1/3(-3x + 6)- 1/8(8-32x) Use the order of operations to simplify.

To simplify the expression \( \frac{1}{3}(-3x + 6) - \frac{1}{8}(8 - 32x) \) using the order of operations, follow the steps below:

### Given Expression:
\[
\frac{1}{3}(-3x + 6) - \frac{1}{8}(8 - 32x)
\]

### Step 1: Parentheses
First, evaluate the expressions inside the parentheses:

- The expression inside the first set of parentheses is \(-3x + 6\).
- The expression inside the second set of parentheses is \(8 - 32x\).

### Step 2: Exponents (No exponents in this expression)

### Step 3: Multiplication and Division
Next, distribute the constants \(\frac{1}{3}\) and \(\frac{1}{8}\) to the terms inside the parentheses:

- Distribute \(\frac{1}{3}\) to the expression \(-3x + 6\):
  \[
  \frac{1}{3}(-3x + 6) = \frac{1}{3}(-3x) + \frac{1}{3}(6) = -x + 2
  \]

- Distribute \(\frac{1}{8}\) to the expression \(8 - 32x\):
  \[
  \frac{1}{8}(8 - 32x) = \frac{1}{8}(8) - \frac{1}{8}(32x) = 1 - 4x
  \]

### Step 4: Subtraction and Addition
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 5: Combine Like Terms
Finally, 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:
\[
3x + 1
\]

This solution follows the order of operations: parentheses, multiplication/distribution, and then addition/subtraction.

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

Here are 5 related questions:
1. How do you use the order of operations (PEMDAS) in algebraic expressions?
2. What is the difference between distribution and combining like terms?
3. How do you handle expressions with multiple variables and coefficients?
4. What are common mistakes when simplifying algebraic expressions?
5. How can you verify your result after simplifying an expression?

**Tip:** When simplifying expressions, always perform operations inside parentheses first and then follow the order of operations systematically.

Learn how to simplify the algebraic expression 1/3(-3x + 6) - 1/8(8 - 32x) step-by-step with detailed explanations. This problem covers key algebraic concepts such as the order of operations, distribution, and combining like terms. Suitable for students in Grades 7-9.

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