Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental concept that pupils are required learn owing to the fact that it becomes more important as you progress to more complex mathematics.
If you see more complex arithmetics, such as differential calculus and integral, on your horizon, then knowing the interval notation can save you hours in understanding these theories.
This article will discuss what interval notation is, what are its uses, 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 refers to the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ signifies infinity.)
Fundamental difficulties you face primarily consists of single positive or negative numbers, so it can be challenging to see the utility of the interval notation from such straightforward applications.
Despite that, intervals are typically employed to denote domains and ranges of functions in higher math. Expressing these intervals can progressively become complicated as the functions become more tricky.
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 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), denoted 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 fixed rules that make writing and comprehending intervals on the number line easier.
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
Various types of intervals place the base for denoting 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 used when the expression do not include the endpoints of the interval. The prior notation is a fine example of this.
The inequality notation {x | -4 < x < 2} express x as being higher than -4 but less than 2, meaning that it excludes 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 not included.
On the number line, an unshaded circle denotes an open value.
Closed
A closed interval is the contrary 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 written 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 stated with brackets, or [-4, 2]. This implies that the interval contains 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 previous 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 2.” This implies that x could be the value -4 but cannot 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 indicates 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 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 examples above, there are numerous symbols for these types under the interval notation.
These symbols build the actual interval notation you create 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 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 among the two. In this instance, 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 denoted with symbols, the different interval types can also be described in the number line using both shaded and open circles, relying on the interval type.
The table below will display 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 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
Transform the following inequality into an interval notation: {x | -6 < x < 9}
This sample question is a simple conversion; just utilize the equivalent symbols when denoting 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 join in a debate competition, they need at least three teams. Express this equation in interval notation.
In this word problem, let x stand for the minimum number of teams.
Since the number of teams required is “three and above,” the number 3 is included on the set, which means that 3 is a closed value.
Additionally, because no upper limit was stated regarding the number of maximum teams a school can send to the debate competition, this number should be positive to infinity.
Thus, 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 called unbounded intervals.
Example 3
A friend wants to participate in diet program constraining their daily calorie intake. For the diet to be successful, they should have at least 1800 calories every day, but maximum intake restricted to 2000. How do you express this range in interval notation?
In this question, the value 1800 is the minimum while the number 2000 is the maximum value.
The question implies that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, written 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 range 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 fundamentally a technique of describing inequalities on the number line.
There are rules of expressing 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 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 just 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 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 excluded from the interval can be written with parenthesis in the notation. A parenthesis implies that you’re expressing an open interval, which means that the value is excluded from the combination.
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