Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that students are required grasp owing to the fact that it becomes more critical as you advance to more difficult math.
If you see advances mathematics, such as differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you time in understanding these ideas.
This article will discuss 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 across the number line.
An interval refers to the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Fundamental problems you face essentially composed of single positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such simple utilization.
Despite that, intervals are typically used to denote domains and ranges of functions in higher math. Expressing these intervals can increasingly become complicated as the functions become progressively more complex.
Let’s take a straightforward compound inequality notation as an example.
x is greater than negative four but less than 2
So far we understand, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. Though, it can also be expressed with interval notation (-4, 2), signified by values a and b separated by a comma.
So far we know, interval notation is a method of writing intervals elegantly and concisely, using set rules that help writing and comprehending intervals on the number line easier.
In the following section we will discuss about the rules 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 writing the interval notation. These interval types are essential to get to know because they underpin the entire notation process.
Open
Open intervals are applied when the expression does not contain the endpoints of the interval. The prior notation is a fine example of this.
The inequality notation {x | -4 < x < 2} describes x as being more than negative four but less than two, meaning that it excludes 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 implies that in a given set of real numbers, such as the interval between negative four and two, 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 exclude the values mentioned, a closed interval does. In word form, a closed interval is expressed as any value “higher 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 would be written as {x | -4 < x < 2}.
In an interval notation, this is stated 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 utilized to describe 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 last example as a guide, if the interval were half-open, it would read as “x is greater than or equal to -4 and less than two.” This states that x could be the value -4 but couldn’t 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 written 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 indicates the value which are not included from the subset.
Symbols for Interval Notation and Types of Intervals
To recap, 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 does not include the other value.
As seen in the prior example, there are different symbols for these types subjected to interval notation.
These symbols build the actual interval notation you develop when plotting points on a number line.
( ): The parentheses are utilized when the interval is open, or when the two endpoints on the number line are excluded from 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 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 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 represented in the number line using both shaded and open circles, depending on the interval type.
The table below will display 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 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
Transform 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 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 means that 3 is a closed value.
Additionally, because no upper limit was referred to regarding the number of 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 also known as unbounded intervals.
Example 3
A friend wants to do a diet program constraining their regular calorie intake. For the diet to be successful, they must have at least 1800 calories regularly, but no more than 2000. How do you describe this range in interval notation?
In this question, the value 1800 is the minimum while the value 2000 is the highest value.
The question implies that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, denoted with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is denoted as [1800, 2000].
When the subset of real numbers is confined 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 describing inequalities on the number line.
There are rules of expressing an interval notation to the number line: a closed interval is denoted with a shaded circle, and an open integral is denoted with an unfilled circle. This way, you can promptly see on a number line if the point is excluded or included from the interval.
How Do You Change Inequality to Interval Notation?
An interval notation is basically a different way of expressing an inequality or a combination of real numbers.
If x is greater than or less 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 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 utilized.
How To Rule Out Numbers in Interval Notation?
Values 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 set.
Grade Potential Could Guide You Get a Grip on Math
Writing interval notations can get complicated fast. There are multiple difficult topics within this area, such as those dealing with the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and more.
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