Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial concept that students should grasp because it becomes more essential as you grow to more difficult math.
If you see higher arithmetics, something like differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you time 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 way to express a subset of all real numbers across 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 encounter primarily composed of single positive or negative numbers, so it can be challenging to see the utility of the interval notation from such effortless applications.
Though, intervals are generally used to denote domains and ranges of functions in advanced math. Expressing these intervals can increasingly become complicated as the functions become further complex.
Let’s take a straightforward compound inequality notation as an example.
x is greater than negative 4 but less than 2
As we understand, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be written with interval notation (-4, 2), denoted by values a and b segregated by a comma.
So far we understand, interval notation is a method of writing intervals elegantly and concisely, using set principles that help writing and comprehending intervals on the number line less difficult.
In the following section we will discuss regarding the principles 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 writing the interval notation. These interval types are necessary to get to know because they underpin the complete notation process.
Open
Open intervals are used when the expression does not comprise the endpoints of the interval. The prior notation is a great example of this.
The inequality notation {x | -4 < x < 2} describes x as being greater than negative four but less than two, meaning that it excludes 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 represent that in a given set of real numbers, such as the interval between -4 and 2, 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 word 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 stated 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 used 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 as a guide, if the interval were half-open, it would read as “x is greater than or equal to -4 and less than 2.” This means that x could be the value -4 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 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 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 doesn’t include 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 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 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 instance, 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 described in the number line utilizing both shaded and open circles, relying on the interval type.
The table below will display all the different types of intervals as they are described 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
Transform the following inequality into an interval notation: {x | -6 < x < 9}
This sample question is a easy conversion; just use the equivalent symbols when stating 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 stand for the minimum number of teams.
Since the number of teams needed is “three and above,” the number 3 is consisted in the set, which states that 3 is a closed value.
Furthermore, because no upper limit was mentioned regarding the number of teams a school can send to the debate competition, this value should be positive to infinity.
Therefore, 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 also known as unbounded intervals.
Example 3
A friend wants to do a diet program limiting their daily calorie intake. For the diet to be a success, they must have at least 1800 calories every day, but maximum intake restricted to 2000. How do you describe this range in interval notation?
In this word problem, the value 1800 is the minimum while the number 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, 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 called a bounded interval.
Interval Notation Frequently Asked Questions
How Do You Graph an Interval Notation?
An interval notation is basically 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 written 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 Do You Convert Inequality to Interval Notation?
An interval notation is basically a different technique of expressing an inequality or a set 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 higher 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 prior to see how these symbols are utilized.
How Do You Exclude Numbers in Interval Notation?
Numbers excluded from the interval can be denoted with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which means that the value is ruled out from the set.
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