BTGmoderatorDC wrote:Is the range of the integers 6, 3, y, 4, 5, and x greater than 9?
(1) y > 3x
(2) y > x > 3
Source: Official Guide
$$A = \left\{ {3,4,5,6} \right\} \cup \left\{ {x,y} \right\}\,\,\,\,\,\left[ {{\rm{ints}}} \right]$$
$${R_A} = {\max _A} - {\min _A}\,\,\mathop > \limits^? \,\,9$$
$$\left( 1 \right)\,\,\,y > 3x\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {0,1} \right)\,\,\,\, \Rightarrow \,\,\,\,{R_A} = 6 - 0\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {0,10} \right)\,\,\,\, \Rightarrow \,\,\,\,{R_A} = 10 - 0\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\,y > x > 3\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {4,5} \right)\,\,\,\, \Rightarrow \,\,\,\,{R_A} = 6 - 3\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr
\,{\rm{Take}}\,\,\left( {x,y} \right) = \left( {4,13} \right)\,\,\,\, \Rightarrow \,\,\,\,{R_A} = 13 - 3\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,y\mathop > \limits^{\left( 1 \right)} 3x\mathop \ge \limits^{\left( 2 \right)\,\,x \ge 4} 12\,\,\,\, \Rightarrow \,\,\,\,y \ge 13\,\,\,\,\,\, \Rightarrow \,\,\,\,{R_A} \ge 13 - 3 = 10\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle $$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio. 
