Eva and Karishma,
Thanks for your detailed reply. I am sorry for late reply. You are correct in that one shouldn't conclude from A^2 > B^2 that A > B. That's not infer-able. I think that in this discussion, we are confusing "given" and "question" -- While solving or simplifying the inequality or any question , we have to assume that the equation/inequality holds good.
For instance, (given some condition); is it true that 4x + 4 > 3x ? The first step would be to simplify the equation to x>-4? Correct?
OR If the inequality requires cross multiplication, say, 4/x > 3/y; y>0; x!=0; Here, I would solve the equation with an assumption that x >0; I wouldn't assume that's THE ONLY possibility. I would also solve the equation with an assumption that x<0;
In the above question, I was trying to solve the equation and see under what conditions that equation holds good. It turns out, from our discussion, that the second inequality doesn't hold good. Let's assume that the inequality holds good i.e. xv>0. However, there is also an underlying assumption that x>y (we assumed that x-y> square of a number -- I think that this assumption is CRUCIAL which I missed initially. ); Hence, even though xy> 0 is true, that doesn't mean that xy>0 is a sufficient condition because the necessary condition x>y may not satisfied.
Here's an equation that's fun to solve.
2.5(x-2)> sqrt (x) [this is from GMATClub m05/q15]
if-x-is-a-positive-integer-is-sqrt-x-2-5-x-5-1-x-91414.htmlMethod1: One of the methods is to plugin number.
MEthod2: Another would be to draw a line + parabolic curve. However, finding focus points etc would be difficult.
MEthod3:
Let's try by squaring. (Let's hold on to this)
Method4 :Let's first analyze the inequality to see what's going on:
Possible values of x : X<0 |X=0| 2>X>0 |2|X>2
X<0 => not possible
0<x<2 => (x is positive integer => x=1) => substitute in the equation => 1<2.5(1-2) => not possible.
X=2 => sqrt (2) ~1.4 <0 not possible
x>=3 => sqrt(3) < 2.5(3)-5 -> yes, true.
You see that by solving this inequality using analytical method, we can easily arrive at the answer even without looking at the two answer choices DS question! I didn't have to worry anything about primes etc.
Now, let's try my favorite square method (squaring an inequality is like an attempt to play blues chords on a major scale on your musical instrument--it's fun. Just as not all songs permit us to do that, we shouldn't do it. Now onward, I will refrain from squaring an inequality. I would probably try to do number plugging first. )
2.5(x-2) > sqrt(x)
6.25(x-2)^2 > x
Solving and re-arranging
x=2.65 or x=1.5; From our analysis, we know that 1.5 is not possible; hence, X> 2.65! This is also same as X>=3 (x = integer)
Thoughts?