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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