Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that learners need to grasp owing to the fact that it becomes more essential as you advance to more complex math.
If you see advances math, something like integral and differential calculus, on your horizon, then knowing the interval notation can save you hours in understanding these theories.
This article will discuss what interval notation is, what it’s used for, and how you can understand it.
What Is Interval Notation?
The interval notation is simply a way to express a subset of all real numbers across 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 problems you encounter essentially composed of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such effortless applications.
Though, intervals are usually employed to denote domains and ranges of functions in more complex mathematics. Expressing these intervals can increasingly become complicated as the functions become progressively more complex.
Let’s take a simple compound inequality notation as an example.
x is greater than negative four but less than two
As we understand, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. However, it can also be denoted with interval notation (-4, 2), signified by values a and b separated by a comma.
So far we understand, interval notation is a method of writing intervals concisely and elegantly, using set principles that make writing and understanding intervals on the number line easier.
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
Various 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 complete notation process.
Open
Open intervals are used when the expression do not include the endpoints of the interval. The last notation is a fine example of this.
The inequality notation {x | -4 < x < 2} express x as being greater than negative four but less than two, which means that it does not contain either of the two numbers mentioned. 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 opposite of the previous 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 “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 can be written as {x | -4 < x < 2}.
In an interval notation, this is expressed with brackets, or [-4, 2]. This implies that the interval contains those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized 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 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 two.” This states that x could be the value negative four but couldn’t 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 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 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 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 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 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 utilized 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 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 included in the set, while the right endpoint is not included. This is also called a right-open interval.
Number Line Representations for the Different Interval Types
Aside from being denoted with symbols, the various interval types can also be represented 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 represented 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
Convert the following inequality into an interval notation: {x | -6 < x < 9}
This sample question is a simple conversion; just 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 expressed as (-6, 9].
Example 2
For a school to join in a debate competition, they require 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 number 3 is consisted in the set, which implies that 3 is a closed value.
Plus, because no upper limit was stated regarding the number of 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, 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 minimum of 1800 calories regularly, but no more than 2000. How do you describe this range in interval notation?
In this word problem, the value 1800 is the lowest while the number 2000 is the highest 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.
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 Frequently Asked Questions
How To Graph an Interval Notation?
An interval notation is basically a way 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 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 Do You Transform 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 higher than or lower than a value (not equal to), then the number should be expressed with parentheses () in the notation.
If x is greater than or equal to, or less than or equal to, then the interval is expressed with closed brackets [ ] in the notation. See the examples of interval notation above to check how these symbols are used.
How Do You Exclude Numbers in Interval Notation?
Numbers ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which means that the value is ruled out from the combination.
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