M21-30

Official Solution:


Is \(x^2*y^3*z > 0\)?

For the inequality \(x^2*y^3*z > 0\) to hold true, the following conditions must be met:

1. Both \(y\) and \(z\) must either be positive or negative, meaning they must share the same sign. This would ensure that \(y^3*z\) is positive;

AND

2. \(x\) cannot be zero, to ensure that \(x^2\) is positive.

(1) \(yz > 0\).

The above implies that \(y\) and \(z\) are either both positive or both negative, which satisfies the first condition. However, information about \(x\) (the second condition) is lacking: if \(x = 0\), \(x^2*y^3*z\) would be 0, not more than 0. Not sufficient.

(2) \(xz < 0\).

The above implies that \(x \ne 0\), thus satisfying the second condition. But we don't have information about the signs of \(y\) and \(z\) (the first condition). This too, is insufficient.

(1)+(2) When combined, both conditions are met. Therefore, it is sufficient.


Answer: C