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a. Explain the error completely.

b. Rework the problem correctly.

a. **Error Analysis**:

The error occurs in **Step 2**. Let's break down why:

- In **Step 1**, the equation is correctly written as \( 2(x + 3) = 5(x - 1) + 6 \).
- When expanding both sides in **Step 2**, it is expected to distribute the constants:
  - The left-hand side should become \( 2x + 6 \), which is correct.
  - However, the right-hand side should have been:
    \[
    5(x - 1) + 6 = 5x - 5 + 6 = 5x + 1
    \]
  But the equation in **Step 2** incorrectly shows \( 5x - 5 + 6 \) as \( 5x - 5 + 6 \), without further simplification to \( 5x + 1 \).

Thus, the correct simplification was missed. This led to an error in Step 3 and beyond, impacting the final solution.

---

b. **Reworking the problem correctly**:

Let's re-solve the equation step by step.

1. Start with the original equation:
   \[
   2(x + 3) = 5(x - 1) + 6
   \]

2. Distribute the constants on both sides:
   \[
   2x + 6 = 5x - 5 + 6
   \]

3. Simplify the right-hand side:
   \[
   2x + 6 = 5x + 1
   \]

4. Move all terms involving \(x\) to one side and constants to the other:
   \[
   2x - 5x = 1 - 6
   \]
   Simplify:
   \[
   -3x = -5
   \]

5. Divide both sides by \(-3\):
   \[
   x = \frac{-5}{-3}
   \]
   Simplify:
   \[
   x = \frac{5}{3}
   \]

So, the correct solution is \( x = \frac{5}{3} \).

---

Do you have any questions or need further details?

Here are 5 related questions to help deepen your understanding:

1. How can you identify distribution errors when simplifying equations?
2. Why is it important to simplify constants before moving terms?
3. What are the potential consequences of making small simplification errors in multi-step problems?
4. How can we check if the solution we obtained is correct?
5. Can you solve a similar equation \( 3(x + 2) = 4(x - 3) + 7 \)?

**Tip**: Always simplify each side of an equation fully before moving terms to avoid common errors like the one in this example.

Explain the error completely and rework the problem correctly.

Learn how to identify and correct errors in solving linear equations, focusing on distributive property and simplification. This problem demonstrates key algebraic concepts for students in Grades 7-9.

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