If a 0, is, is 1/a > a/(b^4 + 3) ?

If a ≠ 0, is \(\frac{1}{a} > \frac{a}{b^4 +3}\)?

Notice that we cannot cross-multiply \(\frac{1}{a} > \frac{a}{b^4 +3}\), since we don't know whether \(a>0\) or \(a<0\), for the first case we should keep the same sign but for the second case we should flip the sign (when multiplying by negative number we should flip the sign of the inequity).

Also notice that we don't have the same issue with \(b^4 +3\), since this expression is always positive: \(b^4 +3=nonnegative+positive=positive\).


(1) a = b^2:

The question becomes:

Is \(\frac{1}{b^2} > \frac{b^2}{b^4 +3}\)?

Now, since \(b^2>0\) then here we can safely cross-multiply and the question becomes:

Is \(b^4+3>b^4\)?
Is \(3>0\)?

Answer to this question is YES. Sufficient.

(2) a^2 = b^4:

The question becomes:

Is \(\frac{1}{a} > \frac{a}{a^2 +3}\)?

We have the same problem: we cannot cross-multiply since we don't know the sign of \(a\). If \(a=1\) then the answer to the question would be YES but if \(a=-1\) then the answer to the question would be NO. Not sufficient.

Answer: A.