Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental concept that pupils should learn due to the fact that it becomes more critical as you advance to more difficult arithmetic.
If you see advances math, something like integral and differential calculus, on your horizon, then being knowledgeable of interval notation can save you hours in understanding these concepts.
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 simply a method to express a subset of all real numbers along the number line.
An interval refers to 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 one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such simple utilization.
However, intervals are typically employed to denote domains and ranges of functions in more complex arithmetics. Expressing these intervals can increasingly become complicated 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 written as: {x | -4 < x < 2} in set builder notation. Though, it can also be expressed with interval notation (-4, 2), denoted by values a and b separated by a comma.
As we can see, interval notation is a method of writing intervals concisely and elegantly, using predetermined rules that help writing and understanding intervals on the number line simpler.
The following sections will tell us more regarding the principles of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Many types of intervals lay the foundation for writing the interval notation. These kinds of interval are important to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are applied when the expression does not contain the endpoints of the interval. The prior notation is a great example of this.
The inequality notation {x | -4 < x < 2} express x as being higher than negative four but less than two, which means that it excludes neither 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 means that in a given set of real numbers, such as the interval between negative four and two, those two 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 “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 2.”
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 implies that the interval consist of those two boundary values: -4 and 2.
On the number line, a shaded circle is employed to represent an included open value.
Half-Open
A half-open interval is a combination of prior types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the last 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 implies that x could be the value negative four but couldn’t possibly be equal to the value 2.
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 excluded from the subset.
Symbols for Interval Notation and Types of Intervals
In brief, 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 excludes the other value.
As seen in the last example, there are different symbols for these types under the 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 not included in the subset.
[ ]: The square brackets are used 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 included. Also called 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 included in the set, while the right endpoint is not included. This is also known as 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 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’ve understood everything you need 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 straightforward 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 written as (-6, 9].
Example 2
For a school to participate in a debate competition, they require at least three teams. Express this equation in interval notation.
In this word question, let x be the minimum number of teams.
Since the number of teams needed is “three and above,” the value 3 is included on the set, which means that three is a closed value.
Furthermore, since no maximum number was mentioned regarding the number of maximum teams a school can send to the debate competition, this number should be positive to infinity.
Thus, 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 participate in diet program limiting their daily calorie intake. For the diet to be successful, they should have minimum of 1800 calories every day, but maximum intake restricted to 2000. How do you express 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 suggest that both 1800 and 2000 are included in the range, so the equation is a close interval, written 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 variation 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 technique of describing inequalities on the number line.
There are laws to writing an interval notation to the number line: a closed interval is expressed with a shaded 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 Convert Inequality to Interval Notation?
An interval notation is basically a different technique of describing an inequality or a combination of real numbers.
If x is higher than or lower than 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 expressed with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are utilized.
How Do You Rule Out Numbers in Interval Notation?
Numbers ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which states that the value is excluded from the combination.
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