To solve the expression \( \frac{1}{3}(-3x + 6) - \frac{1}{8}(8 - 32x) \), follow these steps:

### Step 1: Distribute the constants
Distribute \(\frac{1}{3}\) and \(\frac{1}{8}\) across the terms inside the parentheses.

- Distributing \(\frac{1}{3}\) across \((-3x + 6)\):
  \[
  \frac{1}{3}(-3x + 6) = \frac{1}{3}(-3x) + \frac{1}{3}(6) = -x + 2
  \]

- Distributing \(\frac{1}{8}\) across \((8 - 32x)\):
  \[
  \frac{1}{8}(8 - 32x) = \frac{1}{8}(8) - \frac{1}{8}(32x) = 1 - 4x
  \]

### Step 2: Combine like terms
Now substitute these distributed terms back into the original expression:
\[
-x + 2 - (1 - 4x)
\]

Distribute the negative sign in front of \((1 - 4x)\):
\[
-x + 2 - 1 + 4x
\]

### Step 3: Simplify the expression
Combine the like terms:
- Combine the \(x\)-terms: \(-x + 4x = 3x\)
- Combine the constants: \(2 - 1 = 1\)

Thus, the simplified expression is:
\[
3x + 1
\]

### Final Answer:
\[
3x + 1
\]

Would you like more details or have any questions about this solution?

Here are 5 related questions:
1. How do you distribute fractions over expressions with variables?
2. How do you combine like terms in algebraic expressions?
3. What happens when you distribute a negative sign over an expression?
4. How do you handle complex expressions involving multiple fractions?
5. Can you simplify expressions with more than two terms?

**Tip:** Always carefully distribute signs when simplifying algebraic expressions to avoid mistakes.

Learn how to simplify the algebraic expression \( \frac{1}{3}(-3x + 6) - \frac{1}{8}(8 - 32x) \) step-by-step. This problem involves distributing fractions over expressions with variables and combining like terms. Suitable for high school students.

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