Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential principle that pupils are required grasp due to the fact that it becomes more critical as you advance to higher arithmetic.
If you see more complex math, such as differential calculus and integral, in front of you, then being knowledgeable of interval notation can save you time in understanding these concepts.
This article will talk about what interval notation is, what are its uses, and how you can understand it.
What Is Interval Notation?
The interval notation is merely a way to express a subset of all real numbers across the number line.
An interval refers to the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Basic difficulties you face mainly consists of single 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 generally employed to denote domains and ranges of functions in higher arithmetics. Expressing these intervals can progressively become difficult as the functions become further tricky.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative 4 but less than two
Up till now we know, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be expressed with interval notation (-4, 2), signified 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 predetermined rules that make writing and understanding intervals on the number line simpler.
The following sections will tell us more 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 essential to get to know because they underpin the entire notation process.
Open
Open intervals are applied when the expression do not contain the endpoints of the interval. The last 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 does not contain 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 represent that in a given set of real numbers, such as the interval between -4 and 2, those 2 values are not included.
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 not contain 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 last example was a closed interval, it would read, “x is greater than or equal to negative four and less than or equal to two.”
In an inequality notation, this would 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 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 last 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 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 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
In brief, 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 examples above, 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 utilized 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 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 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 Various Interval Types
Aside from being written with symbols, the different interval types can also be represented in the number line utilizing both shaded and open circles, relying 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’ve understood everything you are required 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 problem is a simple conversion; simply 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 join in a debate competition, they should have a at least 3 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 number 3 is consisted in the set, which states that three is a closed value.
Plus, since no upper limit was referred to with concern to 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 also known as unbounded intervals.
Example 3
A friend wants to do a diet program constraining their regular calorie intake. For the diet to be a success, they must have minimum of 1800 calories regularly, but maximum intake restricted to 2000. How do you express this range in interval notation?
In this word problem, the value 1800 is the minimum while the number 2000 is the highest 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.
Therefore, the interval notation is denoted 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 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 of expressing an interval notation to the number line: a closed interval is written with a filled circle, and an open integral is expressed with an unshaded circle. This way, you can promptly see on a number line if the point is included or excluded from the interval.
How To Change 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 higher than or lower than 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 less than or equal to, then the interval is written with closed brackets [ ] in the notation. See the examples of interval notation prior to see how these symbols are utilized.
How To Rule Out Numbers in Interval Notation?
Values excluded from the interval can be written with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which means that the number is ruled out from the combination.
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