We are given the equation:

$\frac{x + 9}{18} = \frac{7}{9} + \frac{x - 3}{3}$

Let's solve for $x$.

Step 1: Simplify each fraction.

1. On the left side, the expression is already simplified:

$\frac{x + 9}{18}$

2. On the right side, simplify the fractions:

$\frac{7}{9} + \frac{x - 3}{3}$

Notice that $\frac{x - 3}{3}$ can be rewritten as:

$\frac{x - 3}{3} = \frac{x}{3} - 1$

So, the right side becomes:

$\frac{7}{9} + \frac{x}{3} - 1$

Step 2: Find a common denominator on the right side.

The least common denominator (LCD) of 9 and 3 is 9. Rewrite $\frac{x}{3}$ with denominator 9:

$\frac{x}{3} = \frac{3x}{9}$

Now the right side is:

$\frac{7}{9} + \frac{3x}{9} - 1$

To combine these terms, first express $-1$ as a fraction with denominator 9:

$-1 = \frac{-9}{9}$

So the right side becomes:

$\frac{7}{9} + \frac{3x}{9} - \frac{9}{9} = \frac{3x - 2}{9}$

Step 3: Now rewrite the equation:

$\frac{x + 9}{18} = \frac{3x - 2}{9}$

Step 4: Cross-multiply to eliminate the fractions:

$9(x + 9) = 18(3x - 2)$

Step 5: Distribute on both sides:

$9x + 81 = 54x - 36$

Step 6: Move all terms involving $x$ to one side and constants to the other side:

$81 + 36 = 54x - 9x$

$117 = 45x$

Step 7: Solve for $x$ by dividing both sides by 45:

$x = \frac{117}{45}$

Step 8: Simplify the fraction:

$x = \frac{39}{15} = \frac{13}{5}$

Final Answer:

$x = \frac{13}{5} \text{ or } 2.6$

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you determine the least common denominator when adding fractions?
2. How can you simplify complex algebraic fractions?
3. What is the process for cross-multiplying to solve equations?
4. Why is it important to combine like terms when solving equations?
5. How do you check your solution once you've solved for $x$?

Tip: Always simplify fractions early in the problem to make the calculations easier later on!