Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential concept that pupils are required grasp owing to the fact that it becomes more essential as you grow to more complex math.
If you see more complex math, something like integral and differential calculus, on your horizon, 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 simply a way to express a subset of all real numbers through 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 problems you encounter mainly composed of single positive or negative numbers, so it can be challenging to see the utility of the interval notation from such effortless applications.
Despite that, intervals are typically employed to denote domains and ranges of functions in higher math. Expressing these intervals can progressively become difficult as the functions become more complex.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative 4 but less than 2
So far we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. Though, it can also be expressed with interval notation (-4, 2), signified by values a and b separated by a comma.
So far we understand, interval notation is a method of writing intervals concisely and elegantly, using set rules that make writing and comprehending intervals on the number line simpler.
In the following section we will discuss 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 necessary to get to know because they underpin the complete notation process.
Open
Open intervals are used when the expression do not include the endpoints of the interval. The prior notation is a good example of this.
The inequality notation {x | -4 < x < 2} describes x as being more than -4 but less than 2, meaning that it excludes 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 implies 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 exclude 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 last example was a closed interval, it would read, “x is greater than or equal to negative four and less than or equal to two.”
In an inequality notation, this can be written as {x | -4 < x < 2}.
In an interval notation, this is written 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 describe an included open value.
Half-Open
A half-open interval is a blend 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 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 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 signifies 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 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 does not include the other value.
As seen in the prior example, there are different symbols for these types subjected to interval notation.
These symbols build the actual interval notation you create 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 employed 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 not included 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 among the two. In this case, 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
Apart 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’ve understood 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 problem is a easy conversion; simply utilize 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 join 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.
Since the number of teams required is “three and above,” the value 3 is consisted in the set, which implies that three is a closed value.
Plus, since no upper limit 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.
Therefore, the interval notation should be denoted as [3, ∞).
These types of intervals, where there is 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 constraining their regular calorie intake. For the diet to be successful, they should have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you write this range in interval notation?
In this question, the number 1800 is the minimum while the value 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, written with the inequality 1800 ≤ x ≤ 2000.
Therefore, the interval notation is written as [1800, 2000].
When the subset of real numbers is restricted to a variation between two values, and doesn’t stretch to either positive or negative infinity, it is called a bounded interval.
Interval Notation FAQs
How Do You 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 written with a filled circle, and an open integral is written with an unshaded circle. This way, you can promptly see on a number line if the point is excluded or included from the interval.
How Do You Change Inequality to Interval Notation?
An interval notation is just a diverse way 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 greater 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 prior to check how these symbols are utilized.
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 value is ruled out from the combination.
Grade Potential Can Guide You Get a Grip on Mathematics
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