Question

Solve the equation.
StartFraction 3 left parenthesis x minus 5 right parenthesis Over 4 EndFraction minus x equals 1 minus StartFraction x Over 8 EndFraction

Answer 1

To solve the equation

\[
\frac{3(x - 5)}{4} - x = 1 - \frac{x}{8},
\]

we first eliminate the fractions by finding a common denominator. The least common multiple of the denominators (4 and 8) is 8. We can multiply the entire equation by 8 to eliminate the fractions:

\[
8 \left(\frac{3(x - 5)}{4}\right) - 8x = 8\left(1 - \frac{x}{8}\right).
\]

This simplifies to:

\[
2 \cdot 3(x - 5) - 8x = 8 - x.
\]

Now, distributing on the left side:

\[
6(x - 5) - 8x = 8 - x.
\]

Distributing the 6:

\[
6x - 30 - 8x = 8 - x.
\]

Combining like terms on the left side gives:

\[
-2x - 30 = 8 - x.
\]

Now, add \(x\) to both sides:

\[
-2x + x - 30 = 8,
\]

which simplifies to:

\[
-x - 30 = 8.
\]

Next, add 30 to both sides:

\[
-x = 38.
\]

Finally, multiply both sides by -1:

\[
x = -38.
\]

Thus, the solution to the equation is

\[
\boxed{-38}.
\]