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3. 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
No solution.

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

The equation solved correctly would show that it has
solution(s).

1 answer

Let's analyze the student's steps in solving the equation:

  1. The original equation written by the student is: \[ 4(2 - 3x) = x - 2(2x + 1) \]

  2. The student simplified the left side: \[ 4(2 - 3x) = 8 - 12x \] And the right side: \[ x - 2(2x + 1) = x - 4x - 2 = -3x - 2 \]

    Therefore, the equation becomes: \[ 8 - 12x = -3x - 2 \]

  3. The student combined terms and rearranged: \[ 8 - 12x = -3x - 2 \]

    This step appears incorrect in reaching the conclusion of "No solution." The next step should have been combining like terms, not reaching an immediate conclusion.

If we continue solving from \( 8 - 12x = -3x - 2 \):

  1. Add \( 12x \) to both sides: \[ 8 = 9x - 2 \]

  2. Add 2 to both sides: \[ 10 = 9x \]

  3. Finally, divide both sides by 9: \[ x = \frac{10}{9} \]

Thus, the correct solution to the equation shows that it has one solution, namely \( x = \frac{10}{9} \).

In summary, based on the student's work, the equation solved correctly would show that it has one solution.