Manhattan Inequalities Chapter 12, DS Problem 12

[jr1988]

I have some doubts about problem 12 on page 18 of the Manhattan Gmat Guide#3:

If x≠0, is (x2 + 1) / x > y?

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

Can someone please explain why the first statement is not sufficient? I know that the strategy tells us not to multiply by a variable if it is unclear whether the number it stands for is positive or negative. However, I don't see how the statement x=y is not sufficient since the square of a number is always positive regardless of its sign, and any number plus one is always greater than the number itself, so by knowing that x=y I know that (x2 + 1)>yx = (x2 + 1)>x2. Am I missing something? Please tell how my reasoning is flawed since according to the answer key it should be C.

[vkb001]

I guess you were careful in defining your strategy, but multiplied anyhow.

We know that x≠0. This means x < 0 or x > 0.

(i) If x = y,

(x^2+1)/x = (y^2+1)/y = y + 1/y

If x > 0, or y > 0 (since x=y), then 1/y > 0 ==> y + 1/y > y
If x < 0, or y < 0 (since x=y), then 1/y < 0 ==> y + 1/y < y

It is because of this uncertainty that (i) is not sufficient. The value of (x^2+1)/x depends on the sign of x.

[Tommy Wallach]

Hey jr,

vkb is exactly right here. It's always important to understand why a given rule is a rule. The reason you're not supposed to multiply by a variable if you don't it's sign is that, whenever you multiply both sides of an inequality by a negative number, the sign flips. So, just as vkb showed here, you'll end up with two possible equations.

Good luck!

-t

[jr1988]

Thank you for your replies! I just took another look at the problem, and it is quite clear now...I don't know what was going through my head yesterday, thanks!