souravp94 wrote:
\(|x|y > x|y|\)
Quantity A |
Quantity B |
\((x+y)^2\) |
\((x-y)^2\) |
Nice question!!
The given information, \(|x|y > x|y|\), provides some very useful information about x and y.
First recognize that:
i) If x and y are both POSITIVE, then \(|x|y = x|y|\)
ii) If x and y are both NEGATIVE, then \(|x|y = x|y|\)
iii) If x is POSITIVE and y is NEGATIVE, then \(|x|y < x|y|\)
iv) If x is NEGATIVE and y is POSITIVE, then \(|x|y > x|y|\)You can verify this by testing various values of x and yNotice that case iv is the only case that satisfies the given inequality \(|x|y > x|y|\)
So, it must be the case that
x is NEGATIVE and y is POSITIVENow let's move on to the comparison. We have:
Quantity A: \((x+y)^2\)
Quantity B: \((x-y)^2\)
Expand and simplify both quantities to get:
Quantity A: \(x^2 + 2xy + y^2\)
Quantity B: \(x^2 - 2xy + y^2\)
Subtract \(x^2\) and \(y^2\) from both quantities to get:
Quantity A: \(2xy\)
Quantity B: \(-2xy\)
Add \(2xy\) to both quantities to get:
Quantity A: \(4xy\)
Quantity B: \(0\)
Since we already know
x is NEGATIVE and y is POSITIVE, then we also know that \(xy\) is NEGATIVE, which also means \(4xy\) is NEGATIVE.
Since 0 > some NEGATIVE value, the correct answer is B
Cheers,
Brent