If x and y are numbers such that x < - 10 and y > 6, w

[BTGmoderatorDC]

If x and y are numbers such that x < - 10 and y > 6, which of the following expressions is true?

I. |x+4| + | y + 4| + | y-4| + |x - 4| > 32
II. |x - 4| + |y + 4| - | x + 4| - | y - 4| < 0
III. |x + 4| + |y + 4| - |x - 4| - |y - 4| = 0

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II and III

OA D

Source: e-GMAT

[fskilnik@GMATH]

If x and y are numbers such that x < - 10 and y > 6, which of the following expressions is true?

I. |x+4| + | y + 4| + | y-4| + |x - 4| > 32
II. |x - 4| + |y + 4| - | x + 4| - | y - 4| < 0
III. |x + 4| + |y + 4| - |x - 4| - |y - 4| = 0

(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II and III
Source: e-GMAT

$$?\,\,\,:\,\,\,\,{\rm{I,II,III}}\,\,\,{\rm{true}}$$
$$x < - 10\,\,\, \Rightarrow \,\,\,\left\{ \matrix{
\,x + 4 < - 10 + 4 = - 6\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left| {x + 4} \right| > 6 \hfill \cr
\,x - 4 < - 10 - 4 = - 14\,\,\,\,\, \Rightarrow \,\,\,\,\,\left| {x - 4} \right| > 14 \hfill \cr} \right.$$
$$y > 6\,\,\, \Rightarrow \,\,\,\left\{ \matrix{
\,y + 4 > 6 + 4 = 10\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left| {y + 4} \right| > 10 \hfill \cr
\,y - 4 > 6 - 4 = 2\,\,\,\,\, \Rightarrow \,\,\,\,\,\left| {y - 4} \right| > 2 \hfill \cr} \right.$$
$$\left( {\rm{I}} \right)\,\,\,\left| {x + 4} \right| + \,\left| {y + 4} \right| + \,\left| {y - 4} \right| + \left| {x - 4} \right| > 6 + 10 + 2 + 14 = 32\,\,\,\, \Rightarrow \,\,\,\,{\rm{I}}.\,\,{\rm{TRUE}}\,\,\,\, \Rightarrow \,\,\,\,\left( B \right),\left( C \right)\,\,{\rm{refuted}}$$
$$\left( {{\rm{II}}} \right)\,\,\left| {x - 4} \right| + \,\left| {y + 4} \right| - \,\left| {x + 4} \right| - \left| {y - 4} \right|\,\,\mathop < \limits^? \,\,0\,$$
$${\rm{Take}}\,\,\left( {x,y} \right) = \left( { - 11,7} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\,\,\, \Rightarrow \,\,\,\,\,\,\left( E \right)\,\,{\rm{refuted}}$$
$$\left( {{\rm{III}}} \right)\,\,\,\left| {x + 4} \right| + \,\left| {y + 4} \right| - \,\left| {x - 4} \right| - \left| {y - 4} \right|\,\,\, = - \left( {x + 4} \right) + \left( {y + 4} \right) - \left[ { - \left( {x - 4} \right)} \right] - \left( {y - 4} \right) = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{III}}.\,\,{\rm{TRUE}}\,\,\,\, \Rightarrow \,\,\,\,\left( D \right)\,$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.