If |x| |y|, does |xy| = 15

Given: \(|x| ≠ |y|\)

Target question: Does \(|xy| = 15\)

Statement 1: \((x + 5)(y – 3) = 0\)
There are infinitely many values of x and y that satisfy statement 1. Here are two:
Case a: \(x = -5\) and \(y = 3\). In this case, the answer to the target question is YES, \(|xy| = 15\)
Case b: \(x = 0\) and \(y = 3\). In this case, the answer to the target question is NO, \(|xy|\) does not equal \(15\)
Since we can’t answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: \((x + y)^x = y^x\)
Aside: The equation indirectly tells us that y cannot equal 0.
Why?
If \(y = 0\), then \(y^x = 0^x = 0\), which means our equation becomes \((x + y)^x = 0\), and the only way this can be true is if \((x+y)=0\), which can only happen if \(x = 0\), but we're specifically told that \(|x| ≠ |y|\)

Since y cannot equal 0, we can safely divide both sides of the equation by \(y^x\) to get: \(\frac{(x + y)^x}{y^x} = 1\)

Rewrite the left side of the equation as follows: \((\frac{x + y}{y})^x = 1\)

Finally, we can simplify the fraction as follows: \((\frac{x}{y} + 1)^x = 1\)

At this point, we can see that the only wait for this equation to hold true is if \(x = 0\)

If \(x = 0\), then \(|xy|\) can never equal 15, which means the answer to the target question is NO, \(|xy|\) does not equal \(15\)
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer: B