Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential topic that learners need to learn due to the fact that it becomes more important as you grow to more difficult mathematics.
If you see higher math, something like integral and differential calculus, in front of you, then knowing the interval notation can save you time in understanding these ideas.
This article will talk in-depth what interval notation is, what it’s used for, 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 across 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 consists of single positive or negative numbers, so it can be challenging to see the utility of the interval notation from such simple utilization.
Though, intervals are generally used to denote domains and ranges of functions in more complex math. Expressing these intervals can progressively become difficult as the functions become more tricky.
Let’s take a simple compound inequality notation as an example.
x is greater than negative 4 but less than 2
So far we understand, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. However, it can also be expressed 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 set principles that make writing and comprehending intervals on the number line simpler.
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 lay the foundation 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 prior notation is a good example of this.
The inequality notation {x | -4 < x < 2} describes x as being higher than negative four but less than two, meaning that it does not contain neither of the two numbers referred to. 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 contain the values mentioned, a closed interval does. In word form, a closed interval is written 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 -4 and less than or equal to two.”
In an inequality notation, this can be written as {x | -4 < x < 2}.
In an interval notation, this is written 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 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 prior example for assistance, if the interval were half-open, it would be expressed as “x is greater than or equal to negative four and less than 2.” This states that x could be the value -4 but couldn’t possibly be equal to the value two.
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 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
In brief, there are different types of interval notations; open, closed, and half-open. An open interval excludes 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 numerous symbols for these types under the interval notation.
These symbols build the actual interval notation you create 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 employed 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 between the two. In this instance, the left endpoint is not excluded 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
Aside from being denoted with symbols, the different interval types can also be described 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 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 simple conversion; simply use 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 should have a at least 3 teams. Represent this equation in interval notation.
In this word question, let x stand for the minimum number of teams.
Because the number of teams needed is “three and above,” the value 3 is included on the set, which means that three is a closed value.
Additionally, since no upper limit was stated regarding 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, when one side of the interval that stretches to either positive or negative infinity, are called unbounded intervals.
Example 3
A friend wants to undertake a diet program constraining their daily calorie intake. For the diet to be a success, they must have at least 1800 calories regularly, but no more than 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 problem suggest that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, written 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 technique of describing inequalities on the number line.
There are rules to writing an interval notation to the number line: a closed interval is written with a filled circle, and an open integral is denoted 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 Do You Transform Inequality to Interval Notation?
An interval notation is just 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 lower than or equal to, then the interval is written 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 stated with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which means that the number is ruled out from the combination.
Grade Potential Could Guide You Get a Grip on Math
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