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!

Learn how to solve equations involving fractions step-by-step with a detailed guide. This problem involves simplifying fractions, cross-multiplication, and finding common denominators. Suitable for students in Grades 7-9.

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