Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial principle that pupils are required understand owing to the fact that it becomes more essential as you progress to more complex mathematics.
If you see advances arithmetics, something like differential calculus and integral, in front of you, then being knowledgeable of interval notation can save you hours in understanding these concepts.
This article will discuss what interval notation is, what it’s used for, and how you can understand 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 values between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Fundamental difficulties you encounter essentially consists of one positive or negative numbers, so it can be challenging to see the utility of the interval notation from such straightforward applications.
Though, intervals are typically used to denote domains and ranges of functions in more complex arithmetics. Expressing these intervals can increasingly become difficult as the functions become further complex.
Let’s take a simple compound inequality notation as an example.
x is higher than negative four but less than 2
Up till now we know, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be expressed with interval notation (-4, 2), signified by values a and b separated by a comma.
As we can see, interval notation is a way to write intervals elegantly and concisely, using predetermined principles that help writing and comprehending intervals on the number line less difficult.
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 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 last 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 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 -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 opposite of the last type of interval. Where the open interval does exclude the values mentioned, a closed interval does. In text form, a closed interval is expressed as any value “greater 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 negative four and less than or equal to 2.”
In an inequality notation, this can be written as {x | -4 < x < 2}.
In an interval notation, this is expressed with brackets, or [-4, 2]. This means that the interval consist of 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 for assistance, if the interval were half-open, it would read 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 written 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 included in the interval, and the unshaded circle denotes the value which are not included from the subset.
Symbols for Interval Notation and Types of Intervals
To summarize, 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 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 employed 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 included. Also called 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 not excluded in the set, while the right endpoint is not included. This is also known as a right-open interval.
Number Line Representations for the Various Interval Types
Aside from being written with symbols, the various interval types can also be represented 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 are required 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 problem is a simple 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 written as (-6, 9].
Example 2
For a school to take part in a debate competition, they need at least three teams. Express this equation in interval notation.
In this word question, let x stand for the minimum number of teams.
Since the number of teams needed is “three and above,” the number 3 is included on the set, which states that three is a closed value.
Furthermore, since no maximum number was referred to with concern to the number of maximum teams a school can send to the debate competition, this number should be positive to infinity.
Therefore, 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 participate in diet program limiting their daily calorie intake. For the diet to be successful, they must have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you write this range in interval notation?
In this question, the value 1800 is the minimum while the number 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.
Therefore, the interval notation is denoted as [1800, 2000].
When the subset of real numbers is restricted 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 Do You Graph an Interval Notation?
An interval notation is simply 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 shaded circle, and an open integral is expressed with an unfilled circle. This way, you can quickly see on a number line if the point is included or excluded from the interval.
How Do You Transform Inequality to Interval Notation?
An interval notation is just 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 number should be written with parentheses () in the notation.
If x is greater than or equal to, or less 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 utilized.
How To Rule Out Numbers in Interval Notation?
Values excluded from the interval can be written with parenthesis in the notation. A parenthesis implies that you’re writing an open interval, which states that the value is excluded from the combination.
Grade Potential Can Guide You Get a Grip on Arithmetics
Writing interval notations can get complicated fast. There are many nuanced topics within this area, such as those dealing with the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and many more.
If you want to conquer these concepts fast, you need to review them with the professional help and study materials that the professional teachers of Grade Potential provide.
Unlock your arithmetics skills with Grade Potential. Connect with us now!
