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

1 answer

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:

  1. Start with the original equation: \[ 4(2 - 3x) = x - 2(2x + 1) \]

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

  3. Next, the student appears to have made an arithmetic mistake in the document: The correct manipulation from here should look like: \[ 8 - 12x = -3x - 2 \]

  4. To isolate \(x\), add \(12x\) to both sides: \[ 8 = 9x - 2 \]

  5. Then, add \(2\) to both sides: \[ 10 = 9x \]

  6. 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.