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

RobertVericPrep · Answer

When you read a "must be true" question on the GMAT involving inequalities or absolute values, your immediate goal should be to try and prove the statements FALSE. The easiest way to do that is to pick a smart, simple number that fits the initial condition.

And when absolute values are involved you absolutely want to begin by testing negative numbers! Let's use -2. Also be aware that squaring a number (or raising it to any other even exponent) and the absolute value of a number have something in common: the result is never negative. Unless it's zero, the result is always positive. Always keep this in mind when you look at absolute value questions.

The prompt tells us that w is not equal to 0. This means w can be any positive or negative number, it just can't be zero. Let's test our chosen negative number, w = -2.

Statement I:

|w| / w = 1

If we plug in w = -2, the equation becomes: |-2| / -2 = 2 / -2 = -1

So Statement I is not a "must" be true. It could be true (choose a positive number), but it does not have to be true. We can instantly cross out any answer choices that include I.

Now let's look at Statement II:

(w^2) / |w| = w

Here again, we benefit from having chosen a negative number. The left-hand side of the equation would have to be positive, which means the right-hand side could not be -2. Statement II is not a "must" be true.

Now let's look at Statement III. We'll write "The square root of w squared = |w| " as

sqrt(w^2) = |w|

Important note here: do "squaring" and "square rooting" always cancel each other? Not when a number appears under the square root symbol. If something appears inside the square root symbol, the result is defined as the "positive root."

Suppose x^2 = 25. Well, yes, x itself could be 5 or -5

BUT...

sqrt( (-5)^2 ) = sqrt (25) = positive 5. That's just a rule. A squared number becomes positive under a square root symbol.

Similarly, if we choose -2, the square root of "-2 squared" is the same thing as the square root of positive 4, which equals positive 2.

sqrt( (-2)^2 ) = |-2|

sqrt(4) = 2

In this case, 2 = 2, so it works. Because a squared number becomes positive under a square root, the output will always match the absolute value of the original number, regardless of whether w is positive or negative. It is an algebraic identity.

So Statement III must be true.

Answer B

One additional point - because the question does NOT say that w has to be an integer, we could have also chosen a negative number that is ALSO a fraction. Say, -1/3.

For this question, -2 was all we'll needed, but for hard questions using a negative fraction can be very helpful, as well.

onlyPlanA · Answer

Directly use numbers to check.
Let w = -1

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

\frac{|-1|}{-1} = -1 
eq{1}

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

\frac{(-1)^2}{|-1|} = 1 
eq{-1}

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

\sqrt{(-1)^2} = |-1| = 1

So, III is the only correct option.

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

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