What about when both are negative

If both x and y are negative then the inequality $|x|y>x|y|$ is not true as for x<0 and y<0 we can write $|x|y>x|y|$ as $-xy> - xy$ which is clearly not possible.


The best way to solve this is to rewrite the inequality:

$|x|y>x|y|$ as

$\frac{y}{|y|}>\frac{x}{|x|}$ since $|x|$ and $|y|$ are always positive numbers it wont effect the inequality.

Also $\frac{n}{|n|}$ can be either +1 or -1.

Hence for this inequality to hold y is positive and x is negative.


Once you have the fact that x is negative you can substitute in the equations given in the quantity A and B to calculate which is greater!

For example put $x =-n$ where n is a positive number and y is positive as we have seen before!

Qty A: $(x+y)^2=(y-n)^2$

Qty B: $(x-y)^2=(y+n)^2$

Quantity B is greater!

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