To determine which equation has infinitely many solutions, we need to check if the equations are equivalent or simplify into true statements.
Let's check each option:
A. \( 8x + \frac{3}{2} = 2x + 4 \)
Rearranging gives: \[ 8x - 2x = 4 - \frac{3}{2} \] This simplifies to: \[ 6x = \frac{8}{2} - \frac{3}{2} = \frac{5}{2} \] Thus, \[ x = \frac{5}{12} \] This has a single solution.
B. \( 15x + \frac{5}{3} \)
This is not an equation since there is no equal sign, so we cannot determine if it has solutions.
C. \( 8x + \frac{5}{2} = 8x + 6 \)
If we subtract \( 8x \) from both sides, we get: \[ \frac{5}{2} = 6 \] This statement is false (since \( \frac{5}{2} = 2.5 \)), meaning this equation has no solutions.
D. \( \frac{9x + 6}{3} = 3x + 2 \)
Simplifying the left side: \[ 3x + 2 = 3x + 2 \] This simplifies to the identity \( 3x + 2 = 3x + 2 \), which is always true for all \( x \). Thus, this equation has infinitely many solutions.
Conclusion: The equation with infinitely many solutions is D.