Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial concept that students are required learn due to the fact that it becomes more essential as you grow to more complex arithmetic.
If you see higher arithmetics, 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 talk in-depth what interval notation is, what are its uses, and how you can understand it.
What Is Interval Notation?
The interval notation is merely a way to express a subset of all real numbers through 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 difficulties you encounter essentially consists of single positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such straightforward utilization.
However, intervals are usually used to denote domains and ranges of functions in higher math. Expressing these intervals can progressively become complicated as the functions become progressively more tricky.
Let’s take a straightforward compound inequality notation as an example.
x is greater than negative four but less than two
So far we understand, this inequality notation can be denoted 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 separated by a comma.
So far we understand, interval notation is a way to write intervals concisely and elegantly, using predetermined principles that make writing and comprehending intervals on the number line less difficult.
In the following section we will discuss regarding the principles of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Many types of intervals place the base for writing the interval notation. These interval types are necessary to get to know due to the fact they underpin the complete notation process.
Open
Open intervals are used when the expression does not comprise the endpoints of the interval. The last 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 does not contain neither 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 represent 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 contrary of the last type of interval. Where the open interval does not include the values mentioned, a closed interval does. In word form, a closed interval is written as any value “greater 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 -4 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 implies that the interval consist of 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 as a guide, if the interval were half-open, it would read as “x is greater than or equal to negative four and less than two.” This states that x could be the value negative four but couldn’t possibly be equal to the value two.
In an inequality notation, this would be written 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 present in the interval, and the unshaded circle signifies 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 excludes 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 numerous symbols for these types under the interval notation.
These symbols build the actual interval notation you create when plotting 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 not included 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 within the two. In this instance, the left endpoint is not excluded in the set, while the right endpoint is excluded. This is also called a right-open interval.
Number Line Representations for the Different Interval Types
Apart from being written with symbols, the various interval types can also be described in the number line utilizing both shaded and open circles, depending 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 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
Transform 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 written as (-6, 9].
Example 2
For a school to join in a debate competition, they need minimum of three teams. Express this equation in interval notation.
In this word question, let x be the minimum number of teams.
Since the number of teams required is “three and above,” the value 3 is included on the set, which means that 3 is a closed value.
Furthermore, because no maximum number 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.
Thus, 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 undertake a diet program constraining their regular calorie intake. For the diet to be a success, they should have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you describe this range in interval notation?
In this question, the number 1800 is the minimum while the value 2000 is the maximum value.
The question suggest that both 1800 and 2000 are included in the range, so the equation is a close interval, written with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is denoted 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 also known as a bounded interval.
Interval Notation Frequently Asked Questions
How To Graph an Interval Notation?
An interval notation is basically a technique of describing inequalities on the number line.
There are laws to writing an interval notation to the number line: a closed interval is expressed with a shaded circle, and an open integral is expressed with an unshaded circle. This way, you can quickly check the number line if the point is included or excluded from the interval.
How Do You Transform Inequality to Interval Notation?
An interval notation is basically 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 number should be stated with parentheses () in the notation.
If x is greater than or equal to, or less than or equal to, then the interval is written with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are employed.
How Do You Rule Out Numbers in Interval Notation?
Numbers excluded 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 combination.
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