In the inequality -2(3x - k) - 5(x + 1) > 17, k is an integer constant. If the maximum int… — SAT

In the inequality $-2(3x - k) - 5(x + 1) > 17$ , $k$ is an integer constant. If the maximum integer value of $x$ that satisfies the inequality is $2$ , what is the least possible value of $k$ ?

Answer: 23

Answer

The correct answer is 23.

By simplifying the inequality to

x < \frac{2k - 22}{11}

, we establish that the upper bound of the solution interval must be strictly greater than

2

but less than or equal to

3

for

2

to be the maximum integer solution. Solving the resulting compound inequality

2 < \frac{2k - 22}{11} \le 3

yields

22 < k \le 27.5

. The smallest integer within this range is

23

Step-by-Step Solution

Distribute the constants on the left side of the inequality.

-6x + 2k - 5x - 5 > 17

Applying the distributive property simplifies the expression and removes the parentheses.

Combine like terms on the left side and isolate the

x

term.

-11x + 2k - 5 > 17 \implies -11x > 22 - 2k

Grouping

x

terms together and moving the constant terms to the other side prepares the inequality for division.

Divide both sides by

-11

and flip the inequality sign.

x < \frac{22 - 2k}{-11} \implies x < \frac{2k - 22}{11}

Dividing by a negative number reverses the direction of the inequality sign from greater-than (

>

) to less-than (

<

Set up the inequality for the maximum integer solution to be

2

2 < \frac{2k - 22}{11} \le 3

For

2

to be the largest integer satisfying

x < L

(where

L

is the boundary),

2

must be strictly less than

L

, and

L

must be less than or equal to the next integer,

3

Solve the compound inequality for the parameter

k

22 < 2k - 22 \le 33 \implies 44 < 2k \le 55 \implies 22 < k \le 27.5

Multiplying all parts by 11, adding 22, and dividing by 2 isolates

k

Find the least integer value of

k

in the interval

(22, 27.5]

23

The integers that satisfy

22 < k \le 27.5

are

23, 24, 25, 26,

and

27

. The least of these values is

23

Key Concept

Solving linear inequalities in one variable involving parameter bounds, negative coefficients, and integer solution constraints.

Question

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