Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental concept that pupils are required grasp due to the fact that it becomes more essential as you grow to more difficult arithmetic.
If you see advances mathematics, 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 discuss what interval notation is, what it’s used for, and how you can decipher 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 values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ signifies infinity.)
Fundamental problems you face primarily composed of one positive or negative numbers, so it can be challenging to see the utility of the interval notation from such effortless applications.
Though, intervals are generally employed to denote domains and ranges of functions in more complex math. 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 higher than negative 4 but less than two
As we know, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. Though, 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 rules that make writing and understanding intervals on the number line easier.
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 place the base for writing the interval notation. These interval types are important to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are used when the expression do not include the endpoints of the interval. The previous notation is a fine 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 does not include either of the two numbers referred to. As such, this is an open interval denoted 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 -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 opposite of the previous type of interval. Where the open interval does not contain 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 -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 expressed with brackets, or [-4, 2]. This means that the interval consist of 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 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 read as “x is greater than or equal to -4 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 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 indicates 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 does not include the other value.
As seen in the examples above, there are different 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 utilized when the interval is open, or when the two endpoints on the number line are not included in the subset.
[ ]: The square brackets are utilized 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 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 called 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 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 using 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 know 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 utilize 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 join in a debate competition, they need at least 3 teams. Represent this equation in interval notation.
In this word problem, let x stand for the minimum number of teams.
Since 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 maximum number 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 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 undertake a diet program limiting their daily calorie intake. For the diet to be successful, 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 word problem, the number 1800 is the minimum while the value 2000 is the maximum value.
The question implies that both 1800 and 2000 are included in the range, so the equation is a close interval, written with the inequality 1800 ≤ x ≤ 2000.
Thus, 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 To Graph an Interval Notation?
An interval notation is basically a technique of representing inequalities on the number line.
There are rules of expressing 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 check the number line if the point is excluded or included from the interval.
How Do You Change Inequality to Interval Notation?
An interval notation is basically a different way of describing an inequality or a combination of real numbers.
If x is greater than or less a value (not equal to), then the number 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 check 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 implies that you’re writing an open interval, which means that the number is excluded from the set.
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