Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that students are required understand because it becomes more essential as you grow to higher arithmetic.
If you see more complex math, such as integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you time in understanding these concepts.
This article will discuss 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 values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Fundamental problems you face mainly consists of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such effortless applications.
Though, intervals are typically employed to denote domains and ranges of functions in advanced mathematics. Expressing these intervals can increasingly become complicated as the functions become progressively more complex.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative 4 but less than 2
As we know, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. Though, it can also be denoted with interval notation (-4, 2), denoted by values a and b separated by a comma.
So far we know, interval notation is a way to write intervals concisely and elegantly, using predetermined principles that make writing and comprehending intervals on the number line easier.
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
Several types of intervals place the base for writing the interval notation. These kinds of interval 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 comprise the endpoints of the interval. The last notation is a good example of this.
The inequality notation {x | -4 < x < 2} express x as being more than negative four but less than two, which means that it does not include 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 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 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 “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 -4 and less than or equal to 2.”
In an inequality notation, this would be written as {x | -4 < x < 2}.
In an interval notation, this is stated 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 employed to represent an included open value.
Half-Open
A half-open interval is a combination of previous types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the previous example for assistance, if the interval were half-open, it would read as “x is greater than or equal to -4 and less than 2.” This states 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 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 excludes the other value.
As seen in the examples above, there are various 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 used when the interval is open, or when the two endpoints on the number line are excluded from 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 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 between the two. In this instance, the left endpoint is included in the set, while the right endpoint is excluded. 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 represented in the number line utilizing 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 need 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 question is a straightforward conversion; just 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 expressed as (-6, 9].
Example 2
For a school to join in a debate competition, they require at least 3 teams. Express this equation in interval notation.
In this word question, let x be the minimum number of teams.
Because the number of teams needed is “three and above,” the value 3 is consisted in the set, which means that 3 is a closed value.
Plus, because no maximum number was referred to with concern to 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 denoted as [3, ∞).
These types of intervals, when 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 constraining their regular calorie intake. For the diet to be a success, they must have minimum of 1800 calories regularly, but no more than 2000. How do you express this range in interval notation?
In this word problem, the number 1800 is the lowest while the value 2000 is the maximum 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 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 Frequently Asked Questions
How To Graph an Interval Notation?
An interval notation is basically 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 shaded circle, and an open integral is written with an unshaded circle. This way, you can quickly check the 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 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 number should be expressed 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 see how these symbols are used.
How To Exclude Numbers in Interval Notation?
Values excluded from the interval can be denoted with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which states that the number is ruled out from the combination.
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