Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that students should grasp owing to the fact that it becomes more essential as you grow to more difficult mathematics.
If you see more complex math, such as integral and differential calculus, in front of you, then knowing the interval notation can save you hours in understanding these concepts.
This article will discuss 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 values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)
Fundamental difficulties you face primarily consists of single positive or negative numbers, so it can be difficult to see the utility of the interval notation from such straightforward applications.
Despite that, intervals are generally employed to denote domains and ranges of functions in higher arithmetics. 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 2
Up till now we understand, this inequality notation can be expressed 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 help writing and understanding intervals on the number line simpler.
The following sections will tell us more 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 kinds of interval are essential to get to know because they underpin the entire notation process.
Open
Open intervals are used when the expression does not contain the endpoints of the interval. The prior notation is a great example of this.
The inequality notation {x | -4 < x < 2} express x as being more than -4 but less than 2, which means that it excludes 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 not included.
On the number line, an unshaded circle denotes an open value.
Closed
A closed interval is the opposite of the previous type of interval. Where the open interval does exclude the values mentioned, a closed interval does. In text form, a closed interval is expressed 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 negative four and less than or equal to 2.”
In an inequality notation, this can be written as {x | -4 < x < 2}.
In an interval notation, this is expressed 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 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 prior example as a guide, if the interval were half-open, it would read 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 2.
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 present in the interval, and the unshaded circle indicates the value excluded 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 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 last example, there are various symbols for these types subjected to interval notation.
These symbols build the actual interval notation you develop when expressing points on a number line.
( ): The parentheses are used when the interval is open, or when the two endpoints on the number line are not included in 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 not excluded. 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 included in the set, while the right endpoint is not included. This is also called 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 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 represented in the number line.
Practice Examples for Interval Notation
Now that you know everything you need 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 problem is a easy conversion; simply utilize the equivalent symbols when writing 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 take part in a debate competition, they should have a 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 consisted in the set, which implies that 3 is a closed value.
Plus, because no upper limit was referred to regarding 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 written 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 limiting their regular calorie intake. For the diet to be successful, they must have minimum of 1800 calories regularly, but maximum intake restricted to 2000. How do you write this range in interval notation?
In this word problem, the number 1800 is the minimum while the number 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, written with the inequality 1800 ≤ x ≤ 2000.
Therefore, the interval notation is described 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 called a bounded interval.
Interval Notation FAQs
How Do You Graph an Interval Notation?
An interval notation is fundamentally a technique of representing inequalities on the number line.
There are laws to writing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is written with an unshaded circle. This way, you can quickly see on a number line if the point is excluded or included from the interval.
How To Convert Inequality to Interval Notation?
An interval notation is just a diverse way of describing an inequality or a set of real numbers.
If x is greater than or less a value (not equal to), then the value should be expressed with parentheses () in the notation.
If x is higher than or equal to, or lower 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 ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which means that the value is ruled out from the combination.
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