# Absolute Value Problem Using Only Variables

I recently encountered this problem, and it does not make sense to me. It looks like

Given $$|a(b-cx)|=d$$ , find the value of $$|x-\frac{b}{c}|$$

This was on a multiple choice test awhile ago, and I don't exactly remember the answer choices, but all of them were in simple fraction form, no addition or subtraction. Example $$\frac{ac}{db}$$. That is, of course, probably not the right answer, just an example of how the answer choices looked. Anyway, this problem bothered me so much that I decided to post it. Thanks for the help.

• You can't delete the question. But, a couple of people were able to answer it no problem. – brandon May 21 at 1:55

Assuming $$a,b,c,d>0$$,
$$|a(b-cx)|=d\implies \left|x-\frac bc\right|=\frac d{ac}.$$
It holds for $$a \neq 0$$ and $$c \neq 0$$: $$|a(b-cx)| = d,$$ $$|a| |b-cx| = d,$$ $$|b-cx|=\frac{d}{|a|},$$ $$\left| c \right| \left| x - \frac{b}{c} \right| = \frac{d}{|a|},$$ $$\left| x - \frac{b}{c} \right| = \frac{d}{|a c|}.$$