1. **Problem:** Solve the rational equation $$\frac{1}{x - 1} + 5 = \frac{11}{x - 1}$$ and find restrictions on $x$. 2. **Restrictions:** The denominator $x - 1$ cannot be zero because division by zero is undefined. So, $x - 1 \neq 0 \Rightarrow x \neq 1$. 3. **Rewrite the equation:** $$\frac{1}{x - 1} + 5 = \frac{11}{x - 1}$$ 4. **Subtract $\frac{1}{x - 1}$ from both sides:** $$5 = \frac{11}{x - 1} - \frac{1}{x - 1}$$ 5. **Combine the right side:** $$5 = \frac{11 - 1}{x - 1} = \frac{10}{x - 1}$$ 6. **Multiply both sides by $x - 1$ (remember $x \neq 1$):** $$5(x - 1) = 10$$ 7. **Distribute 5:** $$5x - 5 = 10$$ 8. **Add 5 to both sides:** $$5x = 15$$ 9. **Divide both sides by 5:** $$x = 3$$ 10. **Check restriction:** $x = 3$ is allowed since $3 \neq 1$. **Final answer:** $$x = 3$$