To analyze the steps taken by the student to solve the equation \( 4(2-3x) = x - 2(2x + 1) \) and determine if their conclusion of "No Solution" is accurate, let's go through the steps one by one:
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Start with the original equation: \[ 4(2 - 3x) = x - 2(2x + 1) \]
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Distribute on both sides:
- Left side: \( 4 \times 2 - 4 \times 3x = 8 - 12x \)
- Right side: \( x - 2(2x + 1) = x - 4x - 2 = -3x - 2 \)
So, the equation simplifies to: \[ 8 - 12x = -3x - 2 \]
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Next, the student appears to have made an arithmetic mistake in the document: The correct manipulation from here should look like: \[ 8 - 12x = -3x - 2 \]
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To isolate \(x\), add \(12x\) to both sides: \[ 8 = 9x - 2 \]
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Then, add \(2\) to both sides: \[ 10 = 9x \]
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Finally, divide by \(9\): \[ x = \frac{10}{9} \]
Thus, the solution to the equation is \( x = \frac{10}{9} \).
Conclusion:
Since we found a specific value for \( x \), the student's conclusion of "No Solution" is incorrect. The equation has a unique solution. The correct answer would show that the equation has one solution.