Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential concept that learners should grasp owing to the fact that it becomes more essential as you advance to higher arithmetic.
If you see higher arithmetics, something like differential calculus and integral, on your horizon, then knowing the interval notation can save you hours in understanding these theories.
This article will talk in-depth 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 means the values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)
Basic problems you encounter essentially consists of one positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such effortless utilization.
Though, intervals are typically employed to denote domains and ranges of functions in advanced arithmetics. Expressing these intervals can progressively become complicated as the functions become progressively more tricky.
Let’s take a simple compound inequality notation as an example.
x is higher than negative four but less than 2
So far we know, this inequality notation can be written 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 understand, interval notation is a way to write intervals concisely and elegantly, using set rules that help writing and understanding intervals on the number line simpler.
In the following section we will discuss regarding the rules of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Several types of intervals lay the foundation for writing the interval notation. These interval types are necessary 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 last notation is a great example of this.
The inequality notation {x | -4 < x < 2} describes x as being higher than negative four but less than two, which means that it does not contain either 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 negative four and two, 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 previous type of interval. Where the open interval does not include the values mentioned, a closed interval does. In word form, a closed interval is expressed 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 -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 expressed 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 used to denote an included open value.
Half-Open
A half-open interval is a blend 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 negative four and less than two.” This implies 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 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 present in the interval, and the unshaded circle signifies the value excluded from the subset.
Symbols for Interval Notation and Types of Intervals
In brief, 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 last example, there are various symbols for these types subjected to interval notation.
These symbols build the actual interval notation you develop when stating 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 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 among the two. In this instance, the left endpoint is not excluded in the set, while the right endpoint is excluded. This is also called a right-open interval.
Number Line Representations for the Various Interval Types
Apart from being written with symbols, the various interval types can also be represented in the number line utilizing 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 need 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 easy conversion; simply 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 take part in a debate competition, they need at least three 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 needed is “three and above,” the number 3 is consisted in the set, which means that 3 is a closed value.
Additionally, because 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 denoted 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 undertake a diet program limiting their daily 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 express this range in interval notation?
In this word problem, the value 1800 is the lowest while the value 2000 is the maximum value.
The problem implies that both 1800 and 2000 are included 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 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 Frequently Asked Questions
How Do You Graph an Interval Notation?
An interval notation is fundamentally a technique 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 filled circle, and an open integral is expressed with an unshaded circle. This way, you can quickly see on a number line if the point is included or excluded from the interval.
How Do You Change Inequality to Interval Notation?
An interval notation is basically a diverse way 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 stated with parentheses () in the notation.
If x is higher than or equal to, or lower than or equal to, then the interval is written with closed brackets [ ] in the notation. See the examples of interval notation above to check how these symbols are utilized.
How Do You Exclude 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 means that the number is ruled out from the set.
Grade Potential Can Assist You Get a Grip on Arithmetics
Writing interval notations can get complex fast. There are multiple nuanced topics in this area, such as those working on the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and many more.
If you want to conquer these concepts quickly, you need to review them with the expert assistance and study materials that the expert teachers of Grade Potential delivers.
Unlock your math skills with Grade Potential. Connect with us now!
