Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental topic that students are required understand owing to the fact that it becomes more important as you grow to higher math.
If you see advances math, such as differential calculus and integral, 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 understand it.
What Is Interval Notation?
The interval notation is merely a method to express a subset of all real numbers through 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 composed of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such simple utilization.
Though, intervals are usually used to denote domains and ranges of functions in advanced arithmetics. Expressing these intervals can progressively become complicated as the functions become further complex.
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 written as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be expressed with interval notation (-4, 2), signified by values a and b segregated by a comma.
As we can see, interval notation is a way to write intervals elegantly and concisely, using fixed principles that make writing and understanding intervals on the number line easier.
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 interval types are important to get to know because they underpin the entire notation process.
Open
Open intervals are applied when the expression do not contain the endpoints of the interval. The prior notation is a good example of this.
The inequality notation {x | -4 < x < 2} express x as being more than negative four but less than two, meaning that it does not contain 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 represent that in a given set of real numbers, such as the interval between negative four and two, 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 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 “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 negative four 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 implies that the interval includes those two boundary values: -4 and 2.
On the number line, a shaded circle is employed to denote an included open value.
Half-Open
A half-open interval is a combination of previous 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 states that x could be the value -4 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 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 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 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 last example, there are numerous 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 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 utilized 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 known as 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 called a right-open interval.
Number Line Representations for the Various Interval Types
Apart from being written with symbols, the various interval types can also be represented in the number line using 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 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 easy 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 join in a debate competition, they require minimum of 3 teams. Represent this equation in interval notation.
In this word question, 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 states that 3 is a closed value.
Plus, because no maximum number was mentioned regarding 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 expressed as [3, ∞).
These types of intervals, where there is one side of the interval that stretches to either positive or negative infinity, are also known as unbounded intervals.
Example 3
A friend wants to participate in diet program limiting their regular calorie intake. For the diet to be successful, they must 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 value 2000 is the highest value.
The question 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.
Therefore, 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 called a bounded interval.
Interval Notation FAQs
How To Graph an Interval Notation?
An interval notation is simply 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 denoted with a shaded circle, and an open integral is expressed with an unfilled circle. This way, you can promptly check the number line if the point is included or excluded from the interval.
How To Transform Inequality to Interval Notation?
An interval notation is just a diverse way of describing an inequality or a set of real numbers.
If x is higher than or lower than 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 denoted with closed brackets [ ] in the notation. See the examples of interval notation prior to see how these symbols are employed.
How To Exclude Numbers in Interval Notation?
Values ruled out from the interval can be written with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which means that the number is ruled out from the set.
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