GMAT Practice Question: If \frac{x^3}{y} < 0, and \frac{y}{z} > 0, then which of the following must be less than 1?
Question
If \frac{x^3}{y} < 0, and \frac{y}{z} > 0, then which of the following must be less than 1?
- \sqrt[3]{x}
- \frac{y}{x^2}
- x^3 z^4
- x^2 y z
- x y^2 z^3
Topics: inequalities, must be true statements
Solution
Step 1
From \frac{x^3}{y} < 0, we know the numerator x^3 and denominator y have opposite signs.
x<0 \text{ and } y>0\quad\text{or}\quad x>0 \text{ and } y<0
Step 2
From \frac{y}{z} > 0, we know y and z have the same sign.
y>0 \text{ and } z>0\quad\text{or}\quad y<0 \text{ and } z<0
Step 3
Combining the two results gives two possible sign assignments for (x,y,z).
\text{Case 1: }x<0,\;y>0,\;z>0 \text{Case 2: }x>0,\;y<0,\;z<0
Step 4
For option E, x y^2 z^3 is negative in both cases, and any negative number is less than 1.
\text{Case 1: }x<0,\;y^2>0,\;z^3>0 \implies x y^2 z^3<0<1 \text{Case 2: }x>0,\;y^2>0,\;z^3<0 \implies x y^2 z^3<0<1
Answer
E
