Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental concept that learners are required understand owing to the fact that it becomes more critical as you progress to higher math.
If you see more complex math, something like integral and differential calculus, on your horizon, then being knowledgeable of interval notation can save you time in understanding these ideas.
This article will talk in-depth what interval notation is, what it’s used for, 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 means the values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Fundamental difficulties you face primarily composed of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such straightforward applications.
Despite that, intervals are usually used to denote domains and ranges of functions in more complex mathematics. Expressing these intervals can increasingly become complicated as the functions become further tricky.
Let’s take a simple compound inequality notation as an example.
x is greater 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. However, it can also be denoted 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 elegantly and concisely, using set rules that help writing and comprehending intervals on the number line less difficult.
In the following section we will discuss about the principles of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Several types of intervals place the base for writing the interval notation. These kinds of interval are essential to get to know due to the fact they underpin the complete notation process.
Open
Open intervals are used when the expression does not comprise the endpoints of the interval. The prior notation is a good example of this.
The inequality notation {x | -4 < x < 2} describes x as being more than -4 but less than 2, which means that it does not include 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 represent that in a given set of real numbers, such as the interval between negative four and two, those two 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 exclude the values mentioned, a closed interval does. In word form, a closed interval is written 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 two.”
In an inequality notation, this would be expressed as {x | -4 < x < 2}.
In an interval notation, this is expressed with brackets, or [-4, 2]. This implies 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 blend of previous 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 negative four and less than two.” This states that x could be the value negative four but cannot possibly be equal to the value two.
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 signifies the value which are not included from the subset.
Symbols for Interval Notation and Types of Intervals
To summarize, 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 excludes the other value.
As seen in the prior example, there are numerous symbols for these types subjected to interval notation.
These symbols build the actual interval notation you develop when expressing points on a number line.
( ): The parentheses are used 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 used when the interval is half-open, or when only the left endpoint is not included 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 within the two. In this instance, the left endpoint is included 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 various 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 show 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 prepared 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 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 take part 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.
Because the number of teams required is “three and above,” the number 3 is consisted in the set, which implies that three is a closed value.
Furthermore, since no upper limit was mentioned with concern to the number of teams a school can send to the debate competition, this number 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 limiting their daily calorie intake. For the diet to be a success, they should have minimum of 1800 calories regularly, but maximum intake restricted to 2000. How do you describe 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 inclusive in the range, so the equation is a close interval, expressed with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is written 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 describing inequalities on the number line.
There are laws to writing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is written with an unfilled circle. This way, you can quickly see on a number line if the point is included or excluded from the interval.
How Do You Convert Inequality to Interval Notation?
An interval notation is basically 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 expressed 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 prior to check how these symbols are employed.
How Do You Rule Out Numbers in Interval Notation?
Numbers ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which means that the number is excluded from the set.
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