Bunuel wrote:

If |x−y|=y, what is the value of x?

(1) xy>0

(2) y=6

VERY beautiful problem, Bunuel. (Kudos!)

\(? = x\)

\(\left| {x - y} \right| = y\,\,\,\,\,\, \Rightarrow \,\,\,\,\,y \geqslant 0\,\,\,\,\,\,\,\,{\text{AND}}\,\,\,\,\,\,\,\left\{ \begin{gathered}

\,\,x = 0\,\,\,,\,\,\,y \geqslant 0\,\,\,{\text{free}} \hfill \\

\,\,{\text{OR}}\,\,\,\, \hfill \\

0 \ne x\,\,,\,\,\,{\text{0}}\,\,\mathop {\text{ < }}\limits^{\left( * \right)} \,\,{\text{dist}}\left( {x,y} \right) = {\text{dist}}\left( {y,0} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,0 < y < x = 2y \hfill \\

\end{gathered} \right.\)

\(\left( * \right)\,\,\,0 \ne x = y\,\,\,\,\, \Rightarrow \,\,\,\,\,0 = \left| {x - y} \right| = y = x\,\,\,\,{\text{impossible}}\,\,\,\)

\(\left( 1 \right)\,\,\,xy > 0\,\,\,\,\left\{ \begin{gathered}

\,{\text{Take}}\,\,\left( {x,y} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{2}}\,\, \hfill \\

\,{\text{Take}}\,\,\left( {x,y} \right) = \left( {4,2} \right)\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{4}}\,\, \hfill \\

\end{gathered} \right.\)

\(\left( 2 \right)\,\,\,y = 6\,\,\,\,\left\{ \begin{gathered}

\,{\text{Take}}\,\,\left( {x,y} \right) = \left( {0,6} \right)\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{0}}\,\, \hfill \\

\,{\text{Take}}\,\,\left( {x,y} \right) = \left( {12,6} \right)\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{12}}\,\, \hfill \\

\end{gathered} \right.\)

\(\left( {1 + 2} \right)\,\,\,\,\,\left\{ \begin{gathered}

\,x \ne 0 \hfill \\

\,y = 6 \hfill \\

\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = x = 2y = 12\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.\)

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

Regards,

Fabio.

_________________

Fabio Skilnik :: https://www.GMATH.net (Math for the GMAT)

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