Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that learners need to grasp because it becomes more critical as you advance to more difficult math.
If you see advances arithmetics, something like differential calculus and integral, 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 understand it.
What Is Interval Notation?
The interval notation is simply a method to express a subset of all real numbers along the number line.
An interval means the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)
Basic problems you face primarily consists of one 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 higher arithmetics. Expressing these intervals can increasingly become difficult as the functions become progressively more tricky.
Let’s take a simple compound inequality notation as an example.
x is higher than negative four but less than two
So far we understand, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. Though, it can also be written with interval notation (-4, 2), signified by values a and b separated by a comma.
As we can see, interval notation is a method of writing intervals elegantly and concisely, using predetermined rules that help writing and comprehending intervals on the number line easier.
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
Various types of intervals lay the foundation for denoting the interval notation. These kinds of interval are essential to get to know because they underpin the complete notation process.
Open
Open intervals are applied when the expression do not contain the endpoints of the interval. The previous notation is a good example of this.
The inequality notation {x | -4 < x < 2} describes x as being more than negative four but less than two, meaning that it excludes either 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 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 expressed 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 -4 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 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 utilized to describe 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 prior example as a guide, if the interval were half-open, it would be expressed as “x is greater than or equal to -4 and less than two.” This means that x could be the value -4 but couldn’t possibly be equal to the value 2.
In an inequality notation, this would be denoted as {x | -4 < x < 2}.
A half-open interval notation is denoted 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 signifies 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 doesn’t include the endpoints on the real number line, while a closed interval does. A half-open interval includes one value on the line but excludes the other value.
As seen in the last example, there are various symbols for these types subjected to interval notation.
These symbols build the actual interval notation you develop when plotting points on a number line.
( ): The parentheses are utilized 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 utilized 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 within the two. In this instance, the left endpoint is included in the set, while the right endpoint is not included. This is also called a right-open interval.
Number Line Representations for the Different Interval Types
Aside 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’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
Convert the following inequality into an interval notation: {x | -6 < x < 9}
This sample problem is a simple conversion; simply 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 take part in a debate competition, they should have a minimum of three 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 required is “three and above,” the number 3 is consisted in the set, which means that three is a closed value.
Additionally, since no upper limit 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, when 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 daily calorie intake. For the diet to be successful, they must have at least 1800 calories every day, but no more than 2000. How do you write this range in interval notation?
In this word problem, the number 1800 is the minimum while the value 2000 is the maximum value.
The problem 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.
Thus, 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 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 written with a filled circle, and an open integral is written with an unfilled circle. This way, you can quickly check the number line if the point is excluded or included from the interval.
How To Convert Inequality to Interval Notation?
An interval notation is basically a different way of describing 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 expressed 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 prior to see how these symbols are employed.
How Do You Exclude Numbers in Interval Notation?
Numbers ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which states that the value is excluded from the set.
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