Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential principle that pupils need to understand because it becomes more essential as you grow to higher arithmetic.
If you see advances mathematics, something like 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 discuss what interval notation is, what are its uses, and how you can interpret it.
What Is Interval Notation?
The interval notation is merely a way to express a subset of all real numbers along the number line.
An interval refers to the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Basic difficulties you encounter essentially consists of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such simple applications.
Despite that, intervals are typically used to denote domains and ranges of functions in more complex mathematics. Expressing these intervals can progressively become difficult as the functions become further tricky.
Let’s take a straightforward compound inequality notation as an example.
x is greater than negative four but less than 2
So far we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be written with interval notation (-4, 2), denoted by values a and b separated by a comma.
As we can see, interval notation is a way to write intervals concisely and elegantly, using set principles that help writing and understanding 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
Many types of intervals place the base for denoting 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 applied when the expression does 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 greater than -4 but less than 2, which means that it does not include 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 represent that in a given set of real numbers, such as the interval between -4 and 2, those 2 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 not contain the values mentioned, a closed interval does. In text 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 two.”
In an inequality notation, this can be expressed 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 used 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 prior example for assistance, 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 cannot possibly be equal to the value 2.
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 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 excludes 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 examples above, there are various 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 excluded from the subset.
[ ]: The square brackets are utilized 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 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 case, the left endpoint is not excluded in the set, while the right endpoint is excluded. This is also known as a right-open interval.
Number Line Representations for the Various Interval Types
Aside from being written with symbols, the different interval types can also be described in the number line utilizing both shaded and open circles, relying 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’ve understood everything you are required 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 question is a simple conversion; just use 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 require at least three teams. Represent 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 number 3 is included on the set, which states that three is a closed value.
Additionally, since no maximum number was mentioned with concern to the number of teams a school can send to the debate competition, this value should be positive to infinity.
Thus, the interval notation should be expressed 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 participate in diet program limiting their daily calorie intake. For the diet to be a success, they should have minimum of 1800 calories every day, but no more than 2000. How do you describe 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 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 range 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 describing 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 denoted with an unfilled circle. This way, you can promptly check the number line if the point is excluded or included from the interval.
How To Change Inequality to Interval Notation?
An interval notation is basically a diverse 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 greater 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 check how these symbols are utilized.
How Do You Rule Out Numbers in Interval Notation?
Numbers ruled out from the interval can be stated with parenthesis in the notation. A parenthesis implies that you’re writing an open interval, which means that the value is excluded from the combination.
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