Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a crucial concept that pupils need to learn due to the fact that it becomes more essential as you advance to more complex math.
If you see more complex mathematics, such as differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you time in understanding these ideas.
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 merely a method to express a subset of all real numbers along the number line.
An interval means the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ means infinity.)
Basic problems you encounter essentially consists of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such straightforward applications.
Despite that, intervals are typically used to denote domains and ranges of functions in advanced 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 4 but less than two
As we understand, this inequality notation can be written as: {x | -4 < x < 2} in set builder notation. Despite that, it can also be written with interval notation (-4, 2), denoted by values a and b segregated by a comma.
As we can see, interval notation is a way to write intervals concisely and elegantly, using set principles that make writing and understanding intervals on the number line easier.
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
Several types of intervals place the base for denoting the interval notation. These interval types are necessary to get to know because they underpin the complete notation process.
Open
Open intervals are applied when the expression does not comprise the endpoints of the interval. The last notation is a great example of this.
The inequality notation {x | -4 < x < 2} express x as being greater than negative four but less than two, which means that it excludes either of the two numbers mentioned. As such, this is an open interval denoted with parentheses or a round bracket, such as the following.
(-4, 2)
This represent 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 contain 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 2.”
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 means that the interval includes those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized 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 prior 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 two.” This means that x could be the value -4 but cannot 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 denotes 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 contain 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 prior example, there are different symbols for these types subjected to interval notation.
These symbols build the actual interval notation you develop 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 used 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 excluded in the set, and the right endpoint is included. 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 instance, the left endpoint is not excluded in the set, while the right endpoint is not included. This is also called a right-open interval.
Number Line Representations for the Different Interval Types
Aside from being written 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 display all the different types of intervals as they are represented in the number line.
Practice Examples for Interval Notation
Now that you’ve understood everything you need 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 easy conversion; just utilize 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 join in a debate competition, they need at least three teams. Represent this equation in interval notation.
In this word question, 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 three is a closed value.
Plus, since no maximum number was mentioned 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 also known as unbounded intervals.
Example 3
A friend wants to undertake a diet program constraining their daily calorie intake. For the diet to be a success, 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 word problem, the value 1800 is the lowest while the number 2000 is the highest 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 described 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 To Graph an Interval Notation?
An interval notation is basically a technique of representing inequalities on the number line.
There are rules of expressing an interval notation to the number line: a closed interval is denoted with a shaded circle, and an open integral is written with an unshaded circle. This way, you can promptly 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 just a diverse way 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 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 written with closed brackets [ ] in the notation. See the examples of interval notation prior to check how these symbols are employed.
How Do You Exclude Numbers in Interval Notation?
Numbers excluded from the interval can be denoted 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.
Grade Potential Can Assist You Get a Grip on Mathematics
Writing interval notations can get complex fast. There are multiple nuanced topics within this concentration, such as those dealing with the union of intervals, fractions, absolute value equations, inequalities with an upper bound, and many more.
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