Answer: Let's solve the given equation step by step:
3(x - 6) + 12 = 6((x + 1) - 3x)
Distribute the constants inside the parentheses:
3x - 18 + 12 = 6(x + 1 - 3x)
Simplify each side of the equation:
3x - 6 = 6(x + 1 - 3x)
Distribute the 6 on the right side:
3x - 6 = 6x + 6 - 18x
Combine like terms:
3x - 6 = -12x + 6
Add 12x to both sides:
15x - 6 = 6
Add 6 to both sides:
15x = 12
Divide by 15:
x = 12 / 15
x = 4 / 5
So, the solution for the equation is x = 4/5.
Regarding the number of solutions, there is only one solution for this equation. The equation is linear and both sides are equal, so there is a single value of x that satisfies the equation. Therefore, it has exactly one solution.