Prob with inequalities ??

[prempeter]

I have an inequality :

r/(b+w) > w/(b+r)

cross multiplying :

r(b+r) > w(b+w)

rb + r^2 > wb + w^2

r^2 - w^2 > wb - rb

(r+w)(r-w) > b(w-r) ----- (A)

Now, I have two ways of proceeding from here, and I get two different contradicting answers. I am not sure where I am going wrong

Route 1:

Multiply (A) by -1 on both sides

-1(r+w)(r-w) < -b(w-r)

(r+w)(w-r) < -b(w-r)

Cancelling (w-r) on both sides

r+w < -b ---------- (Result 1)

Route 2:

Multiply (A) by -1 on both sides

-1(r+w)(r-w) < -b(w-r)

-1(r+w)(r-w) < b(r-w)

Cancelling (r-w) on both sides

-1(r+w) < b

-r-w < b

-r < b+w

-r-b < w

-b < w+r ------------(Result 2)

Note that the equality is reversed between result 1 and result 2. I have gone nuts trying to figure this out.....Any help would be much appreciated.

[OutOfTime]

I am no expert but you can't cross multiply when you have inequalities.

[Guest]

I think the problem is that in route 1, you're assuming (w - r) > 0 and in route 2, you're assuming (r - w) > 0. Since your assumptions on the inequality between w and r are different, your results are different.

[prempeter]

Thanks everyone, for the reply.

Thanks Guest. I think you may be onto something. Can you please explain further ?

Andrew, I did reverse the inequality when multiplying by -1.

This is just one section of a problem that I am working on. I am just curious about the different results. Obviously, I am doing something wrong, and would like to correct it.

[Chris Ryan]

Hi all,

The whole key here is that EVERY TIME you multiply or divide an inequality by a variable or variable expression, you must EITHER consider both cases (expression is positive; expression is negative) OR make an EXPLICIT assumption about the sign of the expression. (The third case, when the variable expression = Zero, tends not to be so important -- it's often ruled out by the presence of division. But you should recognize the case in theory.)

For example, right at the start, you made two manipulations without acknowledging the necessary assumptions:

Stage A: r/(b+w) > w/(b+r)

Manipulation: Multiply both sides by (b+w). [do the cross-multiplication one step at a time -- it's legal, but you must do it this way]

Question: What is the SIGN of (b+w)?

Stage B:
Case 1. IF (b+w) is POSITIVE, then the result is this: r > (b+w)*w/(b+r)
Case 2. IF (b+w) is NEGATIVE, then the result is this: r < (b+w)*w/(b+r)

Of course, the expressions on each side look the same -- what changes is the direction of the inequality symbol.

Considering Case 1 only:

Manipulation: Multiply both sides by (b+r).

Stage C:
Case 1.a. IF (b+r) is POSITIVE, then the result is this: (b+r)*r > (b+w)*w
Case 1.b. IF (b+r) is NEGATIVE, then the result is this: (b+r)*r < (b+w)*w

We can combine with the 2 subcases from Case 2 to get 2 combined cases:
Combined Case 1: IF (b+r) and (b+w) have the SAME SIGN, then we have this: (b+r)*r > (b+w)*w
Combined Case 2: IF (b+r) and (b+w) have OPPOSITE SIGNS, then we have this: (b+r)*r < (b+w)*w

This seems really complex, I know, but the point is that you CAN'T ignore the cases. The situation is inherently complex. Often, the solution to the Data Sufficiency problem (since these are usually DS) *revolves* around the cases. For instance, one of the statements will often be one of the conditions. Or the conditions might arise out of both statements together. Look at OG 11th edition, DS #143 and #145 for examples.

So, to your question!

Route 1: When you say you CANCEL (w-r) on both sides, you are ASSUMING w-r>0 -- because you didn't change the direction of the inequality symbol when you DIVIDED by w-r! (Canceling = dividing.) So you've assumed w>r.

Route 2: In contrast, you CANCEL (r-w) -- and therefore you are assuming r-w>0, given the fact that you didn't change the direction of the inequality symbol. So you've assumed r>w.

And of course, you can't assume both at the same time!

I hope this is helpful.

Best,
Chris Ryan
Director of Product and Instructor Development
ManhattanGMAT

[prempeter]

Thanks Chris. That is a very detailed, well written explanation.