Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental concept that learners are required understand because it becomes more important as you grow to more complex math.
If you see more complex arithmetics, something like differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you hours in understanding these theories.
This article will talk in-depth what interval notation is, what it’s used for, and how you can understand it.
What Is Interval Notation?
The interval notation is simply a way to express a subset of all real numbers through 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 problems you face essentially consists of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such simple utilization.
Despite that, intervals are typically used to denote domains and ranges of functions in advanced mathematics. Expressing these intervals can increasingly become difficult as the functions become more complex.
Let’s take a straightforward compound inequality notation as an example.
x is greater than negative 4 but less than 2
As we know, 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 separated by a comma.
So far we understand, interval notation is a method of writing intervals concisely and elegantly, using fixed rules that make writing and comprehending intervals on the number line easier.
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
Several types of intervals place the base for writing the interval notation. These kinds of interval are important to get to know because they underpin the complete notation process.
Open
Open intervals are applied when the expression do not contain the endpoints of the interval. The previous notation is a great example of this.
The inequality notation {x | -4 < x < 2} express x as being higher than -4 but less than 2, which means that it does not contain either of the two numbers referred to. 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 contrary of the last 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 -4 and less than or equal to 2.”
In an inequality notation, this would be written as {x | -4 < x < 2}.
In an interval notation, this is expressed with brackets, or [-4, 2]. This means that the interval contains 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 combination of previous types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the previous 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 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 expressed 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 signifies 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 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 does not include the other value.
As seen in the prior example, there are numerous symbols for these types subjected to 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 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 included. Also called 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 known as a right-open interval.
Number Line Representations for the Various Interval Types
Aside from being written with symbols, the different interval types can also be represented in the number line employing 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 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
Transform the following inequality into an interval notation: {x | -6 < x < 9}
This sample question is a straightforward conversion; just utilize 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 participate in a debate competition, they should have a at least three teams. Represent this equation in interval notation.
In this word problem, 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 states that 3 is a closed value.
Additionally, because no maximum number was referred to regarding the number of maximum 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, when 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 undertake a diet program limiting their regular calorie intake. For the diet to be successful, 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 value 1800 is the lowest while the number 2000 is the maximum 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 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 also known as a bounded interval.
Interval Notation Frequently Asked Questions
How Do You Graph an Interval Notation?
An interval notation is fundamentally a technique of describing inequalities on the number line.
There are laws to writing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is written with an unshaded circle. This way, you can quickly check the number line if the point is excluded or included from the interval.
How To Convert Inequality to Interval Notation?
An interval notation is just a different way 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 lower 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 Do You Rule Out 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 states that the number is excluded from the combination.
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