MathRevolution wrote:
[
Math Revolution GMAT math practice question]
(absolute value) Is \(ab<0\)?
\(1) |a+b| = - ( a + b )\)
\(2) |a+b| + 1 = |a| + |b|\)
Excellent problem, Max. Congrats (and kudos)!
\(ab\mathop {\,\, < }\limits^? \,\,0\)
\(\left( 1 \right)\,\,\,\left| {a + b} \right| = - \left( {a + b} \right)\,\,\,\, \Leftrightarrow \,\,\,\,\,a + b \le 0\)
\(\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {0,0} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( { - 2, 1} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr} \right.\)
\(\left( 2 \right)\,\,\left| {a + b} \right| + 1 = \left| a \right| + \left| b \right|\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,ab < 0\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle\)
\(\left( * \right)\,\,ab \ge 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left| {a + b} \right| = \left| a \right| + \left| b \right|\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left| {a + b} \right| + 1 \ne \left| a \right| + \left| b \right|\,\,\,\,,\,\,\,{\rm{impossible}}\)
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
_________________
Fabio Skilnik ::
GMATH method creator (Math for the GMAT)
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