kelly_jacques
Bunuel I understand the logic of not cross multiplying given you don't know
the value of r. However, I cross multiplied and just did two scenarios (1) if r was positive (2) if r was negative and deemed first question insufficient. The second question tells you its negative, so I can select the correct equation of the two and answer together is sufficient.
My confusion is just that cross-multiplying doesn't give the same formula. It gives 0>-2. or 0<-2 Is that a problem ? Is my method still a mistake?
Your question is a bit challenging to follow entirely, but here's my response:
Firstly, it's crucial to understand that during cross-multiplication, we multiply by the denominators of the fractions. So, when cross-multiplying \(\frac{1}{p} > \frac{r}{r^2 + 2}\), we multiply by r^2 + 2 and p, not by r.
Secondly, if p is positive, after cross-multiplication (and keeping the sign as is), the question evolves to "
is \(r^2 +2 > pr\)?". Conversely, if p is negative, then after cross-multiplication (and flipping the sign), the question becomes "
is \(r^2 +2 < pr\)?".
For (1) that says p = r, if p (or r, since they're equal) is positive, the question simplifies to "
is \(2 > 0\)?", hence the answer would be YES. For negative p, the question simplifies to "
is \(2 < 0\)?", thus the answer would be NO. However, as we lack information about the sign of p (or r), we cannot say which case we have. .
(2) is obviously useless alone, but when paired with (1), it supplies the missing information: r, and thereby, p is positive. So, the question simplifies to "
is \(2 > 0\)?" to which the answer is clearly YES.
Generally, you should never multiply an inequality by a variable when its sign is unknown.