https://file.aiquickdraw.com/mathbot/file/17236015459502rmh6hnu.jpg

The problem in the image asks to identify the error in Alice's simplification of the given algebraic expression. Here's a breakdown of the steps:

### Expression Given:
\[
\frac{1}{2} \cdot 2x + 4x \cdot \frac{1}{2} + 8x
\]

### Simplified Expression by Alice:
\[
9x + 1
\]

### Let's Simplify the Expression Step by Step:

1. **Distribute**: Multiply each term as indicated:
   \[
   \frac{1}{2} \cdot 2x = x
   \]
   \[
   4x \cdot \frac{1}{2} = 2x
   \]
   \[
   8x \text{ remains } 8x
   \]

2. **Add the Terms**:
   \[
   x + 2x + 8x = 11x
   \]

### Comparison to Alice's Simplified Expression:
- Alice's simplified expression is \(9x + 1\), but the correct simplified expression should be \(11x\).
- The options provided to identify the error:
  1. Alice did not add the first and third terms correctly.
  2. Alice combined some terms that are not like terms.
  3. Alice forgot to add the 1x to the other like terms.

Based on the steps above, **Alice did not add the terms correctly** and possibly made an error by introducing a constant \(+1\) that shouldn't be there. Therefore, the correct option is likely **Option 1** or **Option 3**.

### Conclusion:
The error was in the addition of terms; the correct expression should be \(11x\).

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

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Here are 8 related questions to further your understanding:

1. What are "like terms" in algebraic expressions?
2. How can you identify errors in algebraic simplifications?
3. What does it mean to "distribute" a term in algebra?
4. Can you always combine like terms in any algebraic expression?
5. What happens if you incorrectly add or subtract terms in an algebraic expression?
6. How would you simplify the expression if there were parentheses involved?
7. What are the common mistakes made when simplifying algebraic expressions?
8. How would you verify your simplification of an algebraic expression?

**Tip:** Always double-check each step when simplifying expressions to ensure that terms are combined correctly and that no additional terms are incorrectly introduced.

Understand how to identify errors in simplifying algebraic expressions with a detailed explanation of Alice's mistake. Explore step-by-step breakdowns and common errors in algebraic simplification. Suitable for students in Grades 7-9.

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