Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that students need to learn because it becomes more critical as you progress to more complex math.
If you see advances mathematics, such as differential calculus and integral, on your horizon, then knowing the interval notation can save you hours in understanding these ideas.
This article will talk about what interval notation is, what are its uses, and how you can decipher it.
What Is Interval Notation?
The interval notation is simply 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 ∞ signifies infinity.)
Fundamental problems you face mainly composed of single positive or negative numbers, so it can be challenging to see the utility of the interval notation from such straightforward applications.
Though, intervals are generally employed to denote domains and ranges of functions in higher math. Expressing these intervals can increasingly 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 2
So far we know, 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), denoted by values a and b separated by a comma.
So far we know, interval notation is a way to write intervals elegantly and concisely, using fixed rules that make writing and comprehending intervals on the number line less difficult.
In the following section we will discuss about the principles 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 denoting the interval notation. These interval types are important to get to know due to the fact they underpin the complete notation process.
Open
Open intervals are applied when the expression do not contain the endpoints of the interval. The prior notation is a great example of this.
The inequality notation {x | -4 < x < 2} describes x as being more than negative four but less than two, which means that it does not contain 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 represent that in a given set of real numbers, such as the interval between negative four and two, 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 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 expressed as any value “greater 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 -4 and less than or equal to 2.”
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 states that the interval contains 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 prior types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the previous 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 states that x could be the value -4 but cannot 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 included in the interval, and the unshaded circle indicates 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 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 different symbols for these types subjected to 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 excluded from the subset.
[ ]: The square brackets are utilized 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 utilized when the interval is half-open, or when only the left endpoint is excluded 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 among the two. In this case, the left endpoint is not excluded in the set, while the right endpoint is not included. This is also called a right-open interval.
Number Line Representations for the Various Interval Types
Apart from being written with symbols, the different interval types can also be represented in the number line using both shaded and open circles, relying on the interval type.
The table below will display all the different types of intervals as they are represented in the number line.
Practice Examples for Interval Notation
Now that you know 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 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 expressed as (-6, 9].
Example 2
For a school to join in a debate competition, they need at least 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 needed is “three and above,” the number 3 is included on the set, which states that three is a closed value.
Plus, since no upper limit was stated regarding the number of maximum teams a school can send to the debate competition, this value should be positive to infinity.
Therefore, 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 called unbounded intervals.
Example 3
A friend wants to undertake a diet program limiting their regular 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 value 1800 is the minimum while the number 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, expressed with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is denoted as [1800, 2000].
When the subset of real numbers is restricted 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 Do You Graph an Interval Notation?
An interval notation is basically a way of representing inequalities on the number line.
There are laws of expressing an interval notation to the number line: a closed interval is expressed with a filled circle, and an open integral is expressed with an unfilled circle. This way, you can promptly 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 describing an inequality or a combination 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 expressed with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are employed.
How To Exclude Numbers in Interval Notation?
Values ruled out from the interval can be stated with parenthesis in the notation. A parenthesis implies that you’re writing an open interval, which states that the value is excluded from the combination.
Grade Potential Could Guide You Get a Grip on Math
Writing interval notations can get complicated fast. There are more nuanced topics in this concentration, such as those working on the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and many more.
If you want to conquer these concepts fast, you are required to review them with the expert assistance and study materials that the expert instructors of Grade Potential provide.
Unlock your mathematic skills with Grade Potential. Book a call now!
