BTGmoderatorDC wrote:Is x + y > 0 ?
(1) x - y > 0
(2) x^2 - y^2 > 0
Source: Manhattan Prep
$$x + y\,\,\mathop > \limits^? \,\,0$$
$$\left( 1 \right)\,\,x > y\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {0, - 1} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\left| x \right| > \left| y \right|\,\,\,\left\{ \matrix{
\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {x,y} \right) = \left( {1,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( { - 1,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\left\{ \matrix{
\,\left( 1 \right)\,\,\,x - y > 0 \hfill \cr
\,\left( 2 \right)\,\,\, \Rightarrow \,\,\,\left( {x + y} \right)\left( {x - y} \right) > 0 \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,x + y > 0\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio. 
