This problem type gives us practice solving word problems that require use of the absolute value function.
The basic format of the problems you will see in this problem type is something like this:
- You are told that a man starts at a certain distance from home. He then moves a distance x, either toward his home or away from his home.
- You are then told that if the man had instead moved a different distance
(perhaps 3x instead of x) then he would be m miles from home.
- Find the distance x.
Solving this kind of problem should not be too difficult with a little practice. Most of the variations of this problem can be solved by setting up and solving an equation of this sort:
|c - ax| = B where C = original distance between the man
and his home, A = a multiple of x that the man moved
from his original position, and B = the final distance
between the man and his home
The reason for the absolute value function in the equation above will now be explained.
In solving these problems, there is one possibility you have to take into account: that after traveling the distance the man could end up either on the same side of the house as he started, or on the other side. To give an example, let's assume that the man started 5 miles away from his home. He then travels a distance and ends up 1 mile from home. How many miles did he travel? Our first guess is 4 miles, which we get by subtracting 5 - 1. But there is another possibility: he may have traveled 5 miles to his home, and traveled 1 more mile past his home. He still ends up 1 mile from home, but in this case he traveled a total of 6 miles.
Because of the possibility just described, we have to use the absolute value function when we set up an equation to solve these problems. That is because when we subtract the distance traveled form the starting distance, we may get a positive number (if the man did not pass his home) or a negative number (if the man did pass his home).
A man begins 12 miles from his home. He then moves a distance of x miles toward home. If he had gone a distance of 5x miles
toward his home he would be 1 mile from home. What is the distance x?
Let's use the formula shown earlier to solve this problem. We will use these variables:
x = the distance the man traveled toward home.
We are solving for this value.
C = 12 (original distance from home)
A = 5 (the multiple of the distance x)
B = 1 (the final distance between the man and his home)
Our equation is this:
|12 - 5x| = 1
We solve an equation involving the absolute value function by setting up and solving two other equations:
Equation 1: 12 - 5x = 1 Solution: x = 11/5
Equation 2: 12 - 5x = -1 Solution: x = 13/5
We see, then, that there are two solutions to this problem.