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

Answer 1

Let's analyze the steps the student took to solve the equation \(4(2 - 3x) = x - 2(2x + 1)\).

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

2. Distributing on both sides: \[ 4 \cdot 2 - 4 \cdot 3x = x - 2 \cdot 2x - 2 \cdot 1 \] This simplifies to: \[ 8 - 12x = x - 4x - 2 \]

3. Combine like terms on the right side: \[ 8 - 12x = -3x - 2 \]

4. Now let's isolate the variable \(x\). We can move all the \(x\) terms to one side by adding \(12x\) to both sides: \[ 8 = 12x - 3x - 2 \] Which simplifies to: \[ 8 = 9x - 2 \]

5. Now, add 2 to both sides: \[ 8 + 2 = 9x \] This simplifies to: \[ 10 = 9x \]

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

Thus, the solution to the equation is: \[ x = \frac{10}{9} \]

Therefore, the student's conclusion that the equation has "no solution" is incorrect. The correct solution shows that the equation has a solution, specifically \(x = \frac{10}{9}\).