GRE Problem - What am I doing wrong>

[doubled]

I attached a pic of a practice GRE problem.

The answer is D) relationship cannot be determined.
I understand this answer and it is actually what I picked, but my 2 methods of doing this problem is contradicting each other.

If you use the method of plugging in random #s you'll discover that the answer is D.


However, my other method is trying to match A&B.
For quantity A:
we get (w+z)/(w-z)

For B:
(z+w)/(z-w)=(w+z)/(z-w) <<<<<Can I not do this flip for some reason? It is the only place where I can see a source of error
=(w+z)/-(w-z)

Divide both A&B by w+z and multiplying by w-z we getL
A=1, B= -1

So this would mean that A>B, however this clearly contradicts my other answer and I can't identify why.

 

[ehild]

(z+w)/(z-w) can be both positive and negative, according to the absolute value of the negative z. Interchanging z and w , the sign of the "product" flips. Of course, the positive is greater, but can you state which one is positive, w¤z or z¤w in general?

ehild

 

[doubled]

(z+w)/(z-w) can be both positive and negative, according to the absolute value of the negative z. Interchanging z and w , the sign of the "product" flips.ehild

Sorry, I don't undertand what you mean here.

Of course, the positive is greater, but can you state which one is positive, w¤z or z¤w in general?
ehild

I believe you cannot since the sign in the numerator can be +/- for A&B depending on the magnitude of w&z.

 

[D H]

Divide both A&B by w+z and multiplying by w-z we getL
A=1, B= -1

So this would mean that A>B, however this clearly contradicts my other answer and I can't identify why.

One of those operations (dividing both sides by w+z, multiplying both sides by w-z) was illegal. Which one, and why?

Example: It's obvious that 4 > -2. Dividing both sides by 2 yields 2 > -1, which is still true. Dividing both sides by -2 yields -2 > 1, which is obviously false. Multiplying (or dividing) both sides of an inequality by a positive number preserves the inequality, but multiplying (or dividing) both sides by a negative number reverses the sense of the inequality. What if you don't know the sign of some quantity? Multiplying (or dividing) both sides of an inequality by that unknown quantity destroys the inequality. It's an illegal operation.

In this case, since w>0>z, w-z is a positive quantity, so multiplying both sides by w-z preserves the inequality. This yields w+z : -(w+z). What about w+z? You don't know it's sign. It might be positive, negative, or even zero. Dividing both sides of the inequality by that unknown quantity destroyed the inequality.

Just work with w+z : -(w+z). You don't know whether w+z is positive, zero, or negative, so the answer is D.