push12345 wrote:

If a and b are non-zero numbers, is a > b ?

(1) |a| = b

(2) |b|*a < |a*b|

\(a,b\,\, \ne 0\,\,\,\left( * \right)\)

\(a\,\,\mathop > \limits^? \,\,b\)

\(\left( 1 \right)\,\,\,\left| a \right|\,\,\, = b\,\,\,\, \Rightarrow \,\,\,\,\,\,a\,\,\mathop > \limits^? \,\,\left| a \right|\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\,\,\,\,\,\,\left[ {\,x \le \left| x \right|\,\,{\rm{for}}\,\,{\rm{all}}\,\,{\rm{values}}\,\,{\rm{of}}\,\,x\,} \right]\)

\(\left( 2 \right)\,\,a\left| b \right|\,\, < \,\,\left| {ab} \right| = \left| a \right| \cdot \left| b \right|\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left| b \right|\,\, > \,\,0\,\,\left( * \right)} \,\,\,a < \left| a \right|\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,a < 0\,\,\,\)

\(\left\{ \matrix{

\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( { - 1,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr

\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( { - 1, - 2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.\,\,\,\,\)

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

Regards,

Fabio.

_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

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