VJesus12 wrote:If x and y are nonzero integers, is $$\left(x^{-1}+y^{-1}\right)^{-1}\ >\left(x^{-1}\cdot y^{-1}\right)^{-1}\ ?$$
(1) x = 2y
(2) x + y > 0
Source: Manhattan GMAT
$$x,y\,\, \ne 0\,\,\,{\rm{ints}}\,\,\,\left( * \right)$$
$$\frac{1}{{\frac{1}{x} + \frac{1}{y}}}\,\,\mathop > \limits^? \,\,\frac{1}{{\frac{1}{{xy}}}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\boxed{\,\,\,\,\frac{{xy}}{{y + x}}\,\,\,\mathop > \limits^? \,\,\,xy\,\,\,}$$
$$\left( 1 \right)\,\,x = 2y\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{{2{y^2}} \over {3y}}\,\,\,\mathop > \limits^? \,\,\,2{y^2}\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)\,\,\,{y^2} > \,\,0} \,\,\,\,\,{2 \over {3y}}\,\,\,\mathop > \limits^? \,\,\,2$$
$$\left\{ \matrix{
\,y \le - 1\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\, \hfill \cr
\,y \ge 1\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$
$$\left( 2 \right)\,\,x + y > 0\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x;y} \right) = \left( {1;1} \right)\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x;y} \right) = \left( {3; - 1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio. 
