To solve the equation:

\[
\frac{2(1 - x)}{3} - 8 = \frac{1}{6}x - 2x - \frac{3}{3}
\]

let’s go through it step-by-step.

### Step 1: Simplify Each Term

1. Expand and simplify terms where possible.
   - The left side has \(\frac{2(1 - x)}{3} = \frac{2 - 2x}{3}\).
   - The right side has \(\frac{3}{3} = 1\).

The equation now becomes:

\[
\frac{2 - 2x}{3} - 8 = \frac{1}{6}x - 2x - 1
\]

### Step 2: Eliminate Fractions by Finding a Common Denominator

The denominators here are \(3\) and \(6\). The least common multiple of \(3\) and \(6\) is \(6\). Multiply both sides of the equation by \(6\) to eliminate the fractions:

\[
6 \cdot \left(\frac{2 - 2x}{3} - 8\right) = 6 \cdot \left(\frac{1}{6}x - 2x - 1\right)
\]

Expanding both sides:

\[
2(2 - 2x) - 48 = x - 12x - 6
\]

Simplifying this gives:

\[
4 - 4x - 48 = -11x - 6
\]

### Step 3: Combine Like Terms

Combine constants and \(x\)-terms:

\[
-44 - 4x = -11x - 6
\]

### Step 4: Solve for \(x\)

1. Add \(11x\) to both sides to get all \(x\)-terms on one side:

   \[
-44 + 7x = -6
\]

2. Add \(44\) to both sides to isolate the \(x\)-term:

   \[
7x = 38
\]

3. Divide by \(7\):

   \[
x = \frac{38}{7}
\]

### Final Answer

\[
x = \frac{38}{7}
\]

Would you like additional details or have any questions?

---

1. How do we handle equations with fractions in general?
2. How can we check the solution by plugging \(x\) back into the equation?
3. What strategies work best for isolating variables?
4. How do we determine the least common multiple for fractions?
5. Can similar methods be used for inequalities?

**Tip:** To avoid mistakes, simplify and clear fractions early on in an equation.

Learn how to solve the equation with fractions: 2(1-x)/3 - 8 = 1/6x - 2x - 3/3. This step-by-step solution covers key algebraic methods, including clearing fractions, combining like terms, and isolating variables. Ideal for Grades 9-10 students.

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