Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental concept that pupils need to learn because it becomes more essential as you grow to more complex arithmetic.
If you see advances math, something like differential calculus and integral, in front of you, then knowing the interval notation can save you hours in understanding these concepts.
This article will talk in-depth what interval notation is, what are its uses, and how you can decipher it.
What Is Interval Notation?
The interval notation is simply a method 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 ∞ signifies infinity.)
Basic difficulties you encounter mainly consists of single positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such straightforward utilization.
Despite that, intervals are generally employed to denote domains and ranges of functions in more complex arithmetics. Expressing these intervals can increasingly become complicated as the functions become more complex.
Let’s take a simple compound inequality notation as an example.
x is higher than negative 4 but less than two
As we understand, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. However, it can also be written 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 concisely and elegantly, using predetermined rules that make 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
Several types of intervals place the base for writing the interval notation. These interval types are important to get to know because they underpin the complete notation process.
Open
Open intervals are used when the expression does not comprise the endpoints of the interval. The previous notation is a good 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 neither of the two numbers referred to. 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 two values are excluded.
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 contain the values mentioned, a closed interval does. In word form, a closed interval is written 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 can be expressed as {x | -4 < x < 2}.
In an interval notation, this is expressed 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 blend of previous types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the prior 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 2.” This states that x could be the value -4 but cannot 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 indicates the value excluded 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 include 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 last example, there are different 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 employed 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 used 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 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 represented in the number line employing both shaded and open circles, depending 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 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 problem is a easy conversion; simply use the equivalent symbols when stating 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 should have a minimum of three 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 needed is “three and above,” the value 3 is included on the set, which implies that three is a closed value.
Plus, since no upper limit was referred to regarding the number of maximum teams a school can send to the debate competition, this value should be positive to infinity.
Thus, the interval notation should be written as [3, ∞).
These types of intervals, where there is one side of the interval that stretches to either positive or negative infinity, are called unbounded intervals.
Example 3
A friend wants to participate in diet program constraining their regular calorie intake. For the diet to be successful, they should have at least 1800 calories regularly, but no more than 2000. How do you describe this range in interval notation?
In this question, the value 1800 is the minimum while the value 2000 is the highest 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 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 Frequently Asked Questions
How Do You 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 written with a filled circle, and an open integral is written with an unshaded circle. This way, you can quickly check the number line if the point is included or excluded from the interval.
How To Change Inequality to Interval Notation?
An interval notation is just a different technique of expressing an inequality or a set of real numbers.
If x is greater than or less a value (not equal to), then the value should be stated with parentheses () in the notation.
If x is greater than or equal to, or lower 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 used.
How To Rule Out Numbers in Interval Notation?
Values excluded from the interval can be stated with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which means that the number is excluded from the set.
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