enigma123 wrote:
w, x, y, and z are integers. If z > y > x > w, is |w| > x^2 > |y| > z^2?
(1) wx > yz
(2) zx > wy
Responding to a pm:
When you see z > y > x > w, you could consider a number line:
--------w------------x----------y---------z-----------
The four integers divide the number line into five parts. 0 could be anywhere on the five parts.
Question: is \(|w| > x^2 > |y| > z^2\)?
(1) wx > yz
Think of the number line. When will the product of the 2 smallest numbers be more than the two largest numbers? Say, when all numbers are negative.
------ (-100) -------- (-8) -------- (-5) ----- (-2) ------
Here, wx > yz
Is \(|w| > x^2 > |y| > z^2\)? Yes.
Now we can make a small change to this and get a No as answer.
------ (-10) -------- (-8) -------- (-5) ----- (-2) ------
Is \(|w| > x^2 > |y| > z^2\)? No.
So this statement is clearly not sufficient.
(2) zx > wy
Now when will the product of the largest and third largest number be greater than the product of the second largest and the smallest number? That's easy, right? One case we can think of right away is when all numbers are positive.
--------w------------x----------y---------z-----------
--------1------------2----------3---------4-----------
Is |w| > x^2 > |y| > z^2? No.
Let's now try to get a Yes case.
When will \(|w| > x^2\)? w must be negative with a large absolute value. If x is 2, w must be -5 or smaller. Let's go on. Now \(x^2\) is already greater than |y|. But can |y| be greater than \(z^2\) in this case? No.
For |y| to be greater than \(z^2\), y should be negative with a large absolute value (compared with z). So is it possible that w, x and y are all negative and z is positive or 0? No. In that case zx will be negative (or 0) while wy will be positive.
Hence, it is not possible to get a "Yes" case. Sufficient.
Answer (B)