Let c be a constant such that c < 0. If c(x - 3) \leq 4(x + c), which of the following ine… — SAT

Let $c$ be a constant such that $c < 0$ . If $c(x - 3) \leq 4(x + c)$ , which of the following inequalities must be true?

A
$x \leq \frac{7c}{c - 4}$
B
$x \geq \frac{c}{c - 4}$
$x \geq \frac{7c}{c - 4}$ Answer
D
$x \geq \frac{7c}{c + 4}$

Answer

x \geq \frac{7c}{c - 4}

The correct answer is obtained by first distributing terms on both sides to get

cx - 3c \leq 4x + 4c

. Rearranging the terms to isolate the variable

x

on one side yields

(c - 4)x \leq 7c

. Since

c

is a negative constant (

c < 0

), the coefficient

(c - 4)

must also be negative. Dividing both sides of the inequality by this negative coefficient requires reversing the direction of the inequality sign, which yields the final result.

Step-by-Step Solution

Distribute the constants on both sides of the inequality.

cx - 3c \leq 4x + 4c

This simplifies the parentheses so that variable terms can be grouped.

Group all terms with

x

on the left side and terms with

c

on the right side.

cx - 4x \leq 7c

Isolating the variable terms on one side makes it possible to factor and solve for

x

Factor out

x

on the left side of the inequality.

(c - 4)x \leq 7c

This expresses the left side as a product of

x

and a single coefficient.

Analyze the sign of the coefficient

(c - 4)

given that

c < 0

c - 4 < 0

Since

c

is less than

0

, subtracting

4

from

c

must result in a value less than

-4

, which is strictly negative.

Divide both sides of the inequality by

(c - 4)

and flip the inequality symbol.

x \geq \frac{7c}{c - 4}

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

Key Concept

Solving linear inequalities by isolating the variable, and correctly reversing the inequality direction when multiplying or dividing by a negative variable parameter.

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