If x^3/y 0, then which of the following must be less than 1?

Question

If x^3/y < 0, and y/z > 0, then which of the following must be less than 1?

GMAT Panda · Accepted Answer

E From x^3/y < 0, we know the numerator x^3 and denominator y have opposite signs. x<0 and y>0 x>0 and y<0 From y/z > 0, we know y and z have the same sign. y>0 and z>0 y<0 and z<0 Combining the two results gives two possible sign assignments for (x,y,z). Case 1: x<0, y>0, z>0 Case 2: x>0, y<0, z<0 For option E, x y^2 z^3 is negative in both cases, and any negative number is less than 1. Case 1: x<0, y^2>0, z^3>0 x y^2 z^3<0<1 Case 2: x>0, y^2>0, z^3<0 x y^2 z^3<0<1

Explanation	Calculations	Help
$From \frac{x ^{3}}{y} < 0, we know the numerator x^{3} and denominator y have opposite signs.$	$x < 0 and y > 0 or x > 0 and y < 0$
$From \frac{y}{z} > 0, we know y and z have the same sign.$	$y > 0 and z > 0 or y < 0 and z < 0$
$Combining the two results gives two possible sign assignments for (x, y, z) .$	$Case 1: x < 0, y > 0, z > 0 Case 2: x > 0, y < 0, z < 0$
$For option E, x y^{2} z^{3} is negative in both cases, and any negative number is less than 1.$	$Case 1: x < 0, y^{2} > 0, z^{3} > 0 ⟹ x y^{2} z^{3} < 0 < 1 Case 2: x > 0, y^{2} > 0, z^{3} < 0 ⟹ x y^{2} z^{3} < 0 < 1$

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