Solve the equation.

StartFraction 5 left parenthesis x minus 1 right parenthesis Over 6 EndFraction minus x equals 1 minus StartFraction x Over 8 EndFraction

1 answer

To solve the equation

\[ \frac{5(x - 1)}{6} - x = 1 - \frac{x}{8}, \]

we start by clearing the fractions. The least common multiple of the denominators 6 and 8 is 24. We can multiply the entire equation by 24 to eliminate the fractions:

\[ 24 \left( \frac{5(x - 1)}{6} \right) - 24x = 24 \left( 1 - \frac{x}{8} \right). \]

Simplifying each term gives:

\[ 4 \cdot 5(x - 1) - 24x = 24 - 3x. \]

This simplifies to:

\[ 20(x - 1) - 24x = 24 - 3x. \]

Expanding the left side:

\[ 20x - 20 - 24x = 24 - 3x. \]

Combining like terms on the left gives:

\[ -4x - 20 = 24 - 3x. \]

Next, we'll get all terms involving \( x \) on one side and constants on the other. Adding \( 4x \) to both sides:

\[ -20 = 24 + x. \]

Now, subtracting 24 from both sides gives:

\[ -20 - 24 = x, \]

which simplifies to:

\[ -44 = x. \]

Thus, the solution to the equation is

\[ \boxed{-44}. \]