If \frac{2}{3}(x - 6) - \frac{1}{2}(x - 4) = 4, what is the value of 3x - 12? — SAT

If $\frac{2}{3}(x - 6) - \frac{1}{2}(x - 4) = 4$ , what is the value of $3x - 12$ ?

The correct answer is 96. Multiplying the entire equation by the common denominator 6 clears the fractions to yield

4(x - 6) - 3(x - 4) = 24

. Distributing the factors gives

4x - 24 - 3x + 12 = 24

, which simplifies to

x - 12 = 24

. Adding 12 to both sides determines that

x = 36

. Evaluating the expression

3x - 12

with

x = 36

yields

3(36) - 12 = 96

Multiply the entire equation by 6 (the least common multiple of the denominators 3 and 2) to eliminate the fractions.

4(x - 6) - 3(x - 4) = 24

Clearing the denominators simplifies the algebraic manipulation and reduces the risk of fractional arithmetic errors.

Distribute the coefficients to the terms inside the parentheses, taking care to distribute the negative sign for the second term.

4x - 24 - 3x + 12 = 24

Applying the distributive property expands the expression so that like terms can be grouped.

Combine like terms on the left side of the equation.

x - 12 = 24

Grouping

4x - 3x

to get

x

and

-24 + 12

to get

-12

simplifies the equation to a single variable and constant.

Isolate the variable

x

by adding 12 to both sides of the equation.

x = 36

Adding the opposite of the constant term isolates the variable on one side.

Substitute

x = 36

into the requested expression

3x - 12

to find its value.

3(36) - 12 = 96

The question asks for the value of the expression

3x - 12

, not just the value of

x

Solving multi-step linear equations in one variable and evaluating algebraic expressions.

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