Question

Question
A student solved the following equation using the following steps:

4(2−3x)=x−2(2x+1)

8−3x=x−4x−2

8−3x=−3x−2

Answer: No Solution

(2 points)
Based on the student’s work, the equation was solved

The equation solved correctly would show that it has

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Answer 1

Let's analyze the student's steps to determine if they accurately solved the equation and what the solution indicates.

The original equation they intended to solve is:

\[ 4(2 - 3x) = x - 2(2x + 1) \]

1. **Distributing both sides:**
- Left side: \( 4(2 - 3x) = 8 - 12x \)
- Right side: \( x - 2(2x + 1) = x - 4x - 2 = -3x - 2 \)

So we rewrite the equation:

\[ 8 - 12x = -3x - 2 \]

2. **Reorganizing it as the student did:**
- This was done correctly as follows:

\[ 8 - 12x = -3x - 2 \]

3. **Moving all terms involving \( x \) to one side:**
- After looking at the student's next step, it appears they did some simplification or reorganization but ended up repeating an incorrect format. Let's set it properly:

\[ 8 - 12x = -3x - 2 \]

Now, adding \( 12x \) to both sides and \( 2 \) to both sides gives:

\[ 8 + 2 = 12x - 3x \]
\[ 10 = 9x \]

Therefore, solving for \( x \):
\[ x = \frac{10}{9} \]

### Conclusion
The final calculation shows there is a solution to the equation, specifically \( x = \frac{10}{9} \).

If the student's final answer states "No Solution," that is incorrect based on this analysis. The original equation indeed has a solution, meaning the student made an error in their simplification steps or reasoning down the line.

The conclusion is that the equation has a solution, and the correct final form of the statement should indicate that the equation solved correctly shows it has one unique solution \( x = \frac{10}{9} \).