Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that students need to understand because it becomes more essential as you grow to more difficult arithmetic.
If you see more complex math, something like integral and differential calculus, in front of you, then knowing the interval notation can save you hours in understanding these concepts.
This article will talk about what interval notation is, what are its uses, and how you can interpret it.
What Is Interval Notation?
The interval notation is merely a method to express a subset of all real numbers across the number line.
An interval means the values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ signifies infinity.)
Fundamental problems you face mainly composed of single positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such simple utilization.
Though, intervals are usually used 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 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 written as: {x | -4 < x < 2} in set builder notation. Though, it can also be denoted with interval notation (-4, 2), signified by values a and b segregated by a comma.
So far we understand, interval notation is a method of writing intervals elegantly and concisely, using predetermined principles 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
Several types of intervals lay the foundation for writing the interval notation. These kinds of interval are essential to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are applied when the expression do not include the endpoints of the interval. The previous notation is a great example of this.
The inequality notation {x | -4 < x < 2} describes x as being more than -4 but less than 2, meaning that it does not include either of the two numbers mentioned. 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 negative four and two, those 2 values are not included.
On the number line, an unshaded circle denotes an open value.
Closed
A closed interval is the contrary of the last 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 would be written as {x | -4 < x < 2}.
In an interval notation, this is stated with brackets, or [-4, 2]. This states that the interval includes 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 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 two.” This implies that x could be the value negative four but cannot 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 written 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 which are not included 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 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 last example, there are various symbols for these types under the interval notation.
These symbols build the actual interval notation you create 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 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 not excluded. Also called 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 Various Interval Types
Aside from being denoted with symbols, the different interval types can also be described in the number line employing both shaded and open circles, relying 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’ve understood 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
Transform the following inequality into an interval notation: {x | -6 < x < 9}
This sample question is a simple conversion; just utilize 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 written as (-6, 9].
Example 2
For a school to join in a debate competition, they should have a 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 required is “three and above,” the number 3 is consisted in the set, which implies that 3 is a closed value.
Furthermore, because no upper limit was stated with concern to the number of maximum teams a school can send to the debate competition, this value should be positive to infinity.
Therefore, the interval notation should be expressed 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 do a diet program limiting their regular calorie intake. For the diet to be successful, they must have at least 1800 calories every day, but no more than 2000. How do you express this range in interval notation?
In this question, the value 1800 is the minimum while the number 2000 is the maximum 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 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 FAQs
How To Graph an Interval Notation?
An interval notation is simply a way of describing inequalities on the number line.
There are laws of expressing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is expressed with an unshaded circle. This way, you can promptly check the number line if the point is included or excluded from the interval.
How To Transform Inequality to Interval Notation?
An interval notation is just a different technique of describing an inequality or a set 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 higher than or equal to, or less 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 employed.
How To Rule Out Numbers in Interval Notation?
Values ruled out from the interval can be stated with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which states that the value is ruled out from the set.
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