notation?
An interval notation is basically a way of representing inequalities on the number line.
There are rules to writing an interval notation to the number line: a closed interval is written with a filled circle, and an open integral is expressed with an unfilled circle. This way, you can promptly see on a number line if the point is included or excluded from the interval.
How Do You Convert Inequality to Interval Notation?
An interval notation is basically a diverse way 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 written 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 see how these symbols are utilized.
How To Exclude Numbers in Interval Notation?
Values excluded from the interval can be written with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which means that the number is excluded from the combination.
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