In the inequality 3(4 - 2x) - 5(x - c) \geq -7, where c is a constant, the solution set co… — SAT

In the inequality $3(4 - 2x) - 5(x - c) \geq -7$ , where $c$ is a constant, the solution set consists of all values of $x$ such that $x \leq 3$ . What is the value of $c$ ?

Answer: 2.8

Answer

2.8

To find the value of

c

, first simplify the inequality

3(4 - 2x) - 5(x - c) \geq -7

by expanding the terms using the distributive property, which yields

12 - 6x - 5x + 5c \geq -7

. Combining like terms gives

-11x + 12 + 5c \geq -7

. Next, isolate the variable term by subtracting

12

and

5c

from both sides to obtain

-11x \geq -19 - 5c

. Dividing both sides of the inequality by

-11

and reversing the inequality sign results in

x \leq \frac{19 + 5c}{11}

. Given that the solution set consists of all values of

x

such that

x \leq 3

, the boundary value

\frac{19 + 5c}{11}

must equal

3

. Solving the equation

\frac{19 + 5c}{11} = 3

gives

19 + 5c = 33

, which simplifies to

5c = 14

and results in

c = 2.8

Step-by-Step Solution

Distribute the constants on the left side of the inequality.

12 - 6x - 5x + 5c \geq -7

Applying the distributive property simplifies the parentheses.

Combine the variable terms.

-11x + 12 + 5c \geq -7

Grouping like terms simplifies the inequality.

Subtract

12

and

5c

from both sides of the inequality.

-11x \geq -19 - 5c

This isolates the term containing

x

Divide both sides by

-11

and reverse the direction of the inequality.

x \leq \frac{19 + 5c}{11}

Dividing by a negative number requires flipping the inequality sign to maintain equivalence.

Set the algebraic boundary

\frac{19 + 5c}{11}

equal to the given boundary value of

3

and solve for

c

c = 2.8

Since the solution set is

x \leq 3

, the boundary values must be equivalent.

Key Concept

Solving multi-step linear inequalities in one variable containing parameters.

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