A student solved the following equation using the following steps:

Question

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
correctly or incorrectly
The equation solved correctly would show that it has
solution(s).
no or one or two or three

Answer 1

The student's work shows that the equation was solved correctly.

Let's analyze the given equation and the steps taken:

1. The equation starts as: \[ 2(2x + 2) = 4(x + 4) \]

2. Distributing both sides:

   • Left side: \(2(2x) + 2(2) = 4x + 4\)
   • Right side: \(4(x) + 4(4) = 4x + 16\)

   So the equation becomes: \[ 4x + 4 = 4x + 16 \]

3. Simplifying it:

   • Subtract \(4x\) from both sides, which gives: \[ 4 = 16 \]

   However, this statement is false.

Essentially, an equation that simplifies to a false statement indicates that there are no solutions to the equation, not infinitely many.

In conclusion, the final result \(4 = 4\) as derived by the student from the distribution step is misleading in this context because it suggests that there are infinitely many solutions. To clarify, the equation does not have \(4 = 4\) from the steps provided, but instead, leads to a false equality from which we conclude that there are no solutions.

So, the correct answer is: The equation was solved incorrectly, indicating it has no solutions.