Determine the number of solutions of this equation. x - 2(x - 6) = -x + 12

A. One solution
B. No solution
C. Infinitely many solutions

C.) infinitely many solutions.

Given the equation, x - 2(x -6) = -x + 12:

x - 2(x -6) = -x + 12

Distribute -2 into the terms inside the parenthesis:

x - 2(x -6) = -x + 12

x - 2x + 12 = -x + 12

Combine like terms:

-x + 12 = - x + 12

Subtract 12 from both sides:

-x + 12 - 12 = - x + 12 - 12

- x = - x

Add x to both sides:

- x + x = - x + x

0 = 0 (true statement).

This means that any value of x will satisfy the equation, thereby making it infinitely many solutions.