Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial principle that students are required learn due to the fact that it becomes more important as you grow to more difficult arithmetic.
If you see advances arithmetics, something like integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you time in understanding these theories.
This article will talk about what interval notation is, what are its uses, and how you can interpret it.
What Is Interval Notation?
The interval notation is simply a way to express a subset of all real numbers across 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 problems you encounter essentially composed of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such effortless utilization.
Despite that, intervals are generally employed to denote domains and ranges of functions in more complex math. Expressing these intervals can progressively become complicated as the functions become progressively more tricky.
Let’s take a simple compound inequality notation as an example.
x is higher than negative 4 but less than 2
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), denoted 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 set principles that help writing and comprehending 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 essential to get to know because they underpin the entire notation process.
Open
Open intervals are applied when the expression does not include the endpoints of the interval. The previous notation is a good example of this.
The inequality notation {x | -4 < x < 2} express x as being higher than negative four but less than two, 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 negative four and two, those 2 values are not included.
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 not contain 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 -4 and less than or equal to 2.”
In an inequality notation, this would be written as {x | -4 < x < 2}.
In an interval notation, this is written with brackets, or [-4, 2]. This implies that the interval consist of those two boundary values: -4 and 2.
On the number line, a shaded circle is employed to denote an included open value.
Half-Open
A half-open interval is a combination of previous 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 be expressed as “x is greater than or equal to -4 and less than 2.” This means 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 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 include 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 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 excluded from the subset.
[ ]: The square brackets are utilized 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 employed 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 instance, 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
Aside 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 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 are required 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 use 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 expressed as (-6, 9].
Example 2
For a school to take part in a debate competition, they require minimum of three teams. Represent this equation in interval notation.
In this word problem, let x be the minimum number of teams.
Because the number of teams required is “three and above,” the value 3 is consisted in the set, which means that 3 is a closed value.
Furthermore, because no upper limit was mentioned with concern to the number of teams a school can send to the debate competition, this value 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 also known as unbounded intervals.
Example 3
A friend wants to undertake a diet program constraining their daily calorie intake. For the diet to be a success, they must have at least 1800 calories regularly, but no more than 2000. How do you describe this range in interval notation?
In this question, the value 1800 is the minimum while the number 2000 is the maximum value.
The problem suggest that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, written 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 FAQs
How To Graph an Interval Notation?
An interval notation is fundamentally a technique of describing inequalities on the number line.
There are rules to writing an interval notation to the number line: a closed interval is expressed with a filled circle, and an open integral is written with an unshaded circle. This way, you can quickly see on a number line if the point is excluded or included from the interval.
How Do You Transform Inequality to Interval Notation?
An interval notation is just a different technique 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 stated with parentheses () in the notation.
If x is higher 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 Rule Out Numbers in Interval Notation?
Numbers excluded from the interval can be stated with parenthesis in the notation. A parenthesis implies that you’re writing an open interval, which means that the value is excluded from the set.
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