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
$x \geq \frac{c}{c - 4}$
$x \geq \frac{7c}{c - 4}$ Cevap
D
$x \geq \frac{7c}{c + 4}$

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

Soru

Cevap

Adım Adım Çözüm

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