VERY nice problem, subh2273. Congrats!subh2273 wrote:Is x + y > xy ?
Statement (1):
x > 0 > y
Statement (2):
|y| = x
$$x + y\,\,\mathop > \limits^? \,\,xy$$
$$\left( 1 \right)\,\,x > 0 > y\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1, - 1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {{1 \over 2}, - 1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,x = \left| y \right|\,\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {0,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,x = \left| y \right|\mathop = \limits^{y\, < \,0} - y$$
$$\left. \matrix{
{\rm{FOCU}}{{\rm{S}}_{\,{\rm{LHS}}}}\,\,\,:\,\,\,\,x + y = 0 \hfill \cr
{\rm{FOCU}}{{\rm{S}}_{\,{\rm{RHS}}}}\,\,\,:\,\,\,\,xy = - {y^2}\,\,\mathop < \limits^{y\,\, \ne \,0} \,\,0\,\,\, \hfill \cr} \right\}\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
The correct answer is therefore [spoiler]__(C)______[/spoiler] .
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
