Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that learners should understand due to the fact that it becomes more essential as you progress to higher math.
If you see more complex math, something like integral and differential calculus, in front of you, 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 understand it.
What Is Interval Notation?
The interval notation is merely a method to express a subset of all real numbers through the number line.
An interval means the values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)
Basic difficulties you encounter primarily composed of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such straightforward utilization.
Despite that, intervals are usually employed to denote domains and ranges of functions in more complex arithmetics. Expressing these intervals can progressively become complicated as the functions become more tricky.
Let’s take a simple compound inequality notation as an example.
x is higher than negative four but less than 2
As we understand, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. Despite that, 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 concisely and elegantly, using set rules that make writing and comprehending 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 lay the foundation for denoting the interval notation. These kinds of interval are important to get to know due to the fact they underpin the complete notation process.
Open
Open intervals are used when the expression does 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 -4 but less than 2, which means that it excludes either of the two numbers referred to. As such, this is an open interval denoted 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 excluded.
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 not include 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 previous 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 includes 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 combination of prior 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 2.” This means 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 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 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 excludes 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 last example, there are numerous symbols for these types under the 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 not included in the subset.
[ ]: The square brackets are used 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 excluded 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 among the two. In this instance, 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 denoted with symbols, the various interval types can also be described in the number line using 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 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 question is a easy conversion; just use 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 written as (-6, 9].
Example 2
For a school to participate in a debate competition, they should have a minimum of 3 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 needed is “three and above,” the value 3 is included on the set, which means that 3 is a closed value.
Additionally, because no maximum number was mentioned with concern to 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 expressed 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 limiting their daily calorie intake. For the diet to be a success, they should have at least 1800 calories every day, but maximum intake restricted to 2000. How do you write this range in interval notation?
In this question, the value 1800 is the lowest while the value 2000 is the highest value.
The problem implies that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, written 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 variation between two values, and doesn’t stretch to either positive or negative infinity, it is also known as a bounded interval.
Interval Notation Frequently Asked Questions
How Do You Graph an Interval Notation?
An interval notation is basically a way of describing inequalities on the number line.
There are laws to writing an interval notation to the number line: a closed interval is written with a shaded circle, and an open integral is written 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 Change Inequality to Interval Notation?
An interval notation is just a different way 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 above to check how these symbols are used.
How To Rule Out Numbers in Interval Notation?
Values ruled out from the interval can be stated with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which means that the number is excluded from the set.
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