Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a essential principle that pupils need to grasp owing to the fact that it becomes more important as you progress to more complex arithmetic.
If you see higher arithmetics, something like differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you time in understanding these concepts.
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 along the number line.
An interval means the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ signifies infinity.)
Basic problems you encounter mainly composed of single positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such simple applications.
Though, intervals are generally used to denote domains and ranges of functions in advanced 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 greater than negative four but less than two
As we understand, this inequality notation can be expressed 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 method of writing intervals concisely and elegantly, using set principles that make 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
Various types of intervals lay the foundation for writing the interval notation. These kinds of interval are necessary to get to know because they underpin the complete notation process.
Open
Open intervals are used when the expression does not contain the endpoints of the interval. The prior notation is a good example of this.
The inequality notation {x | -4 < x < 2} describes x as being higher than negative four but less than two, meaning that it does not contain neither 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 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 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 expressed as any value “higher 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 2.”
In an inequality notation, this can be expressed as {x | -4 < x < 2}.
In an interval notation, this is expressed with brackets, or [-4, 2]. This implies 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 combination of prior 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 be expressed as “x is greater than or equal to -4 and less than two.” This means that x could be the value negative four 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 included 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 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 does not include the other value.
As seen in the prior 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 utilized when the interval is open, or when the two endpoints on the number line are excluded from 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 utilized 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 within 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 different 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 know everything you need 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 easy conversion; just utilize the equivalent symbols when stating 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 participate in a debate competition, they should have a minimum of 3 teams. Express this equation in interval notation.
In this word question, let x be 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 three is a closed value.
Additionally, since no maximum number was stated regarding the number of 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 undertake a diet program limiting their daily calorie intake. For the diet to be a success, they must have minimum of 1800 calories every day, but maximum intake restricted to 2000. How do you write this range in interval notation?
In this question, the number 1800 is the minimum while the value 2000 is the highest value.
The problem suggest that both 1800 and 2000 are included in the range, so the equation is a close interval, expressed with the inequality 1800 ≤ x ≤ 2000.
Therefore, the interval notation is written 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 called a bounded interval.
Interval Notation FAQs
How Do You Graph an Interval Notation?
An interval notation is basically a way of representing inequalities on the number line.
There are rules of expressing an interval notation to the number line: a closed interval is written with a filled circle, and an open integral is expressed with an unfilled circle. This way, you can promptly check the number line if the point is included or excluded from the interval.
How To Transform Inequality to Interval Notation?
An interval notation is just 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 number should be written with parentheses () in the notation.
If x is greater than or equal to, or lower 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 used.
How Do You Exclude Numbers in Interval Notation?
Numbers excluded from the interval can be written with parenthesis in the notation. A parenthesis means that you’re writing an open interval, which states that the value is excluded from the set.
Grade Potential Could 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 more.
If you desire to master these ideas quickly, you are required to review them with the expert help and study materials that the professional teachers of Grade Potential provide.
Unlock your mathematic skills with Grade Potential. Call us now!
