Two pieces of information that we get:
(1) $ a^{2} < b^{2} $
(2) $ \frac{b}{c} < \frac{a}{c}$
My question now refers to the statement (2). In the solution, it is told that (2) is not sufficient since $ \frac{b}{c} < \frac{a}{c}$ can be simplified to $\frac{a-b}{c} > 0$, which is the same information we already got in the question itself. However, why could we not simplify the 2nd statement like this:
$ \frac{b}{c} < \frac{a}{c}$ --> multiplying both sides with c leads to $b<a$? And if $a$ is always larger than $b$, then $c$ MUST be positive, or am I wrong? Then statement (2) would be sufficient for me, since the answer would always be no.
What am I missing? I appreciate your help guys!
If $c >0$, then (2) gives $a >b$. Note this holds regardless of the signs of $a$ and $b$. So you have for example $a = 1 > b = -5$, which also satisfies (1) $a^2 < b^2$. So you found an example where (1) and (2) hold for $c >0$.
This means that $c <0$ is not implied by the conditions.
The trick of this question is to try and trap you into thinking of positive values only.