If \(xy > 0\) and \(yz < 0\), then which of the following must be negative?


A \(xyz\)

B \(xy^2z\)

C \(x^2y^2z\)

D \(x^2y^2z^2\)

E \(\frac{xy}{z}\)
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xy>0 and yz<0.

multiply these 2 .

\(xy^2z<0.\)

xy*yz <0

xy>0

yz<0

one is positive and another one is negative. ultimate result must be negative.

B is the correct answer.

Looking at the first inequality we see that either x and y are both positive or they are both negative.

Combine that with the second inequality. If y is positive, z is negative, and when y is negative, z is positive.

Thus, our scenarios are:

x = pos, y = pos, z = neg

Or

x = neg, y = neg, z = pos

We see that x and z always have opposite signs, and thus, (x)(y^2)(z) is always negative.

Alternate solution:

We can use the following fact: If a > 0 and b < 0, then ab < 0. Therefore, we can multiply the two inequalities to obtain:

x(y^2)z < 0

Answer: B
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