Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial principle that students should grasp due to the fact that it becomes more essential as you advance to more complex math.
If you see higher arithmetics, such as differential calculus and integral, on your horizon, then knowing the 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 merely a way to express a subset of all real numbers along 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 difficulties you encounter essentially consists of single positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such effortless utilization.
Despite that, intervals are usually used to denote domains and ranges of functions in more complex mathematics. Expressing these intervals can progressively become difficult as the functions become further 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 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 segregated by a comma.
As we can see, interval notation is a way to write intervals elegantly and concisely, using fixed principles that help writing and understanding intervals on the number line simpler.
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
Many types of intervals lay the foundation for denoting the interval notation. These kinds of interval are necessary to get to know due to the fact they underpin the entire 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} describes x as being greater than -4 but less than 2, which means that it does not include neither of the two numbers referred to. As such, this is an open interval expressed with parentheses or a round bracket, such as the following.
(-4, 2)
This means that in a given set of real numbers, such as the interval between -4 and 2, those 2 values are excluded.
On the number line, an unshaded circle denotes an open value.
Closed
A closed interval is the opposite 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 “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 expressed as {x | -4 < x < 2}.
In an interval notation, this is stated with brackets, or [-4, 2]. This states that the interval contains 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 prior 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 negative four and less than two.” This means 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 included 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 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 excludes the other value.
As seen in the examples above, there are numerous symbols for these types subjected to 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 not included in the subset.
[ ]: The square brackets are utilized 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 included. 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 included in the set, while the right endpoint is excluded. This is also called a right-open interval.
Number Line Representations for the Different Interval Types
Apart from being written with symbols, the various 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’ve understood everything you need 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. Express 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 value 3 is consisted in the set, which states that 3 is a closed value.
Furthermore, because no maximum number was mentioned with concern to the number of teams a school can send to the debate competition, this number should be positive to infinity.
Therefore, the interval notation should be written 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 limiting their daily calorie intake. For the diet to be successful, they must have at least 1800 calories regularly, but no more than 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 included 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 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 expressed with a filled circle, and an open integral is written with an unfilled circle. This way, you can promptly see on a number line if the point is included or excluded from the interval.
How To Transform Inequality to Interval Notation?
An interval notation is basically a different technique of describing 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 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 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 number is excluded from the set.
Grade Potential Can Assist You Get a Grip on Arithmetics
Writing interval notations can get complicated fast. There are more nuanced topics in this concentration, such as those working on the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and more.
If you want to master these concepts quickly, you need to review them with the professional assistance and study materials that the expert instructors of Grade Potential delivers.
Unlock your math skills with Grade Potential. Book a call now!
