If x, y, and z are non-zero real numbers such that x y z^2 0, and \frac{x}{z} < \frac{y}{z}, which of the following MUST be true?

If x, y, and z are non-zero real numbers such that x y z^2 < 0, x - y > 0, and \frac{x}{z}… — GMAT

If $x$ , $y$ , and $z$ are non-zero real numbers such that $x y z^2 < 0$ , $x - y > 0$ , and $\frac{x}{z} < \frac{y}{z}$ , which of the following MUST be true?

$\frac{x - z}{y} < 0$ Answer
B
$\frac{x - y}{z} > 0$
C
x^2 - y^2 > 0
D
$\frac{y - z}{x} < 0$
E
$\frac{x + z}{y} > 0$

Answer

\frac{x - z}{y} < 0

From

z^2 > 0

, the inequality

x y z^2 < 0

implies

x y < 0

, so

x

and

y

have opposite signs. Since

x - y > 0 \implies x > y

x

must be positive (

x > 0

) and

y

must be negative (

y < 0

). Next, comparing

x > y

with

\frac{x}{z} < \frac{y}{z}

shows that dividing by

z

reversed the inequality sign, which proves

z

is negative (

z < 0

). Evaluating

\frac{x - z}{y}

: the numerator

x - z

is positive minus negative, which equals positive plus positive (strictly positive), while the denominator

y

is strictly negative. A positive value divided by a negative value is always negative, so

\frac{x - z}{y} < 0

MUST be true.

Step-by-Step Solution

Determine the relative signs of

x

and

y

using

x y z^2 < 0

x

and

y

must have opposite signs (

x y < 0

Since

z \neq 0

z^2 > 0

is always positive. Dividing

x y z^2 < 0

z^2

yields

x y < 0

Determine the individual signs of

x

and

y

using

x - y > 0

x > 0

(positive) and

y < 0

(negative).

x - y > 0 \implies x > y

. Since

x

and

y

have opposite signs and

x

is strictly greater than

y

x

must be positive and

y

must be negative.

Determine the sign of

z

using

\frac{x}{z} < \frac{y}{z}

z < 0

(negative).

We know

x > y

. When dividing both sides of

x > y

z

, the inequality direction reverses to

<

. An inequality sign flips if and only if the divisor

z

is negative.

Evaluate the expression

\frac{x - z}{y}

\frac{x - z}{y} < 0

Because

x > 0

and

z < 0

-z > 0

, so the numerator

x - z = x + (-z)

is the sum of two positive numbers, which is positive. The denominator

y

is negative. Dividing a positive number by a negative number yields a negative result.

Key Concept

Deducing variable signs from product conditions and inequality sign reversal rules

Estimated Time:2m 0s

Question

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Step-by-Step Solution

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