Consider the following linear inequality: 3(x - 2) - 7x \geq 2(x + 3) Which of the followi… — Armed Services Vocational Aptitude Battery (ASVAB)

Consider the following linear inequality:

3(x - 2) - 7x \geq 2(x + 3)

Which of the following represents the correct solution set for $x$ ?

The correct solution is

x \leq -2

To find the solution, expand the terms to get

3x - 6 - 7x \geq 2x + 6

. Combine the like terms on the left to form

-4x - 6 \geq 2x + 6

. Subtracting

2x

and adding

6

to both sides isolates the variable, resulting in

-6x \geq 12

. Finally, divide by

-6

. Because we are dividing by a negative number, the inequality sign must flip, giving the correct answer of

x \leq -2

Distribute the constants on both sides of the inequality.

3x - 6 - 7x \geq 2x + 6

Expanding the expressions clears the parentheses and allows us to combine like terms.

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

-4x - 6 \geq 2x + 6

3x - 7x

simplifies to

-4x

Move all terms containing

x

to one side and constants to the other side.

-6x \geq 12

Subtracting

2x

from both sides gives

-6x - 6 \geq 6

. Then, adding

6

to both sides isolates the variable term.

Divide both sides by

-6

and reverse the inequality symbol.

x \leq -2

When multiplying or dividing an inequality by a negative number, the inequality direction must be flipped to remain mathematically true.

Solving Linear Inequalities with Variables on Both Sides

Question