Question

5. A student solved the following equation using the following steps: 2(2x+2)=4(x+4) 4x+4=4x+4 4=4 Infinitely many solutions (2 points) Based on the student's work, the equation was solved incorrectly The equation solved correctly would show that it has no solution(s).

Answer 1

Let's analyze the steps taken by the student to see if the conclusions drawn are appropriate. The student's initial equation is:

\[ 2(2x + 2) = 4(x + 4) \]

We'll simplify both sides carefully:

1. Expand the left side:
\[ 2(2x + 2) = 4x + 4 \]

2. Expand the right side:
\[ 4(x + 4) = 4x + 16 \]

Now we set both sides equal:

\[ 4x + 4 = 4x + 16 \]

Next, let's subtract \(4x\) from both sides:

\[ 4 = 16 \]

The statement \(4 = 16\) is clearly false, indicating that the original equation does not hold true for any values of \(x\). Thus, the conclusion from this step is correct in stating that there are **no solutions** to the equation.

The student's assertion of "infinitely many solutions" is mistaken. The proper interpretation of the solved equation results in a contradiction (4 does not equal 16), confirming that the equation has no solutions instead.

To summarize:
- The equation has no solutions, not infinitely many.
- The steps leading to the conclusion (particularly subtracting \(4x\) and resulting in a false statement) are the source of the misunderstanding.

Therefore, the corrected statement should be:

**The equation has no solutions.**