Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential concept that students are required grasp owing to the fact that it becomes more critical as you progress to more complex math.
If you see higher math, such as differential calculus and integral, in front of you, then being knowledgeable of interval notation can save you time in understanding these concepts.
This article will talk about what interval notation is, what it’s used for, and how you can understand 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 problems you face essentially composed of single positive or negative numbers, so it can be difficult to see the utility of the interval notation from such straightforward applications.
Though, intervals are generally used to denote domains and ranges of functions in advanced math. 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 greater 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. Though, it can also be expressed 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 elegantly and concisely, using predetermined principles that make writing and comprehending intervals on the number line easier.
In the following section we will discuss about 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 kinds of interval are important to get to know because they underpin the entire notation process.
Open
Open intervals are used when the expression do not contain the endpoints of the interval. The previous 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 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 means that in a given set of real numbers, such as the interval between -4 and 2, those 2 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 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 previous 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 can be expressed as {x | -4 < x < 2}.
In an interval notation, this is stated with brackets, or [-4, 2]. This implies that the interval contains 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 prior types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the prior 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 means that x could be the value -4 but cannot 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 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 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 includes 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 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 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 used 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 between the two. In this case, the left endpoint is not excluded in the set, while the right endpoint is not included. This is also known as a right-open interval.
Number Line Representations for the Different Interval Types
Apart from being written with symbols, the various interval types can also be described 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 straightforward conversion; simply 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 expressed as (-6, 9].
Example 2
For a school to join in a debate competition, they should have a at least 3 teams. Represent this equation in interval notation.
In this word problem, let x stand for the minimum number of teams.
Since the number of teams needed is “three and above,” the number 3 is included on the set, which means that 3 is a closed value.
Plus, because no upper limit was referred to with concern to the number of teams a school can send to the debate competition, this number should be positive to infinity.
Thus, 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 also known as unbounded intervals.
Example 3
A friend wants to do a diet program limiting their daily calorie intake. For the diet to be a success, they must have minimum of 1800 calories regularly, 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 number 2000 is the highest value.
The question implies that both 1800 and 2000 are included in the range, so the equation is a close interval, denoted with the inequality 1800 ≤ x ≤ 2000.
Therefore, the interval notation is denoted 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 FAQs
How Do You Graph an Interval Notation?
An interval notation is simply a technique 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 written with an unshaded circle. This way, you can promptly see on a number line if the point is excluded or included from the interval.
How To Transform Inequality to Interval Notation?
An interval notation is just a diverse way 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 number should be written with parentheses () in the notation.
If x is higher 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 check how these symbols are used.
How To Exclude Numbers in Interval Notation?
Values excluded from the interval can be written with parenthesis in the notation. A parenthesis implies that you’re writing an open interval, which states that the value is ruled out from the set.
Grade Potential Could Help You Get a Grip on Math
Writing interval notations can get complicated fast. There are many nuanced topics in this concentration, such as those working on the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and many more.
If you want to master these ideas quickly, you need to revise them with the professional help and study materials that the professional tutors of Grade Potential delivers.
Unlock your math skills with Grade Potential. Call us now!
