If w 0, which of the following must be true?

Question

If  w
eq{0} , which of the following must be true?

I.  \frac{|w|}{w} = 1

II.  \frac{w^{2}}{|w|} = w

III.  \sqrt{w^2} = |w|

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

gmatophobia · Accepted Answer

I.  \frac{|w|}{w} = 1

When  w < 0 ,  \frac{|w|}{w}  = -1.

Example, w = -5
|w| = 5
 \frac{|-5|}{-5} = \frac{5}{-5} = -1

Hence, this statement is not always true.

II.  \frac{w^{2}}{|w|} = w

When  w < 0 ,  \frac{w^{2}}{|w|}  = -w.

Example, w = -5
|w| = 5
 \frac{w^{2}}{|w|} = \frac{25}{5} = 5 = -(-5)

Hence, this statement is not always true.

III.  \sqrt{w^2} = |w|

For all values of w,  \sqrt{w^2} = |w| . Hence, this statement is always true.

Option B

Kinshook · Answer

egobappe Asked: If w eq{0} , which of the following must be true? I. \frac{|w|}{w} = 1 If w>0; |w| = w; \frac{|w|}{w} = 1 But if w<0; |w| = -w; \frac{|w|}{w} = -1 COULD BE TRUE II. \frac{w^{2}}{|w|} = w If w>0; |w| = w; \frac{|w|ˆ2}{|w|} = |w| = w But if w<0; |w| = -w; \frac{|w|ˆ2}{|w|} = |w| = -w COULD BE TRUE III. \sqrt{w^2} = |w| \sqrt{w^2} = |w| is always true MUST BE TRUE (A) II only (B) III only (C) I and II only (D) II and III only (E) I, II, and III IMO B

Archit3110 · Answer

Test with values 1/2,-2,3
Only option 3 is valid..
Option B

Posted from my mobile device

GmatKnightTutor · Answer

Worth remembering that the question informs you w cannot equal zero.

Statement (1) doesn't have to be true because w could be negative (i.e. -6) which wouldn't result in a POSITIVE 1.

Statement (2) doesn’t have to be true either for the same reason. Suppose w = -6. 36/6 would equal 6 not w which is -6.

Statement (3) you don’t really need to solve since this has to be true if the others aren't through process of elimination. That being said, the LHS and RHS will always be positive under the provided conditions and have the same absolute values.

(B) is your answer.

orangebicycle5 · Answer

gmatophobia , do you mind walking me through the solution for the second option? I'm good with 1 and 3.

I test both positive and negative:

(1) positive --> w^2/w = w --> works
(2) negative --> w^2/-w = my brain keeps inputting this as w^2-1 = w^1 = w making it work.

Should I instead write it out as w*w/-w --> factor out the w(1*1)/(-1) = -w?

DonBosco7 · Answer

option I. Only test values 1 & -1 for option I

option II. you can write  \frac{w^{2}}{|w|}  =  \frac{\sqrt{w^{2}} * \sqrt{w^{2}} }{|w|}  =  \frac{|w| * |w|}{|w|}  
 => |w| = w
 => w = + w OR w = -w

option III. directly can be written as 
  \sqrt{w^2} = |w|  
 |w| = |w|.......................... which is always true

Answer B

If w 0, which of the following must be true?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

­­If w 0, which of the following must be true?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

If w 0, which of the following must be true?