Solve for x to determine if the equation has one, none, or infinite solutions. 9(x+1)=3(5x+1)−12 (1 point) Responses The equation has one solution: x=3 . The equation has one solution: x is equal to 3. - no response given The equation has one solution: x=0 .

1 answer

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: \( x = 3 \).