no, you can't do this -- you can't multiply by a denominator whose sign you are unsure of.
i.e., to get to this “rephrase” you have to multiply by (x + y); that is a problem, because we don't know whether (x + y) is positive or negative.
if (x + y) is positive then you get to rephrase shown here, but, if it is negative, then it becomes "<".

PLEASE READ THE THREAD BEFORE YOU POST -- all of this has already been explained earlier on this same page:
post10355.html#p10355

If x is not equal to -y, is (x-y)/(x+y) > 1?

(1) x > 0
(2) y < 0

I performed a few steps to simplify the left side of the inequality
and obtain a certain sign for the denominator.

As one number, x+y multiplied or divided by itself is positive. So, I performed the following operation to only
the left side,

[(x-y)/(x+y)]*[(x+y)/(x+y)] =

(x^2-y^2)/(x^2+2xy+y^2)

The denominator is positive, because it has can be written as n^2. Multiply both sides of the inequality by it

(x-y)/(x+y) > 1.............?
(x^2-y^2)/(x^2+2xy+y^2) > 1
x^2-y^2 > x^2+2xy+y^2

and combine like variables.

0 > 2y^2+2xy.

With a bit more simplification, rephrase the original expression as

0 > y^2 + xy

Is y(y+x) < 0 ?

Viewed as a positive-negative product.
The answer is yes in two situations.
Note that each situation has four conditions.

y > 0, y+x < 0, x < 0 AND x < -y

y < 0, y+x > 0, x > 0 AND x > -y

(1) x > 0. Statement (1) does not allow one
to reach the further conclusions that y < 0 and x > -y.
(2) y < 0. Statement (2) provides no information
on the value of x or the signs of x and y.

We do not have further information that x > -y, as is necessary to conclude that (x-y)/(x+y) > 1.

your steps to manipulate this inequality are okay (i.e. the first line quoted above). the problem is everything after that. regardless of whether y is positive or negative (BTW notice that you haven't accounted for y being 0), we know nothing about x other than that it is not -y. you seem to be taking the question to be a true statement and comparing it against the statements, or reading too much into the relationship between x and y. either way, this analysis is incorrect..

_________________
Tim Sanders
Manhattan GMAT Instructor

I did this problem with flow chart approach learned from advanced quant strategy book.

Basically, if I can rephrase the question starting with a condition:
that x+y is positive, then I can cross-multiply directly.

This condition thus leads to two possiblities, one in which:
x-y > x + y
so it becomes o>2y, thus y is negative

And one in which
x-y < x+y (pretending that the scenario is opposite and x+y is negative, then I had just flipped the sign by cross-multiplying, hence I'm compensating it by flipping the sign later.
then in this case, 0<2y, so y is positive

Therefore, we arrived at two different conclusions, based on whether x+y is positve or negative. And we need a statement that tells us that x+y is positve or negative to target one specific result. Yet, as you can see, given that x is positve and y is negative from combining both statements, there are still different possibilities. x+y can be either positive or negative. THUS E


Similar problems can be found in OG 11th:
question 143 and 145, both can be solved easily without picking numbers if you know the flow chart approach.

rachel, you are doing great work here! Keep it up!

_________________
Jamie Nelson
ManhattanGMAT Instructor