Helpful Tools for Calculus, Chapter 0: Numbers, Notations and Notes
Previous Section: 0.3 Infinity
Absolute Values will be used throughout the text, so it would be helpful to be familiar with how they work and what they represent. The notation for the absolute value of some number x is |x|. This can be formula defined as the following:
Absolute Value and Magnitude
The absolute value represents the magnitude of a number, or its distance away from 0. For example, |3|=3 and |-3|=3. Another way to define an absolute value is so that
because we generally assume that a square root has a positive result, and squaring the number first will not change the outcome when taking the root. Similar to how 
, we can say that 
.
Absolute values are used not only to define the distance away from 0, but also the distance between two numbers. |b–a| represents the distance between two numbers a and b. It is important to note that |b–a|≠|b|-|a| and similarly, |a+b|≠|a|+|b| unless a and b are the same sign, or one of them is 0.
If a and b are opposite signs, then a+b will result in some cancellation between their values. If we think of them as lengths, then a and b having opposite signs is like moving forward a units, then backwards b units. So, |a+b|≤|a|+|b|. This is called the Triangle Inequality. In terms of physics, we can think of this as saying, “The displacement is less than or equal to the total distance travelled.”
In many of the problems we will encounter it is necessary to identify a domain or interval on which our problems will make sense. Absolute values play a big role in doing this for problems involving distance, lengths, shapes, and more. Let’s first talk about intervals before applying absolute values to them.
There are three main types of intervals:
- Both endpoints are included (Closed Interval)
- Both endpoints are not included (Open Interval)
- One endpoint is included and the other is not (Half-Open Interval)
A closed interval is used when the boundary points are meant to be included in the problem. For example, if we ask how much we should cut off a 10 inch piece of paper it is possible to cut between 0 and 10 inches because that would love some paper left. It is also possible to not cut anything off, so we can cut off 0 inches. Similarly, we could remove the whole part which is equivalent to removing all 10 inches. We can describe this interval using square brackets like the following: [0,10].
An open interval is used when the boundary points are not, or cannot, be included in a problem. Take the example of a soda can that holds some amount of liquid. It has a volume, because it holds liquid. Therefore, it cannot have a radius of 0. Similarly, it cannot have a radius that is infinitely large so its radius is represented, with parenthesis, as the open interval (0,∞). Open intervals are commonly used in calculus because any time we describe an interval with infinity an open interval must be used. This is because it is not possible for something to actually equal infinity. So the boundary of infinity cannot be included. For example, the domain of a function might be described as (-∞,∞) because any number, no matter how large, positive or negative, can be plugged in.
A half-open interval is used in cases where one boundary cannot be included and the other can. The included boundary will have a square bracket on it, while the other would have a parenthesis. For example, [5,12) means between 5 and 12, including 5.
Here are some examples with number lines to provide a visual. Empty circles represent boundaries that are not included (open intervals) while filled in circles represent included boundaries (closed intervals):
Graphing Intervals
Here is an example with a number line to provide a visual. Sometimes the interval is drawn on the number line, but often it is drawn above the number line for clarity. Empty circles represent boundaries that are not included (open intervals) while filled in circles represent included boundaries (closed intervals):
So how do these intervals relate to absolute values? The answer lies with inequalities. If we take |x| and make a claim about how it relates to a value, then we have created an interval. For example, if |x| represents the x’s distance away from 0 then |x|≤5 is claiming that x is within 5 units from 0.
Similarly, |x-c| represents the distance between x and c so the inequality |x-c|<7 is claiming that x and c are within 7 units of each other. That is to say, x is within a range of 7 units around c. We often use r to represent this range, like |x-c|<r.
Here are a couple variations of inequalities with absolute values:
|x-c|<r
|x-c|<=r
|x-c|>r
|x-c|≥r
This is helpful for discussing domains and ranges of functions, as well as many other things. In terms of domains, if a function is restricted so that only numbers within 3 units around 5 can be plugged into it we can express the domain as |x-5|<3 or |x-5|≤3 depending on if it is including the bounds. In order to solve this inequality we use a rule about absolute values that says the following: If |x-c|>r then there are two cases: x-c>r or x-c<-r. This makes more sense if you refer to the visual representations above.
Thus, for the example of |x-5|≤3 we can say -3≤x-5≤3 and 2≤x≤8 by adding 5 to all three parts of the inequality. Thus, x∈[2,8].
Review
Absolute Values describe the distance between points, so |5| is the distance from 5 to 0. That is why they are always positive
If a distance is given as a range, points that are included in the range are denoted with brackets and filled in circles
If a distance is given as a range, points that are not included in the range are denoted with parenthesis and empty circles
Absolute Value inequalities can be solved by writing as a new inequality:
If |x-c|<r then -r<x-c<r
If |x-c|>r then there are two cases: x-c>r or x-c<-r.