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

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Being well-dressed gives a feeling of inward tranquillity [that] religion is powerless to bestow.
C.F. Forbes

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!

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Jamie Nelson
ManhattanGMAT Instructor

Hi Rachel / Jamie - yeah this is approach is very good... Can you please confirm in my understanding is correct or am i missing any points here?

(x-y)/(x+y) > 1?

As Rachel has manipulated above we can set up 2 scenarios

1. (x+y) > 0
then x-y > x + y
so it becomes o>2y, thus y is negative

2. (x+y) < 0
then x-y < x + y
so it becomes o<2y, thus y is positive

how do i know which one to prove? i.e if Im able to prove any 1 scenario then would i be able to confirm (x-y)/(x+y) > 1?

Let says Option A and B were x<0 and y < 0 - in this case

x+y is definitely less than 0 so if x+y<0 then I should get y = +ve which is not the case here hence we get a definite NO so is the ANS C?

Let says Option A and B were x>0 and y > 0 - in this case


x+y>o and the result to prove is y = negative which is not the case so we again a definite NO so the ANS is C


Let says Option A and B were x>0 and y < 0 - in this case


x+y can be +ve or -ve so we don't know which one to prove y=-ve or +ve hence the ans is E.

Just wanted to be really sure of this method. My sincere thanks to both.

Cheers

i'm actually going to recommend that you go back to the drawing board on this one. your approach is flawed from the beginning. it appears you are effectively trying to use the question to prove the statements, or some other approach that does not use the statements to evaluate the question. be careful about this sort of thing, as this is one of the easiest traps to fall into..

_________________
Tim Sanders
Manhattan GMAT Instructor

Hi Tim - thanks for your response.

the sol below for this questions is parallel to the sol given for Q13 - Advanced guide Pg132

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

1. x>0
2. y<0

Constrain = x + y not equal to zero

Question rephrase

Since we dont know whether x+y is +ve or -ve so we need to consider 2 scenario

1/ x+y > 0
x-y > x + y => 0>y

2/ x+y < 0
x-y < x+y => 0<y

So now if the statements tell us whether x+y > 0 then the question Is (x-y)/(x+y) > 1 becomes Is y negative

or if the statements tell us whether x+y < 0 then the question Is (x-y)/(x+y) > 1 becomes Is y Positive

Conversely if x+y > 0 and y is negative we can say (x-y)/(x+y) > 1
if x+y < 0 and y is +ve we can say (x-y)/(x+y) > 1

Are the above statements right?

Statement 1 - x>0 we do not know whether x+y is +ve or -ve so we dont know to which one to answer y = +ve or -ve

Statement 2 - y<0 we do not know whether x+y is +ve or -ve so we dont know to which one to answer y = +ve or -ve

Combined - we still cant say whether x+y is +ve or -ve

Therefore E

Does this makes sense?

--------------------- Few doubts related to the same problem --------

If the statements were
1. x>0
2. y>0

Combined we know x+y > 0 so the question now Is (x-y)/(x+y) > 1 can be rephrased to Is y negative

But according to statement 2 y is +ve so the answer is Definite NO hence answer would be C

If the statements were
1. x<0
2. y<0

Adding 2 statements we know x+y < 0 so the question becomes is Y positive
From B we know y is negative hence this is a def no

So the answer is C correct?

Cheers

jp.jprasanna, yes, all of that works.

_________________
Being well-dressed gives a feeling of inward tranquillity [that] religion is powerless to bestow.
C.F. Forbes