Which of the following represents the correct solution set for the inequality below? 4(x -… — Armed Services Vocational Aptitude Battery (ASVAB)

Which of the following represents the correct solution set for the inequality below?

4(x - 3) - 3(2x + 5) \geq x - 12

$x \leq -5$ Answer
B
$x \geq -5$
C
$x \leq 5$
D
$x \geq 5$

Answer

The correct solution is

x \leq -5

The correct answer is derived by carefully distributing the values across the parentheses (yielding

4x - 12 - 6x - 15 \geq x - 12

), combining the like terms to form

-2x - 27 \geq x - 12

, and isolating

x

to get

-3x \geq 15

. Crucially, dividing both sides by

-3

requires reversing the direction of the inequality, which correctly results in

x \leq -5

Step-by-Step Solution

Distribute the coefficients outside the parentheses into the terms inside.

4x - 12 - 6x - 15 \geq x - 12

This removes the parentheses. Care must be taken to multiply -3 by +5 correctly to get -15.

Combine the like terms on the left side of the inequality.

-2x - 27 \geq x - 12

Simplifying both sides makes it easier to isolate the variable.

Move all terms containing

x

to one side and all constant terms to the other side.

-3x \geq 15

Subtracting

x

from both sides gives

-3x

, and adding

27

to both sides gives

15

Divide both sides by -3 and reverse the inequality symbol.

x \leq -5

Whenever you multiply or divide an inequality by a negative number, the direction of the inequality sign must be flipped to maintain a true mathematical statement.

Key Concept

Solving multi-step linear inequalities involving distribution and the inequality sign flip rule.

Question

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Step-by-Step Solution

Key Concept

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