Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential principle that students need to understand due to the fact that it becomes more critical as you advance to more complex math.
If you see higher math, something like integral and differential calculus, on your horizon, then knowing the interval notation can save you time in understanding these theories.
This article will talk about 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 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 simple utilization.
Though, intervals are typically used to denote domains and ranges of functions in higher math. Expressing these intervals can progressively become difficult as the functions become progressively more tricky.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative four but less than 2
As we know, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. Though, it can also be denoted with interval notation (-4, 2), signified by values a and b separated 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 easier.
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 necessary to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are applied when the expression do not comprise 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 -4 but less than 2, which means that it excludes either 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 represent that in a given set of real numbers, such as the interval between negative four and two, those two values are not included.
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 written 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 two.”
In an inequality notation, this can be expressed as {x | -4 < x < 2}.
In an interval notation, this is expressed with brackets, or [-4, 2]. This states that the interval includes those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized to represent an included open value.
Half-Open
A half-open interval is a blend of prior types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the prior example for assistance, if the interval were half-open, it would be expressed as “x is greater than or equal to negative four and less than two.” This implies that x could be the value negative four 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 denotes 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 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 does not include the other value.
As seen in the examples above, there are various symbols for these types subjected to interval notation.
These symbols build the actual interval notation you create when plotting points on a number line.
( ): The parentheses are used when the interval is open, or when the two endpoints on the number line are not included in the subset.
[ ]: The square brackets are utilized 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 not excluded. 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 not included. This is also called 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 using both shaded and open circles, relying on the interval type.
The table below will display 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 ready 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 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 join in a debate competition, they need minimum of 3 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 required is “three and above,” the value 3 is included on the set, which states that 3 is a closed value.
Furthermore, because no upper limit was referred to 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 written as [3, ∞).
These types of intervals, where there is one side of the interval that stretches to either positive or negative infinity, are called unbounded intervals.
Example 3
A friend wants to do a diet program constraining their regular calorie intake. For the diet to be successful, they should have minimum of 1800 calories regularly, but no more than 2000. How do you write this range in interval notation?
In this word problem, the number 1800 is the lowest 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 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 To Graph an Interval Notation?
An interval notation is basically a technique of describing inequalities on the number line.
There are laws of expressing an interval notation to the number line: a closed interval is written with a shaded circle, and an open integral is expressed with an unshaded circle. This way, you can quickly check the number line if the point is included or excluded from the interval.
How Do You Change Inequality to Interval Notation?
An interval notation is just a diverse technique of describing an inequality or a set of real numbers.
If x is greater than or less a value (not equal to), then the number should be expressed with parentheses () in the notation.
If x is greater than or equal to, or lower than or equal to, then the interval is expressed with closed brackets [ ] in the notation. See the examples of interval notation above to see how these symbols are employed.
How To Rule Out Numbers in Interval Notation?
Values ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which states that the number is ruled out from the set.
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