JJ2014 wrote:

Is \((y-10)^2 > (x+10)^2\)?

(1) \(-y > x + 5\)

(2) \(x > y\)

VERY nice problem. Please note that we will use two different rephrasings (one for each (1) and (2) alone, the other for (1+2)):

\({\left( {y - 10} \right)^2}\,\,\mathop > \limits^? \,\,{\left( {x + 10} \right)^2}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left\{ \matrix{

\,\left| {y - 10} \right|\,\,\,\mathop > \limits^? \,\,\,\left| {x + 10} \right|\,\,\,\,\,\,\,\,\left( {\rm{I}} \right) \hfill \cr

\,\,\,{\rm{OR}} \hfill \cr

\,\left( {x + y} \right)\left( {y - x - 20} \right)\,\,\,\mathop > \limits^? \,\,\,0\,\,\,\,\,\,\,\,\left( {{\rm{II}}} \right)\,\,\,\,\,\left( * \right) \hfill \cr} \right.\)

\(\left( * \right)\,\,\,\left\{ \matrix{

{\left( {y - 10} \right)^2}\,\,\mathop > \limits^? \,\,{\left( {x + 10} \right)^2}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,{\left( {y - 10} \right)^2} - \,{\left( {x + 10} \right)^2}\,\,\mathop > \limits^? \,\,\,0 \hfill \cr

\,{\left( {y - 10} \right)^2} - \,{\left( {x + 10} \right)^2} = \left[ {\left( {y - 10} \right) + \left( {x + 10} \right)} \right]\left[ {\left( {y - 10} \right) - \left( {x + 10} \right)} \right] = \left( {x + y} \right)\left( {y - x - 20} \right)\,\,\,\mathop > \limits^? \,\,\,0 \hfill \cr} \right.\)

\(\left( 1 \right)\,\,\,x + y < - 5\,\,\,\,\left( {\rm{I}} \right)\,\,\,\left\{ \matrix{

\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {0, - 6} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr

\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( { - 16,10} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\)

\(\left( 2 \right)\,\,x > y\,\,\,\,\left( {\rm{I}} \right)\,\,\,\,\,\,\left\{ \matrix{

\,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,\left( {x,y} \right) = \left( {0, - 6} \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( {{\rm{II}}} \right)\,\,\,\left\{ \matrix{

\,x + y < 0\,\,\,{\rm{by}}\,\,\,\left( 1 \right) \hfill \cr

\,y - x - 20 = \left( {y - x} \right) - 20\mathop < \limits^{{\rm{by}}\,\,\left( 2 \right)} - 20\, < 0 \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left( {\rm{C}} \right)\,\,\,\,\)

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

Regards,

Fabio.

_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)

Our high-level "quant" preparation starts here: https://gmath.net