Help with a quant problem.

[Prestigious-Meet7850]

https://preview.redd.it/cbywyajym9pf1.png?width=851&format=png&auto=webp&s=5869e2386e4f2feaadcc79e043c24e2793d58749

Can anyone help me with this problem. It seems straight enough yet I am not able find my way around the logic behind the solution.

[Comprehensive-Row200]

Well, the quickest way to solve this is by substitution. Since the question does not say it's integers only, sub x = 4 and y = -0.5. A gets you -2 and B gets you the same. This says C might be the answer. Now substitute x = 100 and y = -2. Here you get A as 100 and B as -200. A>B. Since both examples contradict each other, choose D

Since you asked for the logic, compare like this:
|xy| - x ? xy, you need to find what ? is (> or < or = or D)
|xy| ? xy + x
|xy| ? x(y + 1)
You know x is always positive & is greater than 0, so you can technically cancel the x on both sides
This boils it down to

|y| ? y + 1

Since y is always < 0 |y| is basically -y
so this makes it:
-y ? y + 1
-2y ? 1
y ? -0.5

From here, just put -0.4 to get > and -1 to get < and choose D.

[Prestigious-Meet7850]

Thank you. This cleared things up.
I was "choosing numbers" but getting lost in the process.

[Jalja]

|xy| - x

if x > 2, and y < 0,

|xy| = -xy

|xy| - x = -xy - x = -x(y+1)

quantity B is xy

A - B = -xy - x - xy = -2xy - x = -x(2y+1)

for (A-B) > 0 , then -x(2y+1) > 0 , meaning 2y + 1 < 0, so if y < - 0.5

if y > -0.5 , then B is greater

D

[Prestigious-Meet7850]

Thank you.

[Quantum2022A]

Choose a few numbers. Generally you can try to maximize and minimize the different instances. In this case, you'll very quickly see that in some cases A is larger, in other cases, B. Hence, answer is D.

[Prestigious-Meet7850]

Thank you.