Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential topic that students are required learn because it becomes more essential as you advance to higher math.
If you see higher mathematics, such as integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you hours in understanding these theories.
This article will talk in-depth what interval notation is, what are its uses, and how you can interpret it.
What Is Interval Notation?
The interval notation is simply a way to express a subset of all real numbers across the number line.
An interval means the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)
Fundamental difficulties you face mainly consists of one positive or negative numbers, so it can be challenging to see the utility of the interval notation from such simple utilization.
However, intervals are usually employed to denote domains and ranges of functions in higher mathematics. Expressing these intervals can progressively become difficult as the functions become progressively more complex.
Let’s take a straightforward compound inequality notation as an example.
x is greater than negative 4 but less than 2
Up till now we know, this inequality notation can be denoted 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 separated by a comma.
So far we know, interval notation is a method of writing intervals elegantly and concisely, using predetermined rules that help writing and understanding intervals on the number line less difficult.
The following sections will tell us more about the principles of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Various types of intervals lay the foundation for denoting 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 used when the expression does not comprise 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 greater than negative four but less than two, which means that it excludes either 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 implies that in a given set of real numbers, such as the interval between negative four and two, those 2 values are excluded.
On the number line, an unshaded circle denotes an open value.
Closed
A closed interval is the opposite of the last 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 “greater 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 -4 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 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 utilized to describe 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 for assistance, if the interval were half-open, it would be expressed as “x is greater than or equal to negative four and less than two.” This states that x could be the value -4 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 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 denotes the value which are not included 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 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 does not include the other value.
As seen in the examples above, there are different symbols for these types subjected to interval notation.
These symbols build the actual interval notation you develop when stating 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 not excluded 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 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 not excluded in the set, while the right endpoint is not included. This is also called a right-open interval.
Number Line Representations for the Various Interval Types
Aside from being denoted with symbols, the various interval types can also be represented in the number line employing 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 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 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 participate in a debate competition, they should have a at least 3 teams. Represent 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 number 3 is consisted in the set, which states that three is a closed value.
Furthermore, since 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.
Therefore, the interval notation should be written 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 undertake a diet program constraining their daily calorie intake. For the diet to be successful, they should have at least 1800 calories every day, but no more than 2000. How do you write this range in interval notation?
In this question, the number 1800 is the lowest while the number 2000 is the maximum value.
The problem implies 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 basically a technique 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 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 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 number should be written with parentheses () in the notation.
If x is greater 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 utilized.
How Do You Exclude Numbers in Interval Notation?
Numbers ruled out from the interval can be written with parenthesis in the notation. A parenthesis implies that you’re writing an open interval, which means that the number is ruled out from the combination.
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