Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental principle that learners need to understand because it becomes more important as you progress to more difficult mathematics.
If you see more complex arithmetics, such as differential calculus and integral, 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 understand it.
What Is Interval Notation?
The interval notation is simply a way to express a subset of all real numbers through 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 primarily consists of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such effortless utilization.
Despite that, intervals are typically employed to denote domains and ranges of functions in more complex math. Expressing these intervals can increasingly become difficult as the functions become progressively more tricky.
Let’s take a straightforward compound inequality notation as an example.
x is greater than negative four but less than two
As we understand, 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), signified by values a and b separated by a comma.
So far we know, interval notation is a method of writing 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 rules of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Many types of intervals lay the foundation for denoting the interval notation. These kinds of interval are essential to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are applied when the expression does not comprise the endpoints of the interval. The last 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, which means that it excludes 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 implies that in a given set of real numbers, such as the interval between negative four and two, 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 text form, a closed interval is expressed as any value “greater 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 would be written as {x | -4 < x < 2}.
In an interval notation, this is stated 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 utilized to represent 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 prior example as a guide, if the interval were half-open, it would read as “x is greater than or equal to -4 and less than 2.” This implies 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 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 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 various symbols for these types subjected to interval notation.
These symbols build the actual interval notation you develop when stating 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 used 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 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 case, the left endpoint is included in the set, while the right endpoint is not included. This is also known as 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 represented in the number line employing 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 need 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 straightforward conversion; just 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 written as (-6, 9].
Example 2
For a school to participate in a debate competition, they should have a minimum of three teams. Represent this equation in interval notation.
In this word question, let x be the minimum number of teams.
Because the number of teams required is “three and above,” the value 3 is included on the set, which implies that 3 is a closed value.
Plus, 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, 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 daily calorie intake. For the diet to be a success, they must have at least 1800 calories every day, but maximum intake restricted to 2000. How do you describe this range in interval notation?
In this word problem, the value 1800 is the minimum while the value 2000 is the maximum value.
The problem implies that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, denoted with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is denoted 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 To Graph an Interval Notation?
An interval notation is fundamentally a way of describing inequalities on the number line.
There are laws of expressing 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 included or excluded from the interval.
How To Transform Inequality to Interval Notation?
An interval notation is just a diverse technique of expressing an inequality or a combination 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 greater than or equal to, or lower than or equal to, then the interval is denoted with closed brackets [ ] in the notation. See the examples of interval notation prior to see how these symbols are employed.
How Do You Rule Out Numbers in Interval Notation?
Numbers excluded from the interval can be denoted with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which means that the number is excluded from the set.
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