To determine which equations have infinitely many solutions, we should simplify each equation and check if they hold true for all values of \(x\).
Let's evaluate each equation one by one:
A. \(6 (x - 9) = 6x - 54 + x\)
Left side: \(6(x - 9) = 6x - 54\)
Right side: \(6x - 54 + x = 7x - 54\)
Setting both sides equal: \(6x - 54 = 7x - 54\)
Subtracting \(6x\) from both sides: \(-54 = x - 54\)
So, \(0 = x\)
This equation has a unique solution (\(x = 0\)), not infinitely many solutions.
B. \(5 (x - 8) + 10 = 5 (x + 2)\)
Left side: \(5(x - 8) + 10 = 5x - 40 + 10 = 5x - 30\)
Right side: \(5(x + 2) = 5x + 10\)
Setting both sides equal: \(5x - 30 = 5x + 10\)
Subtracting \(5x\) from both sides: \(-30 = 10\)
This is a contradiction, which means there are no solutions.
C. \(20 - 8x - 6 = 2 (7 - 4x)\)
Left side: \(20 - 8x - 6 = 14 - 8x\)
Right side: \(2(7 - 4x) = 14 - 8x\)
Setting both sides equal: \(14 - 8x = 14 - 8x\)
This equation is always true for all \(x\). Thus, it has infinitely many solutions.
D. \(4 (x + 4) = 2(2x + 5) + 6\)
Left side: \(4(x + 4) = 4x + 16\)
Right side: \(2(2x + 5) + 6 = 4x + 10 + 6 = 4x + 16\)
Setting both sides equal: \(4x + 16 = 4x + 16\)
This equation is also always true for all \(x\). Thus, it has infinitely many solutions.
In conclusion, the equations that have infinitely many solutions are:
C. and D.