Let's analyze the steps the student took to solve the equation and see if they were done correctly.
The original equation is:
\[ 4(2 - 3x) = x - 2(2x + 1) \]
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Distributing on the left side: \[ 4(2) - 4(3x) = 8 - 12x \] So, the left side becomes \( 8 - 12x \).
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Distributing on the right side: \[ -2(2x) - 2(1) = -4x - 2 \] The right side becomes \( x - 4x - 2 \), which simplifies to \( -3x - 2 \).
After combining both sides, we have:
\[ 8 - 12x = -3x - 2 \]
Now let's check if the student's next steps were correct:
The student wrote:
- \( 8 - 3x = x - 4x - 2 \)
This was correct up until this point because \( x - 4x - 2 \) does simplify to \( -3x - 2 \).
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They then wrote:
\[ 8 - 3x = -3x - 2 \]
This is where we need to notice that \( 8 - 12x \) only equals \( -3x - 2 \) when we take into account all terms properly. But the student seems to claim they've simplified it to \( 8 - 3x = -3x - 2 \) incorrectly, as it appears they may have skipped a step or made an algebraic mistake in moving terms.
To actually solve \( 8 - 12x = -3x - 2 \):
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Add \( 12x \) to both sides: \[ 8 = 12x - 3x - 2 \]
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Thus: \[ 8 = 9x - 2 \]
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Add \( 2 \) to both sides: \[ 10 = 9x \]
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Finally, divide by 9: \[ x = \frac{10}{9} \]
This indicates there is exactly one solution to the original equation.
Therefore, based on the work demonstrated by the student, while the final deduction about the equation isn't reached correctly from the earlier incorrect equality they stated, ultimately the resolved answer does indicate one solution. Thus, the equation was solved incorrectly (as their steps did not lead logically), but it reveals one unique solution upon solving correctly.
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