Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental topic that students should understand due to the fact that it becomes more essential as you advance to more complex mathematics.
If you see higher mathematics, such as integral and differential calculus, 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 it’s used for, 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 through 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 mainly composed of single positive or negative numbers, so it can be difficult to see the utility of the interval notation from such straightforward applications.
Though, intervals are usually used to denote domains and ranges of functions in more complex mathematics. Expressing these intervals can increasingly become complicated as the functions become progressively more complex.
Let’s take a simple compound inequality notation as an example.
x is greater than negative 4 but less than two
As we know, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be expressed with interval notation (-4, 2), denoted by values a and b segregated by a comma.
So far we know, interval notation is a method of writing 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 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 essential to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are used when the expression do not contain the endpoints of the interval. The last notation is a great 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 contain 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 -4 and 2, 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 include the values mentioned, a closed interval does. In text 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 negative four and less than or equal to two.”
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 consist of 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 last example as a guide, if the interval were half-open, it would be expressed as “x is greater than or equal to negative four and less than two.” This means 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 denotes the value which are not included 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 doesn’t contain 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 prior example, there are various symbols for these types under the interval notation.
These symbols build the actual interval notation you create when stating points on a number line.
( ): The parentheses are employed 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 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 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 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 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 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 question is a straightforward conversion; just utilize 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 participate in a debate competition, they should have a minimum of three teams. Express 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 implies that 3 is a closed value.
Additionally, because no maximum number was mentioned with concern to 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 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 do a diet program constraining their regular calorie intake. For the diet to be a success, they should have minimum of 1800 calories regularly, but no more than 2000. How do you express this range in interval notation?
In this question, the number 1800 is the lowest while the value 2000 is the highest value.
The problem suggest that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, denoted 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 called a bounded interval.
Interval Notation Frequently Asked Questions
How Do You Graph an Interval Notation?
An interval notation is fundamentally a technique of representing inequalities on the number line.
There are rules 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 unfilled circle. This way, you can promptly see on a number line if the point is excluded or included from the interval.
How Do You Transform Inequality to Interval Notation?
An interval notation is basically a different technique of describing 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 expressed with parentheses () in the notation.
If x is greater than or equal to, or lower 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 used.
How To Exclude Numbers in Interval Notation?
Values ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis implies that you’re writing an open interval, which means that the value is ruled out from the set.
Grade Potential Could Guide You Get a Grip on Mathematics
Writing interval notations can get complicated fast. There are more nuanced topics within this concentration, such as those dealing with the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and more.
If you desire to conquer these ideas fast, you are required to revise them with the expert guidance and study materials that the expert instructors of Grade Potential provide.
Unlock your math skills with Grade Potential. Book a call now!
