To determine which equations have infinitely many solutions, we need to simplify each equation and check if they are equivalent (which means they have infinitely many solutions).
A. \( 3x + x = 4x \)
- Simplifying: \( 4x = 4x \) → True for all \( x \) (infinitely many solutions)
B. \( 3(x + 1) = 3x + 3 \)
- Simplifying: \( 3x + 3 = 3x + 3 \) → True for all \( x \) (infinitely many solutions)
C. \( 2(x + 1) = 2x + 1 \)
- Simplifying: \( 2x + 2 = 2x + 1 \)
- This reduces to: \( 2 = 1 \), which is a contradiction (no solutions)
D. \( 3(x + 1) = 3x + 2 \)
- Simplifying: \( 3x + 3 = 3x + 2 \)
- This reduces to: \( 3 = 2 \), which is a contradiction (no solutions)
E. \( 2(x + 2) = x + 4 \)
- Simplifying: \( 2x + 4 = x + 4 \)
- This reduces to: \( 2x = x \) → \( x = 0 \) (one solution, not infinitely many)
The equations with infinitely many solutions are:
- A. \( 3x + x = 4x \)
- B. \( 3(x + 1) = 3x + 3 \)
So, the correct answers are A and B.