push12345 wrote:

Is |a - b| < |a| + |b| ?

(1) ab< 0

(2) a^b < 0

\(\left| {a - b} \right|\,\,\mathop < \limits^? \,\,\,\left| a \right| + \left| b \right|\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,ab\,\,\mathop > \limits^? \,\,\,0\)

(*) This equivalence will be PROVED at the end of this post. Ignore this proof if you don´t like math!

\(\left( 1 \right)\,\,ab < 0\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\, \Rightarrow \,\,\,\,{\text{SUFF}}.\)

\(\left( 2 \right)\,\,\,{a^b} < 0\,\,\,\left\{ \matrix{

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

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

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

Regards,

Fabio.

POST-MORTEM:

\(\left( * \right)\,\,\,\left\{ \matrix{

\,\left( i \right)\,\,\,\,\left| {a - b} \right|\,\, < \,\,\,\left| a \right| + \left| b \right|\,\,\,\,\,\,\, \Rightarrow \,\,\,\,ab > 0 \hfill \cr

\,\left( {ii} \right)\,\,\,\,ab > 0\,\,\,\, \Rightarrow \,\,\,\,\left| {a - b} \right|\,\, < \,\,\,\left| a \right| + \left| b \right|\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\left| {a - b} \right|\,\, \ge \,\,\,\left| a \right| + \left| b \right|\,\,\,\,\, \Rightarrow \,\,\,\,\,ab \le 0 \hfill \cr} \right.\,\)

\(\left( i \right)\,\,\,\,\left| {a - b} \right|\,\, < \,\,\,\left| a \right| + \left| b \right|\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{squaring}}} \,\,\,\,{\left( {a - b} \right)^2} < \,\,\,{a^2} + 2\left| {ab} \right| + {b^2}\,\,\,\, \Rightarrow \,\,\,\,\, \ldots \,\,\,\,\, \Rightarrow \,\,\,\, - ab < \left| {ab} \right|\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,ab > 0\)

\(\left( {ii} \right)\,\,\,\,\left| {a - b} \right|\,\, \geqslant \,\,\,\left| a \right| + \left| b \right|\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{squaring}}} \,\,\,\,{\left( {a - b} \right)^2} \geqslant \,\,\,{a^2} + 2\left| {ab} \right| + {b^2}\,\,\,\, \Rightarrow \,\,\,\,\, \ldots \,\,\,\,\, \Rightarrow \,\,\,\, - ab \geqslant \left| {ab} \right|\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,ab \leqslant 0\,\,\,\)

_________________

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