Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental concept that pupils need to understand owing to the fact that it becomes more important as you progress to more complex arithmetic.
If you see higher arithmetics, such as differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you time in understanding these ideas.
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 simply a way to express a subset of all real numbers along 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.)
Basic difficulties you face essentially consists of single positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such straightforward applications.
However, intervals are typically used to denote domains and ranges of functions in advanced math. Expressing these intervals can increasingly become difficult as the functions become more tricky.
Let’s take a simple compound inequality notation as an example.
x is higher than negative four but less than two
So far we know, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be written with interval notation (-4, 2), denoted by values a and b separated by a comma.
As we can see, interval notation is a method of writing intervals elegantly and concisely, using set rules that make writing and comprehending intervals on the number line easier.
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
Various types of intervals place the base for denoting the interval notation. These interval types are necessary to get to know due to the fact they underpin the complete notation process.
Open
Open intervals are applied when the expression does not contain the endpoints of the interval. The prior notation is a great example of this.
The inequality notation {x | -4 < x < 2} describes x as being greater than -4 but less than 2, which means that it excludes neither 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 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 include the values mentioned, a closed interval does. In word 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 -4 and less than or equal to 2.”
In an inequality notation, this can be written as {x | -4 < x < 2}.
In an interval notation, this is written with brackets, or [-4, 2]. This means that the interval contains those two boundary values: -4 and 2.
On the number line, a shaded circle is used to denote 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 previous example for assistance, if the interval were half-open, it would read as “x is greater than or equal to -4 and less than two.” This states that x could be the value -4 but cannot possibly be equal to the value 2.
In an inequality notation, this would be written 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 denotes 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 includes one value on the line but does not include the other value.
As seen in the last example, there are numerous 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 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 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 included. Also known as a left open interval.
[ ): This is also a half-open notation when there are both included and excluded values between the two. In this instance, the left endpoint is not excluded in the set, while the right endpoint is excluded. This is also known as 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 described in the number line employing 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 problem is a easy conversion; simply 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 written as (-6, 9].
Example 2
For a school to participate in a debate competition, they require minimum of three teams. Represent this equation in interval notation.
In this word question, let x stand for the minimum number of teams.
Since the number of teams needed is “three and above,” the number 3 is included on the set, which states that three is a closed value.
Furthermore, since no upper limit was mentioned regarding 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 denoted 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 undertake a diet program limiting their regular calorie intake. For the diet to be successful, they must have minimum of 1800 calories regularly, but no more than 2000. How do you describe 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 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.
Therefore, 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 called a bounded interval.
Interval Notation Frequently Asked Questions
How Do You Graph an Interval Notation?
An interval notation is simply a technique of representing inequalities on the number line.
There are laws of expressing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is denoted 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 basically 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 value should be written with parentheses () in the notation.
If x is higher than or equal to, or less than or equal to, then the interval is denoted with closed brackets [ ] in the notation. See the examples of interval notation above to check how these symbols are utilized.
How Do You Exclude Numbers in Interval Notation?
Numbers excluded from the interval can be written with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which means that the value is ruled out from the set.
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