3.6 The Distance Between Two Points
In English, |a-b| represents the distance between the points a and b.
We now know how to find the distance between two points if one of the points happens to be the origin. Can we use the technique that we developed to solve those types of equations to solve distances between any two points? If we could, then it would be a valuable tool in answering our original question, "How many Apps would we have to sell for our profit to be within $10,000 of $500,000?", which can be translated in terms of the distance between two things: "How many Apps would we have to sell so that the distance between .69x-3000 and $500,000 is less than $10,000?"
Let's study specific numerical examples first to get a better understanding of the distance between two objects. Suppose that your friend lives 5 miles East of school and you live 7 miles East of school. If we think of the school as the origin, what is the distance between you and your friend from your perspective? Using the location of the school as our reference point, we can calculate your distance from school less your friend's distance from school. In other words, we get that the distance between the two of you is 7 - 5 = 2 miles. From your friend's perspective, the distance between the two of you would most naturally be thought of as her distance from school minus your distance from school; in other words, we get 5 - 7 = -2 miles. But distance is never negative, so the distance is still 2 miles. What did we just do? We just took the absolute value of the difference between 7 and 5!
|x-y| represents the distance between x and y; it also represents the distance between y and x
Now that we understand this idea using specific numbers, let's play around with the idea.
What does |x-1|=2 mean in English?
Now, let's solve |x-1|=2 for x.
As usual, we have two ways that we can think about this: graphically and algebraically. Graphically, we can plot the point 1 on a number line and then find all of points that are 2 units from 1 since our equation tells us that the distance between 1 and our unknown value(s) is 2 units. Try to sketch the solution before looking below.
Notice that starting from 1, we can move two units either to the left or to the right so that the distance from 1 is 2 units; by my picture, the two solutions are that x is either -1 or 3. Note that we use the English word "or" here as opposed to the English word "and". The connecting word "and" would be incorrect because it would be saying that x is -1 and 3 at the same time, which is not possible.
Another way that we can solve this equation is with the power of algebra. Using a method that we developed earlier, we can set it up using two equations, namely:
x-1=4 or x-1=-4
Using the tools that we developed to solve linear equalities we find that x= or
Now let's try a more interesting problem:
Find all x values that make the distance between 3x-2 and 7 equal to 3.
We could try to solve this graphically, but it would be tough to visualize "3x-2" on the number line, so in this case, we'll just use algebra to solve it. Try to develop an equation before looking below.
We can then solve that x= or
You can see the full solution below.
First, let's combine the terms within the absolute value to transform our equation into something that's easier to work with:
Then, following the same process that we learned when we were first studying absolute value equalities, we can solve the two equations:
3x-9=3 or 3x-9=-3
These two equation are just linear equations. Adding 9 to both sides gives us the two equations 3x=12 or 3x=6. Finally, we can divide both sides by 3 in each equation to get that x=4 or x=2.
We almost have all of the tools necessary to solve the original equation: "How many Apps would we have to sell so that the distance between .69x-3000 and $500,000 is less than $10,000?", since we now have a deep understanding of the distance between two things. However, we still need to better understand how to incorporate the "less than" statement into a mathematical equation. We'll begin our study of absolute value inequalities in the next section.