Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential topic that pupils should grasp owing to the fact that it becomes more essential as you progress to more complex math.
If you see more complex math, such as integral and differential calculus, in front of you, then knowing the interval notation can save you hours in understanding these concepts.
This article will talk about what interval notation is, what are its uses, and how you can interpret it.
What Is Interval Notation?
The interval notation is merely a way 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 ∞ denotes infinity.)
Basic problems you face primarily consists of single positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such simple applications.
However, intervals are generally used to denote domains and ranges of functions in higher mathematics. 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 greater than negative 4 but less than two
Up till now we know, this inequality notation can be denoted 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 understand, interval notation is a way to write intervals concisely and elegantly, using set principles that make writing and comprehending intervals on the number line easier.
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
Various types of intervals place the base for denoting the interval notation. These interval types are essential to get to know because they underpin the entire notation process.
Open
Open intervals are used when the expression does not include the endpoints of the interval. The previous notation is a great example of this.
The inequality notation {x | -4 < x < 2} express x as being more than -4 but less than 2, which means that it does not contain either of the two numbers mentioned. 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 include the values mentioned, a closed interval does. In text form, a closed interval is written 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 -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 implies that the interval includes those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized to represent 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 prior example as a guide, if the interval were half-open, it would be expressed as “x is greater than or equal to -4 and less than 2.” This means that x could be the value negative four 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 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 denotes the value excluded 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 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 excludes 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 develop when expressing points on a number line.
( ): The parentheses are used 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 included 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 excluded in the set, and the right endpoint is included. Also called a left open interval.
[ ): This is also a half-open notation when there are both included and excluded values among the two. In this instance, the left endpoint is included in the set, while the right endpoint is excluded. This is also known as a right-open interval.
Number Line Representations for the Various 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’ve understood 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 problem is a simple conversion; simply utilize the equivalent symbols when denoting 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 minimum of 3 teams. Represent this equation in interval notation.
In this word problem, let x be the minimum number of teams.
Because the number of teams needed is “three and above,” the value 3 is included on the set, which implies that 3 is a closed value.
Plus, because no maximum number was referred to 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 expressed as [3, ∞).
These types of intervals, where there is 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 regular calorie intake. For the diet to be successful, they should have minimum of 1800 calories regularly, but maximum intake restricted to 2000. How do you write 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 inclusive in the range, so the equation is a close interval, denoted with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is described as [1800, 2000].
When the subset of real numbers is confined to a variation 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 way of describing inequalities on the number line.
There are rules to writing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is written with an unshaded circle. This way, you can promptly check the number line if the point is excluded or included from the interval.
How Do You Convert Inequality to Interval Notation?
An interval notation is basically a diverse way of expressing an inequality or a set 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 lower than or equal to, then the interval is denoted with closed brackets [ ] in the notation. See the examples of interval notation above to check how these symbols are used.
How Do You Rule Out Numbers in Interval Notation?
Numbers excluded from the interval can be stated with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which means that the number is excluded from the combination.
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