Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental topic that pupils are required grasp because it becomes more essential as you progress to more complex math.
If you see more complex math, such as differential calculus and integral, in front of you, then being knowledgeable of interval notation can save you hours in understanding these ideas.
This article will discuss what interval notation is, what it’s used for, 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 values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)
Fundamental difficulties you face essentially consists 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 used to denote domains and ranges of functions in advanced math. 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 4 but less than two
So far we know, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. However, it can also be expressed with interval notation (-4, 2), signified 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 make writing and comprehending intervals on the number line simpler.
The following sections will tell us more regarding the rules 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 necessary to get to know because they underpin the complete notation process.
Open
Open intervals are applied when the expression do not comprise the endpoints of the interval. The prior notation is a fine example of this.
The inequality notation {x | -4 < x < 2} describes x as being higher 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 contrary 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 “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 negative four and less than or equal to 2.”
In an inequality notation, this would be written as {x | -4 < x < 2}.
In an interval notation, this is expressed with brackets, or [-4, 2]. This implies that the interval contains those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized to denote an included open value.
Half-Open
A half-open interval is a blend of prior types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the previous example as a guide, if the interval were half-open, it would read as “x is greater than or equal to -4 and less than 2.” This implies 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 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 indicates 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 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 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 excluded from 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 between the two. In this case, the left endpoint is included 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
Aside from being denoted with symbols, the different interval types can also be represented 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 question is a simple conversion; just 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 participate in a debate competition, they need minimum of 3 teams. Represent this equation in interval notation.
In this word problem, 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 states that 3 is a closed value.
Additionally, because no maximum number was mentioned regarding the number of teams a school can send to the debate competition, this number should be positive to infinity.
Therefore, 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 undertake a diet program constraining their regular calorie intake. For the diet to be a success, they should have minimum of 1800 calories regularly, but maximum intake restricted to 2000. How do you write this range in interval notation?
In this question, the value 1800 is the minimum while the value 2000 is the highest value.
The question implies that both 1800 and 2000 are included in the range, so the equation is a close interval, denoted with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is described 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 called a bounded interval.
Interval Notation FAQs
How Do You Graph an Interval Notation?
An interval notation is basically 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 promptly see on a number line if the point is included or excluded from the interval.
How To Transform Inequality to Interval Notation?
An interval notation is just 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 value should be written 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 used.
How To Rule Out Numbers in Interval Notation?
Values ruled out from the interval can be written with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which states that the number is excluded from the set.
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