Question

Solve for ex determine if the equation has one none or infinite solutions
9(x+1)=3(5x+1)-12
The equation has one solution x = 0
The equation has infinite solutions
The equation has no solutions
The equation has one solution x = 3

Answer 1

To solve the equation \( 9(x+1)=3(5x+1)-12 \), let's start by simplifying both sides.

1. Distribute on both sides:
\[
9(x + 1) = 9x + 9
\]
\[
3(5x + 1) - 12 = 15x + 3 - 12 = 15x - 9
\]

2. Now the equation looks like this:
\[
9x + 9 = 15x - 9
\]

3. Next, let's move all terms involving \( x \) to one side and constant terms to the other:
\[
9 + 9 = 15x - 9x
\]
\[
18 = 6x
\]

4. Now, solve for \( x \):
\[
x = \frac{18}{6} = 3
\]

Thus, the equation has one solution: \( x = 3 \).

**Checking for other types of solutions:**
- One solution occurs when the equation simplifies to a single line with a specific point of intersection.
- Infinite solutions occur when both sides of the equation simplify to the same expression (e.g., \( 0 = 0 \)).
- No solutions occur when we end up with a false statement (e.g., \( 0 = 5 \)).

In this case, since we found \( x = 3 \) as the only solution, the correct conclusion is:

**The equation has one solution: \( x = 3 \)**.