If a, b, and c are non-zero real numbers such that a^3 b^2 c 0, and \frac{a}{c} < 0, which of the following expressions MUST be positive?

If a, b, and c are non-zero real numbers such that a^3 b^2 c < 0, a b^3 > 0, and \frac{a}{… — GMAT

If $a$ , $b$ , and $c$ are non-zero real numbers such that $a^3 b^2 c < 0$ , $a b^3 > 0$ , and $\frac{a}{c} < 0$ , which of the following expressions MUST be positive?

$\frac{b - c}{a}$ Answer
B
$\frac{a b}{c}$
C
$c(a + b)$
D
$\frac{a + c}{b}$
E
$b(c - a)$

Answer

\frac{b - c}{a}

Deducing the signs of

a

b

, and

c

reveals two possible cases: either

a > 0, b > 0, c < 0

a < 0, b < 0, c > 0

. In Case 1,

b - c > 0

and

a > 0

, so

\frac{b - c}{a} > 0

. In Case 2,

b - c < 0

and

a < 0

, so

\frac{b - c}{a} > 0

. Therefore,

\frac{b - c}{a}

is strictly positive in all cases.

Step-by-Step Solution

Analyze the sign conditions from the given inequalities.

From

a b^3 > 0

a

and

b

must have the same sign. From

\frac{a}{c} < 0

a

and

c

must have opposite signs. From

a^3 b^2 c < 0

, since

b^2 > 0

for non-zero

b

, we have

a^3 c < 0

, which confirms

a

and

c

have opposite signs.

Odd powers preserve the sign of a variable, whereas even powers are strictly positive for non-zero real numbers.

Determine the two possible sign scenarios for

(a, b, c)

Case 1:

a > 0, b > 0, c < 0

.
Case 2:

a < 0, b < 0, c > 0

Since

a

and

b

share the same sign and

c

has the opposite sign, these are the only two valid assignments.

Evaluate the sign of the numerator and denominator of

\frac{b - c}{a}

in Case 1.

In Case 1 (

a > 0, b > 0, c < 0

b - c = \text{positive} - \text{negative} = \text{positive}

. Denominator

a > 0

. Ratio

\frac{\text{positive}}{\text{positive}} > 0

Subtracting a negative number from a positive number yields a positive result.

Evaluate the sign of the numerator and denominator of

\frac{b - c}{a}

in Case 2.

In Case 2 (

a < 0, b < 0, c > 0

b - c = \text{negative} - \text{positive} = \text{negative}

. Denominator

a < 0

. Ratio

\frac{\text{negative}}{\text{negative}} > 0

Dividing two negative values produces a positive quotient.

Key Concept

Positive and Negative Number Properties in Inequalities

Estimated Time:2m 0s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic