Absolute Value Problem Using Only Variables

Question

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.

Answer 1

Assuming $a,b,c,d>0$,

$$|a(b-cx)|=d\implies \left|x-\frac bc\right|=\frac d{ac}.$$

Answer 2

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