Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential concept that pupils need to understand because it becomes more essential as you progress to more difficult math.
If you see advances math, something like differential calculus and integral, on your horizon, then knowing the interval notation can save you time in understanding these concepts.
This article will talk in-depth what interval notation is, what are its uses, and how you can decipher 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 values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Fundamental difficulties 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 applications.
However, intervals are usually used to denote domains and ranges of functions in advanced math. Expressing these intervals can increasingly become difficult as the functions become more tricky.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative 4 but less than 2
As we know, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. Though, it can also be expressed with interval notation (-4, 2), denoted by values a and b segregated by a comma.
So far we understand, interval notation is a way to write intervals elegantly and concisely, using set principles that help writing and comprehending intervals on the number line simpler.
In the following section we will discuss about the principles of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Various types of intervals place the base for writing the interval notation. These interval types are necessary to get to know due to the fact they underpin the complete notation process.
Open
Open intervals are used when the expression does not include the endpoints of the interval. The prior notation is a good example of this.
The inequality notation {x | -4 < x < 2} describes x as being higher than negative four but less than two, meaning that it does not include neither 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 means that in a given set of real numbers, such as the interval between negative four and two, 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 not include 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 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 states 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 prior types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the previous 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 states that x could be the value -4 but couldn’t possibly be equal to the value two.
In an inequality notation, this would be denoted 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 recap, there are different types of interval notations; open, closed, and half-open. An open interval doesn’t contain 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 prior example, there are numerous symbols for these types subjected to interval notation.
These symbols build the actual interval notation you develop when plotting 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 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 called a left open interval.
[ ): This is also a half-open notation when there are both included and excluded values among 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 Different Interval Types
Aside from being written with symbols, the different interval types can also be described in the number line utilizing both shaded and open circles, depending 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 easy 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 take part in a debate competition, they need at least 3 teams. Represent this equation in interval notation.
In this word problem, let x be the minimum number of teams.
Since the number of teams required is “three and above,” the number 3 is consisted in the set, which states that three is a closed value.
Additionally, since no maximum number was mentioned regarding the number of maximum teams a school can send to the debate competition, this value 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 called unbounded intervals.
Example 3
A friend wants to participate in diet program limiting their regular calorie intake. For the diet to be successful, they should have minimum of 1800 calories regularly, but maximum intake restricted to 2000. How do you describe this range in interval notation?
In this question, the value 1800 is the lowest while the number 2000 is the highest value.
The question implies 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 restricted to a variation 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 way of describing inequalities on the number line.
There are laws to writing an interval notation to the number line: a closed interval is expressed 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 excluded or included from the interval.
How Do You Convert Inequality to Interval Notation?
An interval notation is just a diverse technique of describing an inequality or a combination of real numbers.
If x is higher than or lower than a value (not equal to), then the number should be stated 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 check how these symbols are used.
How To Rule Out Numbers in Interval Notation?
Values ruled out from the interval can be stated with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which means that the value is ruled out from the set.
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