1. **State the problem:** Solve the rational equation $$\frac{5}{x+1} + \frac{1}{x-1} = \frac{x}{x-1}$$ for all values of $x$. 2. **Identify restrictions:** The denominators cannot be zero, so $x \neq -1$ and $x \neq 1$. 3. **Find a common denominator:** The common denominator is $(x+1)(x-1)$. 4. **Multiply both sides by the common denominator to clear fractions:** $$ (x+1)(x-1) \times \left( \frac{5}{x+1} + \frac{1}{x-1} \right) = (x+1)(x-1) \times \frac{x}{x-1} $$ 5. **Distribute and simplify:** $$ 5(x-1) + 1(x+1) = x(x+1) $$ 6. **Expand each term:** $$ 5x - 5 + x + 1 = x^2 + x $$ 7. **Combine like terms on the left:** $$ 6x - 4 = x^2 + x $$ 8. **Bring all terms to one side to set equation to zero:** $$ 0 = x^2 + x - 6x + 4 $$ $$ 0 = x^2 - 5x + 4 $$ 9. **Factor the quadratic:** $$ 0 = (x - 4)(x - 1) $$ 10. **Solve for $x$:** $$ x = 4 \quad \text{or} \quad x = 1 $$ 11. **Check restrictions:** $x = 1$ is not allowed because it makes denominator zero. 12. **Final solution:** $$ \boxed{x = 4} $$