Question

Solve the equation.
StartFraction 3 left parenthesis x minus 3 right parenthesis Over 4 EndFraction minus x equals 1 minus StartFraction x Over 6 EndFraction

Answer 1

To solve the equation

\[
\frac{3(x - 3)}{4} - x = 1 - \frac{x}{6},
\]

we begin by eliminating the fractions. We can do this by finding a common denominator. In this case, the least common multiple of the denominators (4 and 6) is 12. We multiply each term in the equation by 12:

\[
12 \left( \frac{3(x - 3)}{4} \right) - 12x = 12 \left( 1 \right) - 12 \left( \frac{x}{6} \right).
\]

This simplifies to:

\[
9(x - 3) - 12x = 12 - 2x.
\]

Now, distribute on the left side:

\[
9x - 27 - 12x = 12 - 2x.
\]

Combine like terms on the left side:

\[
-3x - 27 = 12 - 2x.
\]

Next, we want to isolate the variable \(x\). First, we can move \(-2x\) to the left side by adding \(2x\) to both sides:

\[
-3x + 2x - 27 = 12.
\]

This simplifies to:

\[
-x - 27 = 12.
\]

Now, add 27 to both sides:

\[
-x = 39.
\]

To solve for \(x\), multiply both sides by \(-1\):

\[
x = -39.
\]

Thus, the solution to the equation is

\[
\boxed{-39}.
\]

We can verify our solution by substituting \(x = -39\) back into the original equation:

\[
\frac{3(-39 - 3)}{4} - (-39) = 1 - \frac{-39}{6}.
\]

Calculating the left side:

\[
\frac{3(-42)}{4} + 39 = \frac{-126}{4} + 39 = -31.5 + 39 = 7.5.
\]

Calculating the right side:

\[
1 + \frac{39}{6} = 1 + 6.5 = 7.5.
\]

Since both sides are equal, our solution \(x = -39\) is confirmed to be correct.