Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential topic that students need to grasp because it becomes more critical as you advance to higher arithmetic.
If you see advances math, something like integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you time in understanding these concepts.
This article will talk in-depth what interval notation is, what it’s used for, and how you can decipher it.
What Is Interval Notation?
The interval notation is simply a method to express a subset of all real numbers along the number line.
An interval means the values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ signifies infinity.)
Fundamental problems you encounter mainly composed of single positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such simple utilization.
However, intervals are generally used to denote domains and ranges of functions in advanced math. Expressing these intervals can increasingly become difficult as the functions become progressively more tricky.
Let’s take a simple compound inequality notation as an example.
x is higher than negative four but less than 2
So far we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. However, it can also be written with interval notation (-4, 2), signified by values a and b separated by a comma.
As we can see, interval notation is a way to write intervals concisely and elegantly, using set rules that make writing and understanding intervals on the number line easier.
In the following section we will discuss about the rules of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Many types of intervals lay the foundation for writing the interval notation. These kinds of interval are necessary to get to know because they underpin the entire notation process.
Open
Open intervals are applied when the expression do not include the endpoints of the interval. The prior notation is a good example of this.
The inequality notation {x | -4 < x < 2} express x as being more than -4 but less than 2, which means that it does not include 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 -4 and 2, 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 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 previous example was a closed interval, it would read, “x is greater than or equal to negative four and less than or equal to 2.”
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 states that the interval includes those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized to denote 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 prior example for assistance, if the interval were half-open, it would be expressed as “x is greater than or equal to -4 and less than 2.” This implies that x could be the value negative four but couldn’t 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 denoted 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 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 excludes 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 last example, there are different symbols for these types under the interval notation.
These symbols build the actual interval notation you develop when plotting points on a number line.
( ): The parentheses are utilized 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 used when the interval is half-open, or when only the left endpoint is not included 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 among the two. In this instance, the left endpoint is included in the set, while the right endpoint is not included. This is also known as a right-open interval.
Number Line Representations for the Various Interval Types
Aside from being denoted with symbols, the different interval types can also be represented in the number line utilizing both shaded and open circles, depending 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 simple conversion; just use 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 join in a debate competition, they require at least three teams. Represent this equation in interval notation.
In this word question, let x be the minimum number of teams.
Since the number of teams needed is “three and above,” the number 3 is consisted in the set, which means that 3 is a closed value.
Furthermore, because no maximum number was mentioned with concern to the number of teams a school can send to the debate competition, this value should be positive to infinity.
Thus, 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 constraining their daily calorie intake. For the diet to be successful, they must have at least 1800 calories every day, but no more than 2000. How do you express this range in interval notation?
In this question, the number 1800 is the lowest while the value 2000 is the highest value.
The question suggest 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 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 basically a way of representing inequalities on the number line.
There are rules to writing an interval notation to the number line: a closed interval is expressed with a shaded circle, and an open integral is written with an unfilled circle. This way, you can promptly check the number line if the point is included or excluded from the interval.
How Do You 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 number should be written 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 prior to see how these symbols are used.
How Do You Exclude Numbers in Interval Notation?
Numbers excluded from the interval can be written with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which states that the number is ruled out from the combination.
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