Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental topic that learners are required understand due to the fact that it becomes more critical as you grow to higher arithmetic.
If you see higher math, such as integral and differential calculus, on your horizon, then knowing the interval notation can save you time in understanding these ideas.
This article will talk in-depth what interval notation is, what it’s used for, and how you can interpret it.
What Is Interval Notation?
The interval notation is merely 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 ∞ signifies infinity.)
Basic difficulties you encounter mainly composed of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such straightforward utilization.
However, intervals are generally used to denote domains and ranges of functions in advanced math. Expressing these intervals can increasingly become difficult as the functions become progressively more tricky.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative four 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 expressed with interval notation (-4, 2), signified by values a and b separated by a comma.
As we can see, interval notation is a method of writing intervals elegantly and concisely, using set principles that help writing and understanding intervals on the number line less difficult.
In the following section we will discuss regarding the rules 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 writing the interval notation. These kinds of interval are essential 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 fine 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 include neither 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 two values are not included.
On the number line, an unshaded circle denotes an open value.
Closed
A closed interval is the contrary of the previous type of interval. Where the open interval does exclude the values mentioned, a closed interval does. In word form, a closed interval is written 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 two.”
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 contains those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized 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 -4 and less than two.” This implies that x could be the value negative four 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 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 signifies the value which are not included from the subset.
Symbols for Interval Notation and Types of Intervals
To recap, 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 prior example, there are numerous 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 employed 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 included 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
Aside from being written with symbols, the different 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 represented 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 prepared for a few practice problems and their accompanying solution set.
Example 1
Convert 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 should have a at least three 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 required is “three and above,” the number 3 is consisted in the set, which implies that 3 is a closed value.
Plus, because no maximum number was stated regarding 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 written as [3, ∞).
These types of intervals, when 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 limiting their regular calorie intake. For the diet to be successful, they should have at least 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 minimum while the value 2000 is the highest value.
The problem implies that both 1800 and 2000 are included in the range, so the equation is a close interval, expressed with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is written as [1800, 2000].
When the subset of real numbers is confined 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 way of representing inequalities on the number line.
There are laws of expressing an interval notation to the number line: a closed interval is expressed with a filled circle, and an open integral is written with an unfilled circle. This way, you can promptly check the number line if the point is excluded or included from the interval.
How Do You Change Inequality to Interval Notation?
An interval notation is just a diverse technique of describing an inequality or a combination of real numbers.
If x is greater than or less a value (not equal to), then the number should be stated with parentheses () in the notation.
If x is greater than or equal to, or lower than or equal to, then the interval is denoted with closed brackets [ ] in the notation. See the examples of interval notation prior to check how these symbols are used.
How To Exclude Numbers in Interval Notation?
Values ruled out from the interval can be stated with parenthesis in the notation. A parenthesis implies that you’re writing an open interval, which means that the value is excluded from the combination.
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