Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential topic that learners should learn owing to the fact that it becomes more essential as you advance to more difficult math.
If you see higher mathematics, something like differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you hours in understanding these concepts.
This article will discuss 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 across the number line.
An interval refers to the values 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 challenging to see the utility of the interval notation from such simple applications.
Despite that, intervals are usually employed to denote domains and ranges of functions in more complex arithmetics. Expressing these intervals can increasingly become difficult 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
Up till now we understand, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. Though, it can also be expressed with interval notation (-4, 2), signified by values a and b separated 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 understanding intervals on the number line simpler.
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
Many types of intervals lay the foundation for writing the interval notation. These interval types are essential to get to know due to the fact they underpin the complete notation process.
Open
Open intervals are used when the expression do not contain the endpoints of the interval. The previous notation is a great example of this.
The inequality notation {x | -4 < x < 2} describes x as being more than -4 but less than 2, which means that it does not include either of the two numbers referred to. As such, this is an open interval expressed with parentheses or a round bracket, such as the following.
(-4, 2)
This represent 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 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 negative four and less than or equal to 2.”
In an inequality notation, this can be expressed as {x | -4 < x < 2}.
In an interval notation, this is written with brackets, or [-4, 2]. This implies 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 previous types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the last example for assistance, 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 -4 but cannot 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 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 includes one value on the line but excludes the other value.
As seen in the last example, there are numerous 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 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 utilized when the interval is half-open, or when only the left endpoint is excluded in the set, and the right endpoint is included. 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 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 various interval types can also be represented in the number line using 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 straightforward conversion; simply use 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 participate in a debate competition, they should have a at least 3 teams. Express this equation in interval notation.
In this word problem, 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 implies that 3 is a closed value.
Additionally, because no upper limit was referred to 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 denoted 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 do a diet program constraining their regular calorie intake. For the diet to be a success, they should have at least 1800 calories regularly, but no more than 2000. How do you describe this range in interval notation?
In this question, the number 1800 is the lowest while the number 2000 is the maximum value.
The problem 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 written 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 describing inequalities on the number line.
There are rules to writing an interval notation to the number line: a closed interval is written 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 Convert Inequality to Interval Notation?
An interval notation is just a different technique of expressing an inequality or a combination of real numbers.
If x is higher than or lower than a value (not equal to), then the value should be written with parentheses () in the notation.
If x is higher 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 above to see how these symbols are utilized.
How To Rule Out Numbers in Interval Notation?
Values excluded from the interval can be written with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which states that the value is excluded from the set.
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