BTGmoderatorLU wrote:Source: Veritas Prep
In the coordinate geometry plane, region P is defined by all the points (x, y) for which 3y + 12 > 2x. Does point (a, b) lie within region P?
1) 4b = a - 8
2) b < 0 and a > 3
$$3y + 12 > 2x\,\,\,\,\, \Leftrightarrow \,\,\,\,\,2x - 3y < 12$$
$$\left( {a,b} \right)\,\,\,\mathop \in \limits^? \:\,{\text{region}}\,P\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\boxed{\,\,\,2a - 3b\,\,\,\mathop < \limits^? \,\,\,\,12\:\,}$$
Let´s BIFURCATE (1+2), that is, present two EXPLICIT VIABLE scenarios satisfying both statements together, each scenario answering our FOCUS differently!
$$\left( {1 + 2} \right)\,\,\,\left( * \right)\,\,\,\left\{ \matrix{
\,a - 4b = 8 \hfill \cr
\,b < 0\,\,,\,\,a > 3 \hfill \cr} \right.\,\,\,\,$$
$$\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {4, - 1} \right)\,\,\,\, \Rightarrow \,\,\,\,\left( * \right)\,\,\,{\rm{ok}}\,\,\,\, \Rightarrow \,\,\,\,\,2\left( 4 \right) - 3\left( { - 1} \right)\,\,\,\,\mathop < \limits^? \,\,\,\,12\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \, \hfill \cr
\,\,{\rm{Take}}\,\,\left( {a,b} \right) = \left( {6, - {1 \over 2}} \right)\,\,\,\, \Rightarrow \,\,\,\,\left( * \right)\,\,\,{\rm{ok}}\,\,\,\, \Rightarrow \,\,\,\,\,2\left( 6 \right) - 3\left( { - {1 \over 2}} \right)\,\,\,\,\mathop < \limits^? \,\,\,\,12\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \, \hfill \cr} \right.$$
The correct answer is therefore (E).
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
