Question

Solve the equation.
StartFraction 5 left parenthesis x minus 2 right parenthesis Over 6 EndFraction minus x equals 1 minus StartFraction x Over 8 EndFraction

Answer 1

To solve the equation

\[
\frac{5(x - 2)}{6} - x = 1 - \frac{x}{8},
\]

we will begin by eliminating the fractions. To do this, we can find a common multiple of the denominators 6 and 8, which is 24. We will multiply every term in the equation by 24:

\[
24 \left( \frac{5(x - 2)}{6} \right) - 24x = 24 \left( 1 - \frac{x}{8} \right).
\]

This simplifies each term as follows:

1. For the left side:
\[
24 \cdot \frac{5(x - 2)}{6} = 4 \cdot 5(x - 2) = 20(x - 2) = 20x - 40.
\]
2. The term \(-24x\) remains as is.

Combining these gives:
\[
20x - 40 - 24x.
\]

So, the left side becomes:
\[
20x - 24x - 40 = -4x - 40.
\]

3. For the right side:
\[
24 \cdot 1 - 24 \cdot \frac{x}{8} = 24 - 3x.
\]

Putting it all together, the equation becomes:
\[
-4x - 40 = 24 - 3x.
\]

Next, we will rearrange this equation to solve for \(x\). First, add \(4x\) to both sides:
\[
-40 = 24 + x.
\]

Now, subtract 24 from both sides:
\[
-40 - 24 = x \Rightarrow -64 = x.
\]

Thus, the solution to the equation is:
\[
\boxed{-64}.
\]