BTGmoderatorDC wrote:If x^3 < x^2, what is the value of the integer x?
(1) |x| < 2
(2) x^3 = x
Source: Veritas Prep
$$x\,\,{\mathop{\rm int}} \,\,\left( * \right)$$
$${x^3} < {x^2}\,\,\,\,\, \Leftrightarrow \,\,\,\,\,{x^2}\left( {x - 1} \right) < 0\,\,\,\,\, \Leftrightarrow \,\,\,\,\,x \ne 0\,\,\,{\rm{and}}\,\,x < 1\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\,x \le - 1\,\,\,\,\left( {**} \right)\,\,$$
$$? = x$$
$$\left( 1 \right)\,\,\left| x \right| < 2\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\,\left| x \right| = 0\,\,\,{\rm{or}}\,\,\,\,\left| x \right| = 1\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\,x = - 1\,\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.\,\,\,\,$$
$$\left( 2 \right)\,\,{x^3} = x\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x\left( {x + 1} \right)\left( {x - 1} \right) = x\left( {{x^2} - 1} \right)\,\, = 0\,\,\,\,\, \Rightarrow \,\,\,\,\,x \in \left\{ {0,-1,1} \right\}\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,\,\,x = - 1\,\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.\,\,$$
The correct answer is (D).
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
