The trick to remember during solving this question is that fractions in the range \((0,1)\) taken to greater power get smaller, for example \(\left(\frac{1}{3}\right)^3 < \left(\frac{1}{3}\right)^2\), etc.
\(\frac{1}{x^5} > \frac{1}{x^3}\) is equivalent to \(x^5 < x^3\). because if two fractions have equal numerators, the fraction with a smaller denominator is the bigger one. Now we need to find the range of values of \(x\) in which \(x^5 < x^3\) holds true. This is possible in \(x \in (0,1)\) for positive \(x\) and in \(x \in (-\infty,-1)\) for negative \(x\).
We can put the values of \(x\) from these ranges to make sure it works:
positive \(x=\frac{1}{2}\):
\(\left(\frac{1}{2}\right)^5 < \left(\frac{1}{2}\right)^3\)
\(\left(\frac{1}{32}\right) < \left(\frac{1}{8}\right)\) -- holds true
negative \(x=-2\):
\((-2)^5 < (-2)^3\)
\(-32 < -8\) -- holds true
In either range the value of \(x\) is smaller than 1, so Statement (2) is sufficient by itself.
Hope this helps.
defoue wrote:
Hi dzyubam, would you please explain again why statement 2 is sufficient
Thx again
what if x=2 in second statement.. i guess then it will not satisfy