If x, y, and z are real numbers such that x z, which of the following expressions MUST be positive?

If x, y, and z are real numbers such that x < 0 < y < z and |x| > z, which of the followin… — GMAT

If $x$ , $y$ , and $z$ are real numbers such that $x < 0 < y < z$ and $|x| > z$ , which of the following expressions MUST be positive?

$\frac{x y}{x + z}$ Answer
B
$(x + z)(y - x)$
C
$\frac{x + y}{z}$
D
$(z - y)(x + z)$
E
$x + y + z$

Answer

\frac{x y}{x + z}

Since

x < 0

and

y > 0

, the numerator

x y

is negative. From

x < 0

and

|x| > z

, we know

-x > z

, which means

x + z < 0

. Dividing the negative numerator by the negative denominator results in a strictly positive quotient. Thus, the expression stating 'x y over (x + z)' MUST be positive.

Step-by-Step Solution

Determine the sign of the numerator

x y

Since

x

is negative (

x < 0

) and

y

is positive (

y > 0

), their product

x y < 0

(negative).

The product of numbers with opposite signs is negative.

Determine the sign of the expression

x + z

using the absolute value condition

Since

x < 0

|x| = -x

. The inequality

|x| > z

becomes

-x > z

, which rearranges to

x + z < 0

(negative).

Subtracting

x

from both sides of

-x > z

yields

0 > x + z

Evaluate the quotient

\frac{x y}{x + z}

\frac{\text{negative}}{\text{negative}} = \text{positive}

Dividing two negative numbers yields a positive value.

Key Concept

Sign determination of products, quotients, and sums involving absolute values and inequalities

Question

Answer

Step-by-Step Solution

Key Concept

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