Equations, Inequalities and VICs, Chapter 5, p. 87

[nvingers]

In the middle of p. 87, there is an example of solving a symmetry problem algebraically:

In one of the steps in solving f(1/x), however, the book goes from "absolute value of x + 1 over 1 - x" to "absolute value of x + 1 over -(1 - x)". Why is there all of a sudden a minus in front of the expression of the denominator of the fraction? Does it have to do with the absolute value signs??

Thanks in advance for your help!

[JonathanSchneider]

Your instinct is entirely correct. Notice that multiplying anything by -1 BEFORE we take the absolute value will have no effect on the end value (after we make it positive). This is simply our way of showing that (1 - x) and (x - 1) have the SAME absolute value. In general, you should recognize that (a - b) is the negative of (b - a). As a result, these forms always have the same absolute value. The problem you cited becomes trickier because the correct answer gives us the opposite from from what we were expecting, but it is still correct, as we are answering for the absolute value.