Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that students should understand because it becomes more essential as you grow to more difficult arithmetic.
If you see higher mathematics, such as differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you time in understanding these theories.
This article will discuss what interval notation is, what it’s used for, 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 mainly consists of single positive or negative numbers, so it can be challenging to see the benefit of the interval notation from such effortless applications.
However, intervals are usually used to denote domains and ranges of functions in higher mathematics. Expressing these intervals can increasingly become difficult as the functions become progressively more complex.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative 4 but less than two
Up till now we know, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. Though, it can also be written with interval notation (-4, 2), denoted 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 fixed rules that make writing and understanding intervals on the number line less difficult.
In the following section we will discuss regarding the principles of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Several types of intervals place the base for writing the interval notation. These kinds of interval are essential to get to know due to the fact they underpin the complete 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} express x as being more than negative four but less than two, which means that it does not include 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 implies 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 contrary 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 “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 two.”
In an inequality notation, this can be expressed as {x | -4 < x < 2}.
In an interval notation, this is written 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 employed to represent 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 for assistance, if the interval were half-open, it would be expressed as “x is greater than or equal to -4 and less than two.” This implies that x could be the value -4 but couldn’t 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 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 last example, there are various symbols for these types under the interval notation.
These symbols build the actual interval notation you create 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 utilized 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 used 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 not excluded in the set, while the right endpoint is excluded. This is also known as a right-open interval.
Number Line Representations for the Different Interval Types
Apart from being denoted with symbols, the various interval types can also be represented 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 know 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 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 participate in a debate competition, they need at least three teams. Represent this equation in interval notation.
In this word problem, let x be the minimum number of teams.
Since the number of teams required is “three and above,” the number 3 is included on the set, which implies that 3 is a closed value.
Plus, because no maximum number was stated with concern to the number of maximum teams a school can send to the debate competition, this value should be positive to infinity.
Therefore, 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 called unbounded intervals.
Example 3
A friend wants to undertake a diet program limiting their regular calorie intake. For the diet to be successful, they must have at least 1800 calories regularly, but no more than 2000. How do you express this range in interval notation?
In this word problem, the number 1800 is the minimum while the value 2000 is the maximum value.
The problem suggest 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 denoted as [1800, 2000].
When the subset of real numbers is restricted to a variation 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 way of representing inequalities on the number line.
There are laws to writing an interval notation to the number line: a closed interval is written with a filled circle, and an open integral is denoted with an unfilled circle. This way, you can quickly check the number line if the point is excluded or included from the interval.
How To Transform Inequality to Interval Notation?
An interval notation is basically a different technique of expressing 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 written with parentheses () in the notation.
If x is higher than or equal to, or less than or equal to, then the interval is written with closed brackets [ ] in the notation. See the examples of interval notation prior to see how these symbols are utilized.
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 expressing an open interval, which means that the value is excluded from the set.
Grade Potential Can Help You Get a Grip on Mathematics
Writing interval notations can get complex fast. There are multiple difficult topics in this concentration, 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 master these ideas fast, you need to review them with the professional help and study materials that the expert tutors of Grade Potential provide.
Unlock your mathematic skills with Grade Potential. Call us now!
