Brent@GMATPrepNow wrote:If y < 0 < x, is x/y > -1?
(1) x + y > 0
(2) 3x < -2y
Source:
www.gmatprepnow.com
VERY important problem, Brent. Congrats!
(I believe it is a 650-700 level, by the way.)
$$y < 0 < x\,\,\,\, \Leftrightarrow \,\,\,\,\,\left\{ \matrix{
\,y < 0\,\,\,\,\left( * \right) \hfill \cr
\,x > 0 \hfill \cr} \right.$$
$$\frac{x}{y}\,\,\,\mathop > \limits^? \,\, - 1\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\boxed{\,\,x\,\mathop < \limits^? \, - y\,\,\,}\,\,\,$$
$$\left( 1 \right)\,\,x + y > 0\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{FOCUS}}\,\,!} \,\,\,\,\,x > - y\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle $$
$$\left( 2 \right)\,\,\,\,3x < - 2y\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{FOCUS}}\,\,!} \,\,\,\,\,x < - {2 \over 3}y\,\,\,\mathop < \limits^{\left( {**} \right)} \,\,\, - y\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,$$
$$\left( {**} \right)\,\,\, - \frac{2}{3} > - 1\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\, - \frac{2}{3}y < - 1 \cdot y\,\,\,\, \Rightarrow \,\,\,\, - \frac{2}{3}y < - y$$
Obs.: this problem is PERFECTLY stated. More explicitly: some people believe (D) must have statements "answering the same" but this belief is simply NOT true.
Proof:
https://www.beatthegmat.com/data-suffic ... 16453.html
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
