To solve the equation:

$\frac{x + 4}{9} = 1 - \frac{x + 4}{5}$

we'll first clear the fractions by finding a common denominator and then solve for $x$.

Step-by-step solution:

1. Rewrite the equation:

   $\frac{x + 4}{9} = 1 - \frac{x + 4}{5}$

2. Combine like terms by getting a common denominator:

   Multiply both sides by $45$ (the least common multiple of 9 and 5):

   $45 \cdot \frac{x + 4}{9} = 45 \cdot \left(1 - \frac{x + 4}{5}\right)$

   This simplifies to:

   $5(x + 4) = 45 - 9(x + 4)$

3. Expand both sides:

   $5x + 20 = 45 - 9x - 36$

   Simplify the right side:

   $5x + 20 = 9 - 9x$

4. Combine like terms:

   Add $9x$ to both sides:

   $5x + 9x + 20 = 9$

   Combine the $x$ terms:

   $14x + 20 = 9$

5. Solve for $x$:

   Subtract 20 from both sides:

   $14x = 9 - 20$

   $14x = -11$

   Divide by 14:

   $x = -\frac{11}{14}$

Solution:

$x = -\frac{11}{14}$

Would you like more details or have any questions about this solution? Here are five related questions for further exploration:

1. How can we check if the solution $x = -\frac{11}{14}$ is correct?
2. What would the equation look like if there were different denominators, such as 7 and 3?
3. How can we solve similar equations that have more complex fractions or multiple terms?
4. What if the fractions had variables in both the numerator and the denominator?
5. How does clearing fractions by multiplying by the least common multiple help simplify equations?

Tip: When dealing with equations involving fractions, always try to clear the fractions by multiplying through by the least common multiple of the denominators. This simplifies the equation and makes it easier to solve!