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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 work step by step.

  1. Starting with the original equation: \[ 4(2-3x) = x - 2(2x + 1) \] Simplifying the right side: \[ 4(2 - 3x) = x - 4x - 2 \] This gives: \[ 4(2 - 3x) = -3x - 2 \]

  2. Now, simplifying the left side: \[ 8 - 12x = -3x - 2 \]

  3. Reorganizing the equation gives: \[ 8 - 12x + 12x = -3x - 2 + 12x \] Simplifying that gives: \[ 8 = 9x - 2 \]

  4. Now, adding 2 to both sides: \[ 8 + 2 = 9x \] This gives: \[ 10 = 9x \]

  5. Dividing both sides by 9: \[ x = \frac{10}{9} \]

This means the original equation indeed has a solution, specifically \( x = \frac{10}{9} \).

Therefore, the student's conclusion of "No solution" is incorrect. The equation solved correctly would show that it has one solution (a unique solution), namely \( x = \frac{10}{9} \).