Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial principle that students should grasp owing to the fact that it becomes more essential as you advance to more complex arithmetic.
If you see more complex arithmetics, such as integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you hours in understanding these ideas.
This article will talk in-depth what interval notation is, what are its uses, and how you can interpret it.
What Is Interval Notation?
The interval notation is simply a method to express a subset of all real numbers along the number line.
An interval means the values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Fundamental problems you encounter mainly composed of one positive or negative numbers, so it can be challenging to see the utility of the interval notation from such simple utilization.
However, intervals are usually used to denote domains and ranges of functions in higher math. Expressing these intervals can progressively become difficult as the functions become progressively more tricky.
Let’s take a simple compound inequality notation as an example.
x is higher than negative 4 but less than two
So far we understand, this inequality notation can be written 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 elegantly and concisely, using predetermined principles that help writing and comprehending intervals on the number line less difficult.
The following sections will tell us more regarding 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 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 does not include the endpoints of the interval. The previous notation is a good example of this.
The inequality notation {x | -4 < x < 2} describes x as being greater than negative four but less than two, meaning that it does not contain neither 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 implies 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 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 “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 2.”
In an inequality notation, this would be expressed as {x | -4 < x < 2}.
In an interval notation, this is stated with brackets, or [-4, 2]. This states that the interval consist of those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized 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 previous example as a guide, if the interval were half-open, it would read 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 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 denotes the value which are not included 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 excludes the endpoints on the real number line, while a closed interval does. A half-open interval includes 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 create when stating 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 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 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 among the two. In this case, the left endpoint is included 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
Apart from being denoted with symbols, the different interval types can also be described in the number line utilizing 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 utilize the equivalent symbols when writing 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 need minimum of 3 teams. Represent this equation in interval notation.
In this word problem, let x stand for the minimum number of teams.
Because 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.
Furthermore, since no upper limit was stated with concern to 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 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 do a diet program constraining their daily calorie intake. For the diet to be a success, they should have minimum of 1800 calories every day, but no more than 2000. How do you write this range in interval notation?
In this question, the value 1800 is the minimum while the number 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.
Thus, the interval notation is written as [1800, 2000].
When the subset of real numbers is restricted to a variation between two values, and doesn’t stretch to either positive or negative infinity, it is called a bounded interval.
Interval Notation FAQs
How To 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 denoted with a shaded circle, and an open integral is written with an unshaded circle. This way, you can promptly check the 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 different technique of describing 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 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 prior to check how these symbols are employed.
How Do You Exclude Numbers in Interval Notation?
Numbers ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which states that the number is excluded from the combination.
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