Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential topic that pupils should grasp owing to the fact that it becomes more important as you grow to more difficult arithmetic.
If you see more complex math, such as integral and differential calculus, on your horizon, 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 are its uses, and how you can decipher it.
What Is Interval Notation?
The interval notation is merely a way to express a subset of all real numbers through the number line.
An interval means the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)
Fundamental difficulties you encounter mainly composed 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 usually used to denote domains and ranges of functions in higher mathematics. Expressing these intervals can increasingly become complicated as the functions become progressively more tricky.
Let’s take a simple compound inequality notation as an example.
x is greater 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 segregated by a comma.
So far we know, interval notation is a way to write intervals elegantly and concisely, using predetermined principles that help writing and understanding intervals on the number line less difficult.
The following sections will tell us more about 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 kinds of interval are important to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are applied when the expression do not include 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 greater than -4 but less than 2, meaning that it does not contain 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 -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 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 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 negative four 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 written with brackets, or [-4, 2]. This means that the interval consist of those two boundary values: -4 and 2.
On the number line, a shaded circle is used to represent 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 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 -4 but cannot 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 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 indicates 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 does not include the other value.
As seen in the examples above, 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 employed when the interval is open, or when the two endpoints on the number line are not included in the subset.
[ ]: The square brackets are employed 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 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 within the two. In this instance, the left endpoint is not excluded 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
Apart from being denoted with symbols, the different interval types can also be represented 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 described in the number line.
Practice Examples for Interval Notation
Now that you’ve understood 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 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 expressed as (-6, 9].
Example 2
For a school to join in a debate competition, they should have a minimum of 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 needed is “three and above,” the number 3 is consisted in the set, which states that 3 is a closed value.
Additionally, because no upper limit 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 written 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 participate in diet program constraining their regular 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 value 1800 is the minimum while the value 2000 is the maximum 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 variation between two values, and doesn’t stretch to either positive or negative infinity, it is also known as a bounded interval.
Interval Notation FAQs
How Do You Graph an Interval Notation?
An interval notation is simply a technique 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 unfilled circle. This way, you can promptly see on a number line if the point is included or excluded from the interval.
How Do You Transform Inequality to Interval Notation?
An interval notation is just a different way of describing an inequality or a set of real numbers.
If x is higher than or lower than 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 ruled out from the interval can be stated with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which means that the value is ruled out from the set.
Grade Potential Can Guide You Get a Grip on Arithmetics
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