If \frac{2}{3}(3x - 4) - \frac{1}{4}(x + 2) = \frac{22}{3}, what is the value of 3x + 2? — ACT

If $\frac{2}{3}(3x - 4) - \frac{1}{4}(x + 2) = \frac{22}{3}$ , what is the value of $3x + 2$ ?

Answer: 20

The correct answer is 20, which is the value of the expression

3x + 2

when

x = 6

The value of the expression

3x + 2

is 20. Solving the linear equation

\frac{2}{3}(3x - 4) - \frac{1}{4}(x + 2) = \frac{22}{3}

yields

x = 6

. Substituting

x = 6

into

3x + 2

gives

3(6) + 2 = 20

Multiply the entire equation by the least common multiple of the denominators (12).

8(3x - 4) - 3(x + 2) = 88

This clears the fractions to make solving the equation simpler.

Distribute the coefficients to remove parentheses.

24x - 32 - 3x - 6 = 88

This allows like terms to be grouped together.

Combine like terms on the left side of the equation.

21x - 38 = 88

Simplifies the equation to prepare for isolating the variable.

Add 38 to both sides of the equation.

21x = 126

Isolates the variable term on one side of the equation.

Divide both sides by 21 to solve for

x

x = 6

Finds the value of the variable

x

Substitute

x = 6

into the target expression

3x + 2

3(6) + 2 = 20

Calculates the final requested value.

Solving multi-step linear equations with fractions and evaluating algebraic expressions.

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