Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that learners should learn because it becomes more important as you grow to more difficult arithmetic.
If you see more complex mathematics, such as 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 about what interval notation is, what are its uses, and how you can decipher it.
What Is Interval Notation?
The interval notation is merely a method to express a subset of all real numbers along the number line.
An interval refers to the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ signifies infinity.)
Fundamental difficulties you face primarily composed of single positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such effortless applications.
Despite that, intervals are typically employed to denote domains and ranges of functions in advanced math. Expressing these intervals can increasingly become complicated as the functions become more complex.
Let’s take a simple compound inequality notation as an example.
x is higher than negative 4 but less than two
Up till now we understand, this inequality notation can be written 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 elegantly and concisely, using fixed rules that make writing and comprehending 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
Several types of intervals place the base for denoting 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 used when the expression does not include the endpoints of the interval. The previous 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 referred to. As such, this is an open interval denoted 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 contain the values mentioned, a closed interval does. In text form, a closed interval is expressed as any value “higher 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 expressed as {x | -4 < x < 2}.
In an interval notation, this is expressed 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 prior 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 negative four and less than 2.” This means that x could be the value -4 but cannot possibly be equal to the value 2.
In an inequality notation, this would be denoted 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 signifies the value excluded 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 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 examples above, there are various 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 employed when the interval is open, or when the two endpoints on the number line are not included in 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 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 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 excluded. This is also called a right-open interval.
Number Line Representations for the Different Interval Types
Apart from being written with symbols, the different 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 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
Transform the following inequality into an interval notation: {x | -6 < x < 9}
This sample question is a straightforward conversion; just 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 expressed as (-6, 9].
Example 2
For a school to participate in a debate competition, they need 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 needed is “three and above,” the number 3 is consisted in the set, which means that three is a closed value.
Plus, since no upper limit was stated with concern to the number of maximum teams a school can send to the debate competition, this value should be positive to infinity.
Therefore, the interval notation should be expressed as [3, ∞).
These types of intervals, when one side of the interval that stretches to either positive or negative infinity, are called unbounded intervals.
Example 3
A friend wants to participate in diet program constraining their daily calorie intake. For the diet to be successful, they must have minimum of 1800 calories every day, but no more than 2000. How do you describe this range in interval notation?
In this word problem, the number 1800 is the minimum while the number 2000 is the highest 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.
Therefore, 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 FAQs
How Do You Graph an Interval Notation?
An interval notation is basically a way of representing inequalities on the number line.
There are laws to writing an interval notation to the number line: a closed interval is written with a shaded circle, and an open integral is denoted with an unfilled circle. This way, you can quickly see on a number line if the point is excluded or included from the interval.
How To Change Inequality to Interval Notation?
An interval notation is just 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 lower 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 used.
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 value is ruled out from the combination.
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