Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial concept that pupils should understand due to the fact that it becomes more critical as you progress to higher math.
If you see more complex arithmetics, such as differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you time in understanding these concepts.
This article will talk about 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 ∞ means infinity.)
Basic problems you face essentially composed of one positive or negative numbers, so it can be difficult to see the utility of the interval notation from such straightforward utilization.
Though, intervals are generally employed to denote domains and ranges of functions in advanced mathematics. Expressing these intervals can increasingly become complicated as the functions become more complex.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative four but less than two
As we understand, 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), signified by values a and b segregated by a comma.
So far we know, interval notation is a way to write intervals elegantly and concisely, using set rules that make writing and understanding intervals on the number line simpler.
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
Many types of intervals place the base for writing the interval notation. These kinds of interval are necessary to get to know because they underpin the entire notation process.
Open
Open intervals are used when the expression do not include the endpoints of the interval. The last notation is a fine example of this.
The inequality notation {x | -4 < x < 2} express x as being more than -4 but less than 2, meaning that it excludes 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 implies that in a given set of real numbers, such as the interval between -4 and 2, those two values are excluded.
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 contain 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 two.”
In an inequality notation, this would 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 describe 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 prior example as a guide, if the interval were half-open, it would be expressed as “x is greater than or equal to negative four and less than two.” This means that x could be the value -4 but couldn’t possibly be equal to the value 2.
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 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 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 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 develop when expressing points on a number line.
( ): The parentheses are utilized 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 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 not included 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 between the two. In this case, 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
Aside from being denoted with symbols, the various interval types can also be described 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 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
Transform the following inequality into an interval notation: {x | -6 < x < 9}
This sample problem is a simple conversion; simply use 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 take part in a debate competition, they require minimum of three teams. Express this equation in interval notation.
In this word problem, 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 3 is a closed value.
Plus, because no upper limit was mentioned regarding the number of teams a school can send to the debate competition, this number should be positive to infinity.
Therefore, the interval notation should be expressed 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 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 write this range in interval notation?
In this question, the number 1800 is the minimum while the number 2000 is the maximum 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.
Thus, the interval notation is written 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 also known as a bounded interval.
Interval Notation FAQs
How To Graph an Interval Notation?
An interval notation is simply a way 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 shaded 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 Transform 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 higher than or lower than a value (not equal to), then the value should be expressed with parentheses () in the notation.
If x is higher 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 prior to see how these symbols are employed.
How Do You Rule Out Numbers in Interval Notation?
Numbers excluded from the interval can be denoted with parenthesis in the notation. A parenthesis means that you’re expressing an open interval, which states that the value is excluded from the combination.
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