What is the complete solution set for the given inequality? \frac{1}{2}(-8x - 10) - \frac{… — Armed Services Vocational Aptitude Battery (ASVAB)

What is the complete solution set for the given inequality?

\frac{1}{2}(-8x - 10) - \frac{2}{3}(9x + 15) \leq -4(x - 2) + 2x + 1

The correct solution is

x \geq -3

By systematically distributing the fractions and negative integers on both sides, we simplify the expression to

-10x - 15 \leq -2x + 9

. Moving the variable terms to the left and constants to the right results in

-8x \leq 24

. Finally, dividing by

-8

isolates

x

and requires flipping the inequality symbol, providing the correct solution.

Distribute the fractions into the terms inside the parentheses on the left side.

(-4x - 5) - (6x + 10) \leq -4(x - 2) + 2x + 1

To eliminate the parentheses and clear the fractions for easier manipulation.

Distribute the

-4

on the right side of the inequality.

-4x - 5 - 6x - 10 \leq -4x + 8 + 2x + 1

To expand all terms fully before attempting to combine them.

Combine the like terms on both the left and right sides independently.

-10x - 15 \leq -2x + 9

To simplify the inequality into a standard two-step format.

Add

2x

to both sides and add

15

to both sides to group variable terms and constant terms.

-8x \leq 24

To isolate the

x

term on one side of the inequality.

Divide both sides by

-8

and reverse the direction of the inequality symbol.

x \geq -3

Multiplying or dividing an inequality by a negative number requires flipping the inequality sign to maintain the mathematical truth.

Solving Complex Linear Inequalities

Soru