Data Sufficiency

wilderness wrote: IMHO the best way to solve 1 is just invert the equations and change the signs (Is there any rule that forbids us from doing so ?? ) .

You can certainly do this if you know everything is positive. If x and y are positive, and you have the inequality:

1/x < 1/y

then you can multiply both sides by xy to get:

y < x

We don't need to worry about whether to reverse the inequality, provided we know that x and y are both positive. The same is true if x and y are both negative: then xy is positive, so we can multiply by xy without reversing the inequality. You do need to be careful, however, when x and y have opposite signs. Then, when you multiply by xy, you are multiplying by a negative, and you must reverse the inequality. If, say:

1/x < 1/y

then x could be negative, and y positive, and we have, multiplying by xy and reversing the inequality:

y > x

So yes, there is a rule one can follow here- but you need to be very careful if you don't know the signs of x and y.

AleksandrM wrote:Hey, I ended up with:

1) 10/(x + 1) - 5 < 0

2) x = 3 x = 4

Both: If x = 4, then 1) does not hold true. If x = 3, then 1 holds true.

And so I went with C.

Is my approach correct?

nope; you left off the left side of the double inequality in statement 1. your version (10/(x+1) - 5 < 0) is a rearrangement of 1/(x+1) < 1/2; you've completely neglected the fact that, per the original problem statement, 1/(x+1) is also greater than 1/5.
this observation totally changes the game, because a number that's stuck between 1/5 and 1/2 must be positive; with just the listed half of the inequality, you can't determine whether 1/(x+1) is positive or negative.

because everything in the double inequality is positive, you can just go ahead and take reciprocals of everything. (RULE: if everything is positive or everything is negative, you can take reciprocals of everything and flip all the inequality signs around.)
this yields 5 > x+1 > 2, or, more traditionally, 2 < x+1 < 5.
therefore, 1 < x < 4.

your rephrase of (2) is correct.

therefore, the answer is still c, although not quite for the reasons you've listed.