BTGmoderatorDC wrote:What is the value of x?
(1) |2x − 1| = 3x + 6
(2) x^2 = 1
\[? = x\]
\[\left( 1 \right)\,\,\left| {2x - 1} \right| = 3x + 6\,\,\,\,\,\, \Rightarrow \,\,\,\,\left\{ \begin{gathered}
\left( {\text{I}} \right)\,\,\,2x - 1 = 3x + 6\,\,\,{\text{when}}\,\,\,x \geqslant \frac{1}{2}\,\,\,\,\,\,\,\,\,\,\left[ {2x - 1 \geqslant 0} \right] \hfill \\
\left( {{\text{II}}} \right)\,\,1 - 2x = 3x + 6\,\,\,{\text{otherwise}} \hfill \\
\end{gathered} \right.\,\,\,\]
\[\left. \begin{gathered}
\left( {\text{I}} \right)\,\,\, \Rightarrow \,\,\,x = - 7\,\,\,{\text{not}}\,\,{\text{acceptable}}\,\,\,\left[ {\left( {\text{I}} \right)\,\,\,{\text{is}}\,\,{\text{for}}\,\,x \geqslant \frac{1}{2}} \right]\,\,\,\,\, \hfill \\
\left( {II} \right)\,\,\, \Rightarrow \,\,\,x = - 1\,\,\, \Rightarrow \,\,\,{\text{unique}}\,\,\left( * \right)\,\,\,\,\, \hfill \\
\end{gathered} \right\}\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{SUF}}.\]
(*) The fact that -1 is less than 1/2 guarantees that this is really viable, but (at least) one possible answer (in "What is the value...?" problems) is an examiner´s burden!
\[\left( 2 \right)\,\,x = 1\,\,\,{\text{or}}\,\,x = - 1\,\,\,\,\,\, \Rightarrow \,\,\,\,{\text{INSUF}}.\]
The above follows the notations and rationale taught in the GMATH method.
