For real numbers x, y, and z, none of which is equal to zero, suppose that x y^3 z > 0, \frac{x^2 z}{y} 0. Which of the following inequalities MUST be true?

For real numbers x, y, and z, none of which is equal to zero, suppose that x y^3 z > 0, \f… — GMAT

For real numbers $x$ , $y$ , and $z$ , none of which is equal to zero, suppose that $x y^3 z > 0$ , $\frac{x^2 z}{y} < 0$ , and $x + y > 0$ . Which of the following inequalities MUST be true?

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

Answer

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

The condition

x y^3 z > 0

reduces to

x y z > 0

because

y^2 > 0

. The condition

\frac{x^2 z}{y} < 0

reduces to

\frac{z}{y} < 0

because

x^2 > 0

, implying

y

and

z

have opposite signs (

y z < 0

). Substituting

y z < 0

into

x (y z) > 0

forces

x < 0

. Furthermore,

x + y > 0

with

x < 0

requires

y > 0

. Then

y > 0

and

y z < 0

force

z < 0

. Consequently,

x

and

z

are both negative numbers, so

x + z

is negative. Dividing this negative sum by the positive number

y

guarantees that

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

MUST be true.

Step-by-Step Solution

Analyze the first given inequality to establish a sign relationship among the variables.

Since

y \neq 0

y^2 > 0

. Dividing

x y^3 z > 0

y^2

gives

x y z > 0

Dividing an inequality by a strictly positive quantity preserves the inequality sign.

Analyze the second given inequality to determine the relationship between

y

and

z

Since

x \neq 0

x^2 > 0

. Dividing

\frac{x^2 z}{y} < 0

x^2

gives

\frac{z}{y} < 0

, which means

z

and

y

must have opposite signs (i.e.,

y z < 0

A quotient of two non-zero numbers is negative if and only if the numerator and denominator have opposite signs.

Deduce the individual signs of

x

y

, and

z

using the results of Steps 1 and 2 and the condition

x + y > 0

From

x(yz) > 0

and

yz < 0

, we deduce

x < 0

. Since

x < 0

and

x + y > 0

y

must be positive (

y > 0

). Because

y > 0

and

yz < 0

z

must be negative (

z < 0

Product of two negative numbers is positive, and adding a positive number larger in magnitude than a negative number yields a positive sum.

Evaluate the expression

\frac{x + z}{y}

using the established signs.

Since

x < 0

and

z < 0

, their sum

x + z < 0

. Dividing the negative sum by positive

y

gives

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

The sum of two negative numbers is negative, and dividing a negative number by a positive number gives a negative quotient.

Key Concept

Deduction of positive and negative variable signs from products, quotients, and sums in inequalities.

Estimated Time:2m 0s

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