w, x, y, and z are integers. If z > y > x > w, is |w| > x^2

Is there any other solution that can be applied to this problem within 2 minutes?

Good question. +1.

Answer to the question is neither E nor D, it's B.

w, x, y, and z are integers. If z > y > x > w, is |w| > x^2 > |y| > z^2?

Given: \(w\), \(x\), \(y\), and \(z\) are integers and \(z>y>x>w\): --w--x--y--z--
Question: is \(|w|>x^2>|y|>z^2\)?

Now, since \(z>y\), then in order \(|y|>z^2\) to hold true \(y\) must be negative and since \(y>x>w\), then \(x\) and \(w\) must be negative. So in order the answer to be YES, \(y\), \(x\) and \(w\) must be negative (notice that the reverse is not always true: they might be negative but the answer could be NO, but if answer is YES then they must be negative). So if we get that either of them is not negative then we could conclude that the answer to the question is NO.

(1) wx > yz --> the product of two smaller integers (w and x) is more than the product of to larger integers (y and z). It's possible for example that all but \(z\) are negative but even in this case we could have two different answers:
If \((z=1)>(y=-2)>(x=-3)>(w=-4)\) --> in this case answer to the question "is \(|w|>x^2>|y|>z^2\)?" will be NO;
If \((z=1)>(y=-2)>(x=-3)>(w=-10)\) --> in this case answer to the question "is \(|w|>x^2>|y|>z^2\)?" will be YES.
Not sufficient.

(2) zx > wy --> \(y\) is positive, because if it's negative then \(x\) and \(w\) are also negative (since \(y>x>w\)) and in this case no matter whether \(z\) is positive or negative: \(zx<wy\) (if z>0 then zx<0<wy and if z<0 then the product of two "more negative" numbers w and y will be more then the product of two "less negative" numbers z and x, so again we would have: zx<wy). Thus we have that \(y\) must be positive, which makes \(|w|>x^2>|y|>z^2\) impossible (as discussed), hence the answer to the question is NO. Sufficient,

Answer: B.

Hope it's clear.

I believe your statement "Now, since \(z>y\), then in order \(|y|>z^2\) to hold true \(y\) must be negative.." is wrong.

if 0 < y < z < 1 , then these y, z also satisfy \(z>y\) and \(|y|>z^2\) (for example y = 0.5 and z = 0.6) so it is not necessary for y to be negative.

Good question. +1.

Answer to the question is neither E nor D, it's B.

w, x, y, and z are integers. If z > y > x > w, is |w| > x^2 > |y| > z^2?

Given: \(w\), \(x\), \(y\), and \(z\) are integers and \(z>y>x>w\): --w--x--y--z--
Question: is \(|w|>x^2>|y|>z^2\)?

Now, since \(z>y\), then in order \(|y|>z^2\) to hold true \(y\) must be negative and since \(y>x>w\), then \(x\) and \(w\) must be negative. So in order the answer to be YES, \(y\), \(x\) and \(w\) must be negative (notice that the reverse is not always true: they might be negative but the answer could be NO, but if answer is YES then they must be negative). So if we get that either of them is not negative then we could conclude that the answer to the question is NO.

(1) wx > yz --> the product of two smaller integers (w and x) is more than the product of to larger integers (y and z). It's possible for example that all but \(z\) are negative but even in this case we could have two different answers:
If \((z=1)>(y=-2)>(x=-3)>(w=-4)\) --> in this case answer to the question "is \(|w|>x^2>|y|>z^2\)?" will be NO;
If \((z=1)>(y=-2)>(x=-3)>(w=-10)\) --> in this case answer to the question "is \(|w|>x^2>|y|>z^2\)?" will be YES.
Not sufficient.

(2) zx > wy --> \(y\) is positive, because if it's negative then \(x\) and \(w\) are also negative (since \(y>x>w\)) and in this case no matter whether \(z\) is positive or negative: \(zx<wy\) (if z>0 then zx<0<wy and if z<0 then the product of two "more negative" numbers w and y will be more then the product of two "less negative" numbers z and x, so again we would have: zx<wy). Thus we have that \(y\) must be positive, which makes \(|w|>x^2>|y|>z^2\) impossible (as discussed), hence the answer to the question is NO. Sufficient,

Answer: B.

Hope it's clear.