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

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

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

A
$a b c$
B
$\frac{a + b}{c}$
$a b^2 c$ Cevap
D
$\frac{b - c}{a}$
E
$a^2 b c$

Cevap

The expression

a b^2 c

must be negative.

From

\frac{a}{b} < 0

a

and

b

have opposite signs. From

b c > 0

b

and

c

have the same sign. Thus,

a

and

c

must have opposite signs. The inequality

a - c > 0

implies

a > c

, so

a

must be positive (

a > 0

) and

c

must be negative (

c < 0

). Since

b

shares the sign of

c

b

is also negative (

b < 0

). Evaluating

a b^2 c

a

is positive,

b^2

is strictly positive for any non-zero real number

b

, and

c

is negative. Therefore,

a b^2 c

is the product of two positive terms and one negative term, which MUST be negative.

Adım Adım Çözüm

Determine the relative signs of

a

b

, and

c

from the given inequalities.

\frac{a}{b} < 0

means

a

and

b

have opposite signs.

b c > 0

means

b

and

c

have the same sign. Therefore,

a

and

c

must have opposite signs.

Quotients of numbers with opposite signs are negative, and products of numbers with the same sign are positive.

Use

a - c > 0

to determine the explicit sign of each variable.

a - c > 0 \implies a > c

. Since

a

and

c

have opposite signs and

a > c

a

must be positive (

a > 0

) and

c

must be negative (

c < 0

). Since

b

has the same sign as

c

b

must also be negative (

b < 0

A positive number is always greater than a negative number.

Evaluate the sign of

a b^2 c

Since

a > 0

b^2 > 0

(as

b \neq 0

), and

c < 0

a b^2 c = (+) \times (+) \times (-) < 0

The product of two positive real numbers and one negative real number is strictly negative.

Anahtar Kavram

Deducing signs of variables using properties of products, quotients, and inequalities.

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