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 essential as you grow to more difficult math.
If you see more complex math, something like integral and differential calculus, in front of you, then knowing the interval notation can save you hours in understanding these ideas.
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 merely a method to express a subset of all real numbers through the number line.
An interval means the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Basic difficulties you encounter primarily composed of one positive or negative numbers, so it can be challenging to see the utility of the interval notation from such simple applications.
However, intervals are usually used to denote domains and ranges of functions in more complex arithmetics. Expressing these intervals can increasingly become difficult as the functions become further complex.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative four but less than 2
As we understand, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be denoted with interval notation (-4, 2), denoted by values a and b separated by a comma.
So far we know, interval notation is a way to write intervals concisely and elegantly, using fixed principles that make writing and comprehending intervals on the number line less difficult.
In the following section we will discuss regarding the principles 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 writing the interval notation. These kinds of interval are necessary to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are applied when the expression do not include the endpoints of the interval. The prior notation is a good example of this.
The inequality notation {x | -4 < x < 2} express x as being more than negative four but less than two, meaning that it excludes either of the two numbers mentioned. 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 excluded.
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 text form, a closed interval is written 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 negative four and less than or equal to two.”
In an inequality notation, this can be expressed as {x | -4 < x < 2}.
In an interval notation, this is stated with brackets, or [-4, 2]. This means that the interval includes those two boundary values: -4 and 2.
On the number line, a shaded circle is used to describe 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 prior example as a guide, 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 implies that x could be the value negative four but couldn’t 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 present in the interval, and the unshaded circle denotes the value excluded 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 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 excludes the other value.
As seen in the last example, there are different 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 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 employed when the interval is half-open, or when only the left endpoint is not included 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 instance, the left endpoint is not excluded 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
Apart from being denoted with symbols, the different 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 display all the different types of intervals as they are described in the number line.
Practice Examples for Interval Notation
Now that you’ve understood 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 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 require 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 number 3 is included on the set, which implies that three is a closed value.
Additionally, since no upper limit was referred to regarding 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 denoted 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 do a diet program constraining their daily calorie intake. For the diet to be successful, they should have minimum of 1800 calories regularly, but no more than 2000. How do you write this range in interval notation?
In this question, the value 1800 is the lowest while the number 2000 is the maximum value.
The question suggest that both 1800 and 2000 are included in the range, so the equation is a close interval, expressed with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is denoted 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 Frequently Asked Questions
How To Graph an Interval Notation?
An interval notation is fundamentally a technique of describing inequalities on the number line.
There are rules 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 excluded or included from the interval.
How Do You Transform Inequality to Interval Notation?
An interval notation is basically a diverse technique 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 value should be stated with parentheses () in the notation.
If x is higher 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 above to check how these symbols are used.
How To Rule Out Numbers in Interval Notation?
Values excluded from the interval can be stated 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 set.
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