Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial principle that learners should learn because it becomes more essential as you grow to more complex mathematics.
If you see higher mathematics, such as differential calculus and integral, in front of you, then knowing the interval notation can save you time in understanding these ideas.
This article will discuss what interval notation is, what are its uses, and how you can interpret it.
What Is Interval Notation?
The interval notation is simply a method 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 problems 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 simple utilization.
Despite that, intervals are usually used to denote domains and ranges of functions in higher mathematics. 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
Up till now we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. However, it can also be written 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 fixed principles that help writing and comprehending intervals on the number line less difficult.
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 interval types are important to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are applied when the expression do not contain the endpoints of the interval. The prior notation is a great example of this.
The inequality notation {x | -4 < x < 2} describes x as being higher than negative four but less than two, meaning that it does not include neither 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 -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 contain the values mentioned, a closed interval does. In word 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 expressed as {x | -4 < x < 2}.
In an interval notation, this is stated 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 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 last example as a guide, if the interval were half-open, it would be expressed as “x is greater than or equal to -4 and less than two.” This means that x could be the value negative four but couldn’t 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 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 includes one value on the line but excludes 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 utilized 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 employed when the interval is half-open, or when only the left endpoint is not included 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 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 Various Interval Types
Apart from being denoted 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 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 ready 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 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 minimum of 3 teams. Express this equation in interval notation.
In this word problem, 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 means that three is a closed value.
Plus, since no upper limit was mentioned 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 called unbounded intervals.
Example 3
A friend wants to undertake a diet program constraining their daily calorie intake. For the diet to be a success, they must have at least 1800 calories regularly, but no more than 2000. How do you write this range in interval notation?
In this word problem, the number 1800 is the lowest while the value 2000 is the highest value.
The question suggest that both 1800 and 2000 are inclusive 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 variation between two values, and doesn’t stretch to either positive or negative infinity, it is also known as a bounded interval.
Interval Notation FAQs
How To Graph an Interval Notation?
An interval notation is simply a way of representing inequalities on the number line.
There are rules of expressing an interval notation to the number line: a closed interval is denoted with a shaded circle, and an open integral is denoted with an unshaded circle. This way, you can promptly check the number line if the point is excluded or included from the interval.
How Do You Transform Inequality to Interval Notation?
An interval notation is basically a different technique of describing an inequality or a set of real numbers.
If x is higher than or lower than a value (not equal to), then the value 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 expressed with closed brackets [ ] in the notation. See the examples of interval notation prior to see how these symbols are used.
How Do You Exclude Numbers in Interval Notation?
Numbers ruled out from the interval can be stated with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which states that the number is ruled out from the combination.
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