Question

Solve for x to determine if the equation has one, none, or infinite solutions 9 (x+1)=3(5x+1)−12

The equation has one solution: x=3.

The equation has one solution: x=0.

Answer 1

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: \( x = 3 \).