A student solved the following equation using the following steps:

Question

4(2−3x)=x−2(2x+1)
8−3x=x−4x−2
8−3x=−3x−2
No solution.

(2 points)
Based on the student's work, the equation was solved

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

Question 4
4. Select the equation that has infinitely many solutions.(1 point)
Responses

3x+5=3x−53 x plus 5 is equal to 3 x minus 5

3−3x=−3(x−1)3 minus 3 x is equal to negative 3 times open paren x minus 1 close paren

3x=2x+13 x is equal to 2 x plus 1

4x+1=4(x+1)4 x plus 1 is equal to 4 times open paren x plus 1 close paren
Question 5
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)

Answer 1

Let's analyze the questions based on the provided information.

Question 4: Select the equation that has infinitely many solutions.

An equation has infinitely many solutions when, after simplification, it results in a tautology, such that both sides are equivalent. Let's examine the options:

1. 3x + 5 = 3x - 5
   Subtract \(3x\) from both sides:
   \(5 = -5\) which is false. (No solutions)

2. 3 - 3x = -3(x - 1)
   Distribute on the right:
   \(3 - 3x = -3x + 3\)
   Add \(3x\) to both sides:
   \(3 = 3\) which is true for all values of \(x\). (Infinitely many solutions)

3. 3x = 2x + 1
   Subtract \(2x\) from both sides:
   \(x = 1\) which is a single solution.

4. 4x + 1 = 4(x + 1)
   Distribute on the right:
   \(4x + 1 = 4x + 4\)
   Subtract \(4x\) from both sides:
   \(1 = 4\) which is false. (No solutions)

The only equation that has infinitely many solutions is:
3 - 3x = -3(x - 1).

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Question 5: The equation solved correctly would show that it has solution(s).

The student correctly solved the equation:
2(2x + 2) = 4(x + 4)

1. Start with the original equation:
   \(4x + 4 = 4x + 16\)

2. Subtract \(4x\) from both sides:
   \(4 = 16\)

This is clearly false, thus the solution indicates there is no solution.

However, in the student’s explanation ("Infinitely many solutions"), they suggest a complete tautology in their work. When \(4 = 4\) is concluded from the proper algebraic manipulation (as seen in the student’s solution), that would indeed indicate infinitely many solutions for a situation where two expressions are equal regardless of the variable.

Please clarify if additional context or specific checks against "solution(s)" is needed regarding the solution in question.

To confirm:

• Question 4 Answer: 3 - 3x = -3(x - 1)
• Question 5 Conclusion: The way it's phrased seems to indicate no solution since \(4 \neq 16\), but the solution indicates a tautology with \(4 = 4\) from a re-written part of the same type of equality would indeed suggest infinitely many solutions if structured that way.

If you have any more questions or need further explanations, feel free to ask!