Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental concept that learners need to grasp due to the fact that it becomes more important as you advance to more difficult math.
If you see higher mathematics, such as integral and differential calculus, on your horizon, then knowing the interval notation can save you hours in understanding these concepts.
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 way 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.)
Fundamental difficulties you encounter essentially consists of single positive or negative numbers, so it can be difficult to see the utility of the interval notation from such effortless applications.
Though, intervals are typically employed to denote domains and ranges of functions in more complex math. Expressing these intervals can progressively become complicated as the functions become further tricky.
Let’s take a straightforward compound inequality notation as an example.
x is greater than negative four but less than 2
So far we know, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. However, it can also be denoted 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 concisely and elegantly, using set principles that make writing and understanding intervals on the number line less difficult.
The following sections will tell us more about 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 interval types are essential to get to know because they underpin the complete notation process.
Open
Open intervals are applied 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 negative four but less than two, meaning that it does not include 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 -4 and 2, those 2 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 not include the values mentioned, a closed interval does. In word form, a closed interval is expressed 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 negative four and less than or equal to two.”
In an inequality notation, this would be written as {x | -4 < x < 2}.
In an interval notation, this is expressed with brackets, or [-4, 2]. This states that the interval includes 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 prior types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the prior example for assistance, if the interval were half-open, it would read as “x is greater than or equal to negative four and less than 2.” This implies that x could be the value -4 but cannot possibly be equal to the value 2.
In an inequality notation, this would be expressed 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 signifies 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 contain 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 under the 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 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 among the two. In this instance, the left endpoint is included 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
Apart from being denoted with symbols, the various interval types can also be represented 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’ve understood everything you are required 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 written as (-6, 9].
Example 2
For a school to join in a debate competition, they should have a minimum of 3 teams. Express this equation in interval notation.
In this word problem, let x be the minimum number of teams.
Since the number of teams needed is “three and above,” the value 3 is consisted in the set, which means that 3 is a closed value.
Plus, because no upper limit was mentioned with concern to 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 expressed 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 participate in diet program limiting their daily calorie intake. For the diet to be a success, they should have minimum of 1800 calories every day, but maximum intake restricted to 2000. How do you write this range in interval notation?
In this question, the number 1800 is the minimum while the number 2000 is the maximum value.
The problem suggest that both 1800 and 2000 are included in the range, so the equation is a close interval, denoted 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 also known as a bounded interval.
Interval Notation FAQs
How To Graph an Interval Notation?
An interval notation is simply a way of representing inequalities on the number line.
There are rules to writing an interval notation to the number line: a closed interval is expressed with a shaded circle, and an open integral is written with an unshaded circle. This way, you can promptly 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 describing 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 stated with parentheses () in the notation.
If x is greater 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 prior to see how these symbols are utilized.
How Do You Exclude Numbers in Interval Notation?
Numbers ruled out from the interval can be stated with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which states that the number is ruled out from the combination.
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