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 one solution(s).

Answer 1

Let's analyze and solve the equation step by step to clarify the student's work.

The original equation given is:

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

Now let's simplify both sides.

1. Distribute on both sides: \[ 2(2x) + 2(2) = 4(x) + 4(4) \] Which gives: \[ 4x + 4 = 4x + 16 \]

2. Next, we isolate the terms involving \(x\) by subtracting \(4x\) from both sides: \[ 4 = 16 \]

At this point, we notice that we have reached a contradiction. The simplified form \(4 = 16\) is false, meaning there are no values of \(x\) that can satisfy the original equation. Therefore, the original equation has no solutions, not infinitely many.

The conclusion would be:

• The student made an error in determining the solutions. The properly analyzed equation shows that there are no solutions, indicating that the equation is inconsistent.

In conclusion, based on the correct simplification:

• The equation \(2(2x + 2) = 4(x + 4)\) does not have infinitely many solutions; it has no solutions.