Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental concept that learners should grasp due to the fact that it becomes more important as you grow to higher math.
If you see more complex arithmetics, such as differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you hours in understanding these concepts.
This article will talk in-depth what interval notation is, what it’s used for, and how you can decipher it.
What Is Interval Notation?
The interval notation is merely 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 ∞ means infinity.)
Basic difficulties you encounter essentially composed of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such simple utilization.
Despite that, intervals are usually employed to denote domains and ranges of functions in advanced math. Expressing these intervals can increasingly become difficult as the functions become progressively more complex.
Let’s take a simple compound inequality notation as an example.
x is higher than negative four but less than 2
As we understand, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. Though, it can also be denoted with interval notation (-4, 2), signified by values a and b separated by a comma.
So far we know, interval notation is a way to write intervals concisely and elegantly, using predetermined rules that help writing and understanding 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
Many types of intervals place the base for denoting the interval notation. These kinds of interval are necessary to get to know due to the fact they underpin the complete notation process.
Open
Open intervals are applied when the expression does not contain the endpoints of the interval. The last notation is a fine example of this.
The inequality notation {x | -4 < x < 2} express x as being greater than -4 but less than 2, meaning that it excludes 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 implies 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 exclude the values mentioned, a closed interval does. In text form, a closed interval is written 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 2.”
In an inequality notation, this can be written as {x | -4 < x < 2}.
In an interval notation, this is expressed with brackets, or [-4, 2]. This implies that the interval includes those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized to denote 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 prior 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 negative four but couldn’t possibly be equal to the value two.
In an inequality notation, this would be expressed 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 indicates the value which are not included from the subset.
Symbols for Interval Notation and Types of Intervals
To summarize, 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 examples above, there are numerous symbols for these types under the 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 employed 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 used 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 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 Various Interval Types
Aside from being denoted 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 show 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 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 take part in a debate competition, they need at least 3 teams. Represent this equation in interval notation.
In this word question, let x be the minimum number of teams.
Because 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.
Furthermore, because no upper limit was mentioned with concern to the number of teams a school can send to the debate competition, this number 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 participate in diet program limiting their regular calorie intake. For the diet to be successful, they should have minimum of 1800 calories every day, but no more than 2000. How do you express this range in interval notation?
In this word problem, the value 1800 is the minimum 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, expressed 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 fundamentally a way 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 unshaded circle. This way, you can promptly see on a number line if the point is included or excluded from the interval.
How To Transform Inequality to Interval Notation?
An interval notation is basically a diverse 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 value should be written 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 prior to check how these symbols are utilized.
How To Exclude Numbers in Interval Notation?
Values ruled out from the interval can be written with parenthesis in the notation. A parenthesis implies that you’re writing an open interval, which states that the number is excluded from the set.
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