A student solved the following equation using the following steps:

Question

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
correctly

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

Answer 1

Let's analyze the student's steps to see if there was an error in the solution.

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

2. Distribute 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 rewrite the equation: \[ 8 - 12x = -3x - 2 \]

5. Next, let's add \(12x\) to both sides: \[ 8 = -3x + 12x - 2 \] Which simplifies to: \[ 8 = 9x - 2 \]

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

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

The final solution \(x = \frac{10}{9}\) indicates that the equation does indeed have one solution.

So, while the student concluded "No solution," this is incorrect. The equation can be solved correctly and has one solution.

Answer 2

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

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

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

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

Answer 3

Based on the original equation \( 4(2 - 3x) = x - 2(2x + 1) \), we found that it can be solved correctly to yield one solution (\( x = \frac{10}{9} \)).

So, the correct statement is:

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

The other options (infinitely many solutions, two solutions, three solutions) are not applicable to this equation. The analysis indicates that the equation is linear, and thus, it can only have one solution unless it simplifies to a contradictory statement or an identity. In this case, it produced a single valid solution.