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

Question

Responses

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

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

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

3−3x=−3(x−1)
3 minus 3 x is equal to negative 3 times open paren x minus 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)
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 given equations to identify which one has infinitely many solutions.

1. 4x + 1 = 4(x + 1)
   Simplifying this:
   \( 4x + 1 = 4x + 4 \)
   Subtracting \( 4x \) from both sides:
   \( 1 = 4 \)
   This is a contradiction; hence, no solution.

2. 3x + 5 = 3x − 5
   Simplifying this:
   \( 3x + 5 - 3x = -5 \)
   \( 5 = -5 \)
   This is also a contradiction; hence, no solution.

3. 3x = 2x + 1
   Simplifying this:
   \( 3x - 2x = 1 \)
   \( x = 1 \)
   This gives us a single solution.

4. 3 − 3x = −3(x − 1)
   Simplifying this:
   \( 3 - 3x = -3x + 3 \)
   Rearranging gives us:
   \( 3 = 3 \)
   This is always true for any value of \( x \); hence, it has infinitely many solutions.

Based on the analysis above, the equation that has infinitely many solutions is:

3 − 3x = −3(x − 1).

Now, regarding Question 5 about the student's work:

The correct interpretation is that the equation \( 2(2x + 2) = 4(x + 4) \) was simplified to \( 4x + 4 = 4x + 4 \), which leads to \( 4 = 4 \), indicating that it holds true for any \( x \).

Therefore, the equation solved correctly would show that it has infinitely many solution(s).