To solve the equation \( 9(x+1) = 3(5x+1) - 12 \), let's first simplify both sides.
Starting with the left side:
\[ 9(x + 1) = 9x + 9 \]
Now, for the right side:
\[ 3(5x + 1) - 12 = 15x + 3 - 12 = 15x - 9 \]
Now we rewrite the equation:
\[ 9x + 9 = 15x - 9 \]
Next, we can move all the terms involving \( x \) to one side and constant terms to the other side. Let's subtract \( 9x \) from both sides:
\[ 9 = 15x - 9x - 9 \] \[ 9 = 6x - 9 \]
Now, add 9 to both sides:
\[ 9 + 9 = 6x \] \[ 18 = 6x \]
Divide both sides by 6 to solve for \( x \):
\[ x = 3 \]
Now, let's determine if there are any other solutions. The solution \( x = 3 \) is a unique solution as there are no variables left in our final equation after simplification.
Thus, the conclusion is that:
- The equation has one solution: \( x = 3 \).