subh2273 wrote:If the product of X and Y is a positive number, is the sum of X and Y a negative number?

(1) X > Y^5
(2) X > Y^6

\[xy > 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left( * \right)\,\,\,\,\left\{ \begin{gathered}
\,\,x > 0\,\,{\text{and}}\,\,y > 0\,\,\,\,\left( {{\text{scenario}}\,\,{\text{I}}} \right) \hfill \\
\,\,\,\,OR\,\,\, \hfill \\
\,\,x < 0\,\,{\text{and}}\,\,y < 0\,\,\,\,\left( {{\text{scenario}}\,\,{\text{II}}} \right) \hfill \\
\end{gathered} \right.\]
\[x + y\,\,\mathop < \limits^? \,\,\,0\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\boxed{\,\,\,?\,\,\,:\,\,\,{\text{scenario}}\,{\text{II}}\,\,}\]
\[\left( 1 \right)\,\,\,x > {y^5}\,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {x,y} \right) = \left( {2,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,\left( {x,y} \right) = \left( { - 1, - 2} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,x > {y^6}\,\, \geqslant 0\,\,\,\, \Rightarrow \,\,\,\,x > 0\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

in the first case x >y^5 can be satisfied when x, y both are negative or positive. But we want that both should be negative.
in the second case x>y^6 is possible only when both are positive. So we now know that x+y can't be negative. sufficient.