GMATH practice exercise (Quant Class 20)



In the rectangular coordinate system, which of the above representations better describes the points (x,y) such that y/(x-y) > 0?

(Alternative choices are presented in the figure.)
_________________

\({\rm{?}}\,\,\,{\rm{:}}\,\,\,{\rm{best}}\,\,{\rm{graph}}\,\,{\rm{representation}}\)

\({y \over {x - y}}\,\,\mathop > \limits^{\left( * \right)} \,\,0\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\left\{ \matrix{
\,y > 0\,\,\,{\rm{and}}\,\,\,x - y > 0 \hfill \cr
\,\,\,{\rm{OR}} \hfill \cr
\,y < 0\,\,\,{\rm{and}}\,\,\,x - y < 0 \hfill \cr} \right.\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\left\{ \matrix{
\,y > 0\,\,\,{\rm{AND}}\,\,\,\underline {x > y} \,\,\,\,\left( {{\rm{hence}}\,\,x > 0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\underline {{\rm{bottom}}\,\,{\rm{half}}} \,\,{\rm{of}}\,\,{\rm{1st}}\,{\rm{quadrant}} \hfill \cr
\,\,\,{\rm{OR}}\,\,\,\,\, \hfill \cr
\,y < 0\,\,\,{\rm{AND}}\,\,\,\underline {x < y} \,\,\,\,\left( {{\rm{hence}}\,\,x < 0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,\underline {{\rm{upper}}\,\,{\rm{half}}} \,\,{\rm{of}}\,\,3{\rm{rd}}\,{\rm{quadrant}} \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{A}} \right)\)

\(\left( * \right)\,\,\,\left\{ \matrix{
\,y \ne 0\,\,\,\, \Rightarrow \,\,\,\,x{\rm{ - axis}}\,\,{\rm{excluded}} \hfill \cr
x - y \ne 0\,\,\,\, \Rightarrow \,\,\,\,{\rm{line}}\,\,y = x\,\,{\rm{excluded}} \hfill \cr} \right.\)


The correct answer is (A).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________