Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that students need to understand because it becomes more essential as you grow to more difficult arithmetic.
If you see advances mathematics, such as integral and differential calculus, on your horizon, then being knowledgeable of interval notation can save you hours in understanding these theories.
This article will discuss 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 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 ∞ means infinity.)
Fundamental difficulties you face essentially composed of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such effortless applications.
Though, intervals are generally used to denote domains and ranges of functions in advanced arithmetics. Expressing these intervals can increasingly become difficult as the functions become progressively more complex.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative four but less than two
As we know, 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), denoted by values a and b segregated by a comma.
So far we know, interval notation is a way to write intervals concisely and elegantly, using set rules that help writing and understanding intervals on the number line simpler.
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 place the base for denoting the interval notation. These interval types are necessary to get to know due to the fact they underpin the entire 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} describes x as being higher than negative four but less than two, which means that it does not contain either 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 negative four and two, those two values are excluded.
On the number line, an unshaded circle denotes an open value.
Closed
A closed interval is the opposite of the last type of interval. Where the open interval does not include 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 two.”
In an inequality notation, this would be expressed as {x | -4 < x < 2}.
In an interval notation, this is stated with brackets, or [-4, 2]. This implies that the interval consist of 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 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 as a guide, if the interval were half-open, it would read as “x is greater than or equal to negative four 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 written 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 included 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
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 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 expressing 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 utilized 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 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 not excluded in the set, while the right endpoint is not included. This is also known as a right-open interval.
Number Line Representations for the Different Interval Types
Apart from being written with symbols, the different 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 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 problem is a straightforward conversion; simply 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 should have a 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 required is “three and above,” the value 3 is included on the set, which means that three is a closed value.
Plus, since no maximum number was mentioned regarding the number of maximum teams a school can send to the debate competition, this number should be positive to infinity.
Therefore, the interval notation should be denoted 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 participate in diet program limiting their regular calorie intake. For the diet to be a success, they must have minimum of 1800 calories regularly, but no more than 2000. How do you express this range in interval notation?
In this question, the number 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, denoted with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is described as [1800, 2000].
When the subset of real numbers is confined to a range 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 fundamentally a way of representing inequalities on the number line.
There are rules of expressing 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 check the number line if the point is included or excluded from the interval.
How To Convert 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 stated with parentheses () in the notation.
If x is higher 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 above to see how these symbols are utilized.
How To Exclude Numbers in Interval Notation?
Values excluded from the interval can be stated with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which states that the value is ruled out from the combination.
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