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.