Q:

Inequality $\frac{1}{x}>\frac{1}{y}$ when $y>0$ has the same solutions as..

Inequality $\frac{1}{x}>\frac{1}{y}$ when $y>0$ has the same solutions as..
a) $x>y$
b) It should say $y>x$
c) $y-x>1$
d) none of three above
Correct answer: d)
My procedure: If x=0,01 and y=0,1 then a) & c) must be false. However, I can not manage to find a counter "example" to b). It seems to me that if the inequality which is posed in the question above should be true then $y>0$ implies that $x>0$. Otherwise the inequality sign does not seem hold. If both variables must be positive then does it not imply that $y>x$? 

A:

The most obvious reason why (b) is wrong is that $y=1$ and $x=0$ is a solution to $y>x$, but is not a solution to $1/x>1/y$. Hence they do not have the same set of solutions.
However, other examples include anything with $y>0$ and $x<0$. That is a solution to $y>x$, but not $1/x>1/y$ with $y>0$.
Explained in another way. It is true that $1/x>1/y$ combined with $y>0$ implies that $x>0$. However, $y>x$ and $y>0$ does not imply that $x>0$, therefore the solution sets cannot be equivalent!

