In the inequality -3(2x - 5) + 8 \geq 2(kx - 3) - 1, k is a constant. If the solution set … — SAT

In the inequality $-3(2x - 5) + 8 \geq 2(kx - 3) - 1$ , $k$ is a constant. If the solution set for $x$ is $x \leq 2$ , what is the value of $k$ ?

Answer: 4.5

Answer

4.5

To find the value of

k

, simplify both sides of the inequality:

-3(2x - 5) + 8 \geq 2(kx - 3) - 1

becomes

-6x + 23 \geq 2kx - 7

. Grouping the variable terms on the left and constants on the right gives

-(6 + 2k)x \geq -30

. Because the solution set is

x \leq 2

, dividing by the negative coefficient reverses the inequality sign to yield

x \leq \frac{30}{6 + 2k}

. Setting the boundary value

\frac{30}{6 + 2k}

equal to 2 gives the equation

30 = 2(6 + 2k) = 12 + 4k

. Subtracting 12 from both sides gives

18 = 4k

, which results in

k = 4.5

Step-by-Step Solution

Simplify both sides of the inequality

-6x + 23 \geq 2kx - 7

Distribute the constants on both sides:

-3(2x - 5) + 8 = -6x + 15 + 8 = -6x + 23

, and

2(kx - 3) - 1 = 2kx - 6 - 1 = 2kx - 7

Isolate the variable terms on the left side and constants on the right side

-(6 + 2k)x \geq -30

Subtract

2kx

and

23

from both sides to get

-6x - 2kx \geq -7 - 23

, then factor out

x

to get

-(6 + 2k)x \geq -30

Relate the inequality to the given solution boundary

x \leq \frac{30}{6 + 2k}

Since the solution is

x \leq 2

, dividing both sides by the negative coefficient

-(6 + 2k)

reverses the inequality sign, yielding

x \leq \frac{-30}{-(6 + 2k)} = \frac{30}{6 + 2k}

Solve for

k

using the boundary equation

k = 4.5

Set the boundary expression equal to 2:

\frac{30}{6 + 2k} = 2

. Multiply both sides by

6 + 2k

to get

30 = 12 + 4k

, which simplifies to

18 = 4k

, giving

k = 4.5

Key Concept

Solving linear inequalities in one variable involving variable coefficients and applying the inequality direction flip when dividing by a negative value.

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