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.
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Fabio Skilnik ::
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