Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial topic that learners need to learn because it becomes more critical as you grow to more difficult math.
If you see higher arithmetics, something like integral and differential calculus, in front of you, then being knowledgeable of interval notation can save you hours in understanding these concepts.
This article will talk about what interval notation is, what are its uses, and how you can interpret it.
What Is Interval Notation?
The interval notation is simply a way to express a subset of all real numbers through 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.)
Fundamental problems you encounter primarily consists of one positive or negative numbers, so it can be difficult to see the utility of the interval notation from such straightforward applications.
Though, intervals are usually used to denote domains and ranges of functions in higher math. Expressing these intervals can progressively become difficult as the functions become further complex.
Let’s take a straightforward compound inequality notation as an example.
x is higher than negative 4 but less than 2
As we understand, this inequality notation can be expressed as: {x | -4 < x < 2} in set builder notation. However, it can also be written 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 concisely and elegantly, using predetermined principles that help writing and understanding 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
Several types of intervals lay the foundation for denoting the interval notation. These interval types are essential to get to know due to the fact they underpin the entire notation process.
Open
Open intervals are applied when the expression does not include the endpoints of the interval. The prior notation is a great 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 referred to. As such, this is an open interval denoted 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 negative four and two, 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 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 -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 states that the interval contains those two boundary values: -4 and 2.
On the number line, a shaded circle is employed 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 previous example as a guide, if the interval were half-open, it would read as “x is greater than or equal to -4 and less than two.” This states that x could be the value -4 but couldn’t possibly be equal to the value two.
In an inequality notation, this would be expressed 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 which are not included 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 includes one value on the line but does not include the other value.
As seen in the examples above, there are different 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 employed when the interval is open, or when the two endpoints on the number line are excluded from the subset.
[ ]: The square brackets are used 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 employed 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 between 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 Various Interval Types
Aside from being written with symbols, the different interval types can also be described in the number line employing both shaded and open circles, relying on the interval type.
The table below will show 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 question is a simple conversion; just 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 should have a minimum of 3 teams. Represent this equation in interval notation.
In this word problem, let x be the minimum number of teams.
Because the number of teams needed is “three and above,” the value 3 is consisted in the set, which implies that 3 is a closed value.
Plus, because 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 expressed 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 constraining their daily calorie intake. For the diet to be successful, they should have at least 1800 calories regularly, but maximum intake restricted to 2000. How do you describe this range in interval notation?
In this word problem, the number 1800 is the lowest 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, written 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 range 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 basically a technique of representing inequalities on the number line.
There are laws of expressing an interval notation to the number line: a closed interval is denoted with a filled circle, and an open integral is expressed with an unfilled circle. This way, you can promptly see on a number line if the point is excluded or included from the interval.
How Do You Change Inequality to Interval Notation?
An interval notation is basically a different technique of expressing an inequality or a set of real numbers.
If x is higher than or lower than a value (not equal to), then the number should be expressed with parentheses () in the notation.
If x is higher than or equal to, or lower 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 utilized.
How Do You Exclude 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 means that the number is ruled out from the set.
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