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

Answer 1

Let’s analyze the steps taken by the student to see if they followed a correct process to solve the equation \(2(2x + 2) = 4(x + 4)\):

1. Start with the original equation:
\[
2(2x + 2) = 4(x + 4)
\]

2. Distributing the terms on both sides:
- Left-hand side:
\[
2(2x) + 2(2) = 4x + 4
\]
- Right-hand side:
\[
4(x) + 4(4) = 4x + 16
\]

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

3. Subtracting \(4x\) from both sides:
\[
4 = 16
\]

This is a contradiction since \(4\) does not equal \(16\). The student mistakenly concluded that the left side and the right side were equal, leading them to say that \(4 = 4\).

In fact, the original equation has no solution because after simplification we reached a false statement. Therefore, the correct conclusion is that the equation does **not** have any solutions (it's not true that there are infinitely many solutions).

So, based on the student's work, the conclusion should not be that it has infinitely many solutions, but rather that the equation has **no solutions** (it is inconsistent).