Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that students need to understand because it becomes more important as you progress to higher arithmetic.
If you see more complex mathematics, something like differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you hours in understanding these theories.
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 simply a way to express a subset of all real numbers through the number line.
An interval refers to the values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Basic problems you face primarily consists of single positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such effortless applications.
Though, intervals are typically used to denote domains and ranges of functions in more complex arithmetics. Expressing these intervals can increasingly become difficult as the functions become more tricky.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative four but less than 2
As we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. However, it can also be denoted with interval notation (-4, 2), denoted by values a and b segregated by a comma.
As we can see, interval notation is a way to write intervals elegantly and concisely, using fixed rules that make writing and understanding intervals on the number line easier.
The following sections will tell us more regarding the rules of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Various types of intervals place the base for writing the interval notation. These interval types are essential to get to know because they underpin the complete notation process.
Open
Open intervals are applied when the expression does not comprise the endpoints of the interval. The previous notation is a great example of this.
The inequality notation {x | -4 < x < 2} describes x as being greater than negative four but less than two, which means that it excludes 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 two 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 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 last 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 would be expressed as {x | -4 < x < 2}.
In an interval notation, this is written with brackets, or [-4, 2]. This states that the interval includes 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 combination of previous types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the previous 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 -4 but cannot possibly be equal to the value two.
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 include 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 last example, there are various symbols for these types under the interval notation.
These symbols build the actual interval notation you create 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 used when the interval is closed, or when the two points on the number line are included 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 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 case, the left endpoint is not excluded in the set, while the right endpoint is excluded. This is also known as a right-open interval.
Number Line Representations for the Different Interval Types
Aside from being written with symbols, the different interval types can also be represented in the number line utilizing both shaded and open circles, relying on the interval type.
The table below will show all the different types of intervals as they are represented in the number line.
Practice Examples for Interval Notation
Now that you’ve understood everything you are required to know about writing things in interval notations, you’re ready 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 written as (-6, 9].
Example 2
For a school to take part in a debate competition, they should have a at least 3 teams. Express this equation in interval notation.
In this word question, let x be 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 3 is a closed value.
Additionally, because no upper limit was mentioned with concern to the number of maximum teams a school can send to the debate competition, this number should be positive to infinity.
Thus, the interval notation should be denoted 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 do a diet program constraining their daily calorie intake. For the diet to be successful, they must have at least 1800 calories every day, but no more than 2000. How do you express this range in interval notation?
In this question, the number 1800 is the lowest while the value 2000 is the highest value.
The problem implies 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 written 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 also known as a bounded interval.
Interval Notation Frequently Asked Questions
How To Graph an Interval Notation?
An interval notation is fundamentally a technique of describing 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 expressed with an unfilled circle. This way, you can quickly check the number line if the point is excluded or included from the interval.
How To Convert Inequality to Interval Notation?
An interval notation is just a diverse 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 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 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 ruled out from the interval can be stated with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which states that the value is excluded from the set.
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