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
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Part 1
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​(Simplify your​ answer.)

Answer 1

To solve the equation

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

let's start by eliminating the fractions. We can do this by finding the least common multiple of the denominators. The denominators are 6 and 8, and the least common multiple (LCM) is 24. We will multiply each term of the equation by 24.

1. Multiply through by 24:

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

2. Simplifying each term, we get:

\[
4 \cdot 5(x - 2) - 24x = 24 - 3x.
\]

This simplifies to:

\[
20(x - 2) - 24x = 24 - 3x.
\]

3. Distributing the 20:

\[
20x - 40 - 24x = 24 - 3x.
\]

4. Combine like terms on the left side:

\[
(20x - 24x) - 40 = 24 - 3x,
\]
which simplifies to:
\[
-4x - 40 = 24 - 3x.
\]

5. To isolate x, first add 3x to both sides:

\[
-4x + 3x - 40 = 24,
\]

which simplifies to:

\[
-x - 40 = 24.
\]

6. Now add 40 to both sides:

\[
-x = 64.
\]

7. Finally, multiply by -1:

\[
x = -64.
\]

Thus, the solution is

\[
\boxed{-64}.
\]