Let's solve the equation step by step to determine the solution for \( x \) and to see if the equation has one, none, or infinite solutions.
The given equation is:
\[ 9 (x + 1) = 3(5x + 1) - 12 \]
First, we'll expand both sides of the equation.
Left side:
\[ 9(x + 1) = 9x + 9 \]
Right side: \[ 3(5x + 1) - 12 = 15x + 3 - 12 = 15x - 9 \]
Now the equation can be rewritten as:
\[ 9x + 9 = 15x - 9 \]
Next, we'll move all terms involving \( x \) to one side and constant terms to the other side. We'll subtract \( 9x \) from both sides:
\[ 9 = 15x - 9x - 9 \]
This simplifies to:
\[ 9 = 6x - 9 \]
Next, add \( 9 \) to both sides of the equation:
\[ 9 + 9 = 6x \] \[ 18 = 6x \]
Now, divide both sides by \( 6 \):
\[ x = \frac{18}{6} = 3 \]
So, we have found one solution:
\[ x = 3 \]
Now, let's summarize:
The equation has one solution, which is \( x = 3 \).
Thus, the answer is:
- The equation has one solution: \( x = 3 \).