Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial concept that learners need to grasp owing to the fact that it becomes more critical as you progress to more difficult arithmetic.
If you see more complex math, something like integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you time in understanding these theories.
This article will discuss what interval notation is, what are its uses, and how you can decipher 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 refers to the values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ signifies infinity.)
Fundamental problems you encounter primarily consists of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such effortless utilization.
Despite that, intervals are typically employed to denote domains and ranges of functions in advanced arithmetics. Expressing these intervals can increasingly become difficult as the functions become more tricky.
Let’s take a simple compound inequality notation as an example.
x is higher than negative 4 but less than two
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 way to write intervals elegantly and concisely, using set principles that make writing and comprehending 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 writing the interval notation. These interval types are essential to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are used when the expression do not comprise the endpoints of the interval. The last notation is a great example of this.
The inequality notation {x | -4 < x < 2} describes x as being greater than -4 but less than 2, meaning that it does not contain neither of the two numbers referred to. 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 -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 opposite 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 expressed as any value “higher than or equal to” or “less than or equal to.”
For example, if the last example was a closed interval, it would read, “x is greater than or equal to negative four and less than or equal to two.”
In an inequality notation, this can be expressed as {x | -4 < x < 2}.
In an interval notation, this is written with brackets, or [-4, 2]. This means that the interval includes those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized 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 last example for assistance, if the interval were half-open, it would read as “x is greater than or equal to -4 and less than 2.” This states that x could be the value negative four 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 indicates the value excluded 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 doesn’t include 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 numerous symbols for these types under the interval notation.
These symbols build the actual interval notation you develop 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 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 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 included. Also called a left open interval.
[ ): This is also a half-open notation when there are both included and excluded values within the two. In this case, the left endpoint is not excluded in the set, while the right endpoint is not included. This is also called 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 described in the number line utilizing both shaded and open circles, relying 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 prepared 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 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 expressed as (-6, 9].
Example 2
For a school to participate in a debate competition, they need minimum of 3 teams. Express this equation in interval notation.
In this word problem, let x stand for the minimum number of teams.
Since the number of teams required is “three and above,” the value 3 is consisted in the set, which implies that three is a closed value.
Furthermore, since no maximum number was stated regarding the number of teams a school can send to the debate competition, this number should be positive to infinity.
Therefore, the interval notation should be expressed as [3, ∞).
These types of intervals, when 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 a success, they must have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you write this range in interval notation?
In this question, the value 1800 is the minimum while the number 2000 is the highest value.
The question implies that both 1800 and 2000 are included in the range, so the equation is a close interval, written with the inequality 1800 ≤ x ≤ 2000.
Therefore, the interval notation is denoted as [1800, 2000].
When the subset of real numbers is restricted to a range between two values, and doesn’t stretch to either positive or negative infinity, it is also known as a bounded interval.
Interval Notation FAQs
How Do You Graph an Interval Notation?
An interval notation is simply a technique of representing inequalities on the number line.
There are laws of expressing an interval notation to the number line: a closed interval is denoted with a shaded circle, and an open integral is written with an unfilled circle. This way, you can quickly 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 different way of expressing an inequality or a set of real numbers.
If x is greater than or less a value (not equal to), then the value 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 denoted with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are employed.
How To Exclude Numbers in Interval Notation?
Values excluded from the interval can be denoted with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which means that the value is excluded from the combination.
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