Question

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
correctly or incorrectly

Answer 1

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

Answer 2

The student's work is correct.

Here’s a breakdown of the steps:

1. The original equation is \(2(2x + 2) = 4(x + 4)\).
2. Distributing on both sides gives:
\[
4x + 4 = 4x + 16.
\]
3. Next, you subtract \(4x\) from both sides, leading to:
\[
4 = 16.
\]

However, that conclusion is incorrect since \(4\) does not equal \(16\). Therefore, the conclusion reached about the original equation having infinitely many solutions is based on an error.

The correct outcome in this case should have indicated that the original equation does not yield a true statement after simplification, signaling no solution instead of infinitely many solutions. The student's final assertion is incorrect.

Thus, the correct evaluation would be: **Incorrect**.

Answer 3

For an equation to have solutions, it typically should provide a definitive statement regarding the variable involved. Let's summarize how to approach the solution and check it properly:

### Original Equation
\[
2(2x + 2) = 4(x + 4)
\]

### Step 1: Expand Both Sides
Distributing on both sides:
\[
4x + 4 = 4x + 16
\]

### Step 2: Move All Terms Involving \(x\) to One Side
Subtract \(4x\) from both sides:
\[
4 = 16
\]

### Interpretation
The statement \(4 = 16\) is **false**. When we reach a false statement after simplifying an equation involving a variable, it indicates that there are **no solutions** to the equation.

If the equation had simplified to a true statement like \(4 = 4\), that would indicate there are **infinitely many solutions**, implying that the equation is an identity.

### Conclusion
Thus, correctly solving the equation shows that it has **no solutions**, and the student's conclusion regarding infinitely many solutions was incorrect.