For the linear inequality -\frac{3}{4}(8x - 12) + 2x > -3(x - 4) - 2x, the value x = 3 is … — Armed Services Vocational Aptitude Battery (ASVAB)

For the linear inequality $-\frac{3}{4}(8x - 12) + 2x > -3(x - 4) - 2x$ , the value $x = 3$ is included in the solution set.

Answer: Answer

False, because the solution set is

x > 3

, which strictly excludes the boundary value of 3.

Simplifying both sides of the inequality yields

-4x + 9 > -5x + 12

. Isolating

x

provides the solution set

x > 3

. Because the inequality is strictly 'greater than', it does not include the boundary value of 3, making the statement false.

Distribute the negative fraction

-\frac{3}{4}

on the left side of the inequality.

-\frac{3}{4}(8x) - (-\frac{3}{4})(12) + 2x = -6x + 9 + 2x

To clear the parentheses and prepare the left side for combining like terms.

Distribute the negative integer

-3

on the right side of the inequality.

-3(x) - (-3)(4) - 2x = -3x + 12 - 2x

To clear the parentheses on the right side of the inequality.

Combine the like

x

terms on both sides of the inequality.

Left side:

-6x + 2x + 9 = -4x + 9

. Right side:

-3x - 2x + 12 = -5x + 12

. The simplified inequality is

-4x + 9 > -5x + 12

To simplify the expression before isolating the variable.

Isolate the variable

x

by adding

5x

and subtracting

9

on both sides.

-4x + 5x > 12 - 9

, which simplifies to

x > 3

To determine the final solution set for the inequality.

Evaluate whether

x = 3

satisfies the solution set

x > 3

Since

3

is not greater than

3

, the value is not included in the solution set.

To definitively determine if the original statement is true or false.

Solving multi-step linear inequalities and determining boundary value inclusion

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