prakhar992 wrote:
Codebug4it wrote:
Is \((q - p) < 0\)?
(1) \((p^3 - q) < 0\)
(2) \((p^2 - q) > 0\)
I would like to showcase two methods here, I will start with the plugging in number method and the 2nd method will simply be comparing the inequalities and attempting to simplify them. The second method will be much faster but requires good understanding on how to simplify inequalities.
Method 1: Plugging in numbersWhen looking at variables in inequalities we have to ask ourselves a couple of questions:
(i) Are the variables positive/negative?
(ii) Are the size of the variables bigger than 1 or less than 1?
It helps to consider how we can make p < q or the other way around. We want to identify the extreme/weird cases, jump to those, and cut any effort in finding the easy cases.
Statement 1: \(p^3 < q\). It should be quick to find a case where \(p^3 < q\) and \(p < q\) are both true, so let's focus on how we can make \(p > q\) and \(p^3 < q\) at the same time. We can have \(0.5^3 < 0.2\) which results in \(p > q\). Since we have both sides of the answers this is insufficient.
Statement 2:\(p^2 > q\). This time we can find many cases where \(p > q\) can be true. So we want to focus on how we can get \(p < q\) from \(p^2 > q\), and we could make a negative p positive q case that satisfies \(p^2 > q\) so this would also be insufficient.
Combined:Let us try to recycle what we did in the earlier statements. p = 0.5 and q = 0.2 can satisfy both statements, hence p > q is possible. We could have the negative p positive q case from statement 2, so p < q is possible. Then combined this is still insufficient.
Method 2: Simplifying inequalitiesWe want to prove p < q or p > q.
Statement 1:\(p^3 < q\) cannot be simplified. Hence we cannot prove p < q from this statement, then this is insufficient.
Statement 2:\(p^2 > q\) cannot be simplified. Hence we cannot prove p < q from this statement, then this is insufficient.
Combined:We have \(p^3 < q\) and \(q < p^2\). Combining the inequalities we can conclude \(p^3 < p^2\) and \(p < 1\). Yet we have no information on q, hence we still don't know the relation between p and q. This is all we can conclude so combined still insufficient.
Ans: E
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