Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
If |x-6|=2x, then x=?
A. -6
B. -4
C. 0
D. 2
E. 6
$$?\,\,\,:\,\,\,\,x\,\,\,{\rm{such}}\,\,{\rm{that}}\,\,\,\,\left| {x - 6} \right| = 2x\,\,\,\,\,\left( {{\rm{Note}}\,\,{\rm{that}}\,\,\,2x \ge 0\,\,,\,\,{\rm{hence}}\,\,x \ge 0} \right)$$
$$\left| {x - 6} \right| = 2x\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{squaring}}} \,\,\,\,\,\,\,{\left( {x - 6} \right)^2} = {\left( {2x} \right)^2}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,3{x^2} + 12x - 36 = 0$$
$$3{x^2} + 12x - 36 = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{x^2} + 4x - 12 = 0\,\,\,\,\,\,\mathop \Rightarrow \limits_{{\rm{product}}\,\, = \,\, - 12}^{{\rm{sum}}\,\, = \,\, - 4} \,\,\,\,\,x = - 6\,\,\,{\rm{or}}\,\,\,x = 2$$
$${\rm{Testing}}\,\,{\rm{each}}\,\,{\rm{potential}}\,\,{\rm{root}}\,\,{\rm{in}}\,\,{\rm{the}}\,\,{\rm{original}}\,\,{\rm{equation}}\,\,\,\left( * \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = x = 2\,\,\,$$
(*) Important: as mentioned in the very beginning, -6 doesn´t need to be tested. Someone (correctly) may argue that 2 does not need, too (otherwise no alternative choice could be chosen).
Anyway, we want to remind you that, whenever an equation is squared - or put to any positive even power - roots may be "created" and, because of that, the "final checking" is an important procedure.
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
