Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential principle that learners need to understand due to the fact that it becomes more essential as you progress to more difficult math.
If you see higher math, something like differential calculus and integral, in front of you, then being knowledgeable of interval notation can save you time in understanding these concepts.
This article will talk in-depth 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 across the number line.
An interval means the values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Basic difficulties you encounter essentially composed of single positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such simple utilization.
Though, intervals are generally used 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 simple compound inequality notation as an example.
x is greater than negative 4 but less than 2
Up till now we understand, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. However, it can also be denoted 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 elegantly and concisely, using set 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
Several types of intervals place the base for writing the interval notation. These interval types are necessary to get to know because they underpin the entire 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} express x as being greater than negative four but less than two, which means that it does not contain neither 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 means 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 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 written 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 -4 and less than or equal to 2.”
In an inequality notation, this can be expressed as {x | -4 < x < 2}.
In an interval notation, this is expressed with brackets, or [-4, 2]. This states that the interval includes those two boundary values: -4 and 2.
On the number line, a shaded circle is employed to describe an included open value.
Half-Open
A half-open interval is a blend of previous 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 be expressed as “x is greater than or equal to -4 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 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 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
In brief, 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 does not include 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 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 used 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 employed 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 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 known as a right-open interval.
Number Line Representations for the Different Interval Types
Apart from being written with symbols, the various 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 know 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 question is a straightforward conversion; just utilize 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 should have a minimum of three teams. Express this equation in interval notation.
In this word problem, 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 implies that 3 is a closed value.
Plus, because no maximum number was referred to 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 denoted 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 undertake a diet program constraining their regular calorie intake. For the diet to be successful, they must have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you express this range in interval notation?
In this word problem, the value 1800 is the lowest while the value 2000 is the highest value.
The problem 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 written 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 also known as a bounded interval.
Interval Notation Frequently Asked Questions
How Do You Graph an Interval Notation?
An interval notation is simply a way of describing inequalities on the number line.
There are rules to writing an interval notation to the number line: a closed interval is denoted 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 Do You Convert 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 greater than or less a value (not equal to), then the number 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 employed.
How Do You Rule Out Numbers in Interval Notation?
Numbers ruled out from the interval can be written with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which states that the value is ruled out from the combination.
Grade Potential Can Help You Get a Grip on Mathematics
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