1. **State the problem:** Solve the equation $$\frac{1}{x+1} + \frac{9}{x+9} = 1$$ for $x$. 2. **Formula and rules:** To solve equations with fractions, find a common denominator to combine terms and clear fractions by multiplying both sides. 3. **Find the common denominator:** The denominators are $x+1$ and $x+9$. The common denominator is $(x+1)(x+9)$. 4. **Multiply both sides by the common denominator:** $$ (x+1)(x+9) \left( \frac{1}{x+1} + \frac{9}{x+9} \right) = (x+1)(x+9) \cdot 1 $$ 5. **Simplify each term:** $$ (x+9) + 9(x+1) = (x+1)(x+9) $$ 6. **Expand terms:** $$ x + 9 + 9x + 9 = x^2 + 10x + 9 $$ 7. **Combine like terms on the left:** $$ 10x + 18 = x^2 + 10x + 9 $$ 8. **Bring all terms to one side:** $$ 0 = x^2 + 10x + 9 - 10x - 18 $$ $$ 0 = x^2 - 9 $$ 9. **Solve the quadratic:** $$ x^2 = 9 $$ $$ x = \pm 3 $$ 10. **Check for restrictions:** The denominators $x+1$ and $x+9$ cannot be zero. - For $x=3$, denominators are $4$ and $12$, valid. - For $x=-3$, denominators are $-2$ and $6$, valid. **Final answer:** $$x = 3 \text{ or } x = -3$$