1. **State the problem:** Simplify the expression $$\frac{x}{x^2-1} - \frac{x-3}{x-1}$$. 2. **Recall the formula and rules:** - Factor the denominator where possible. - Find a common denominator to combine the fractions. - Simplify the resulting expression. 3. **Factor the denominator:** $$x^2 - 1 = (x-1)(x+1)$$ 4. **Rewrite the expression with factored denominators:** $$\frac{x}{(x-1)(x+1)} - \frac{x-3}{x-1}$$ 5. **Find the common denominator:** The common denominator is $$(x-1)(x+1)$$. 6. **Rewrite the second fraction with the common denominator:** $$\frac{x-3}{x-1} = \frac{(x-3)(x+1)}{(x-1)(x+1)}$$ 7. **Combine the fractions:** $$\frac{x}{(x-1)(x+1)} - \frac{(x-3)(x+1)}{(x-1)(x+1)} = \frac{x - (x-3)(x+1)}{(x-1)(x+1)}$$ 8. **Expand the numerator:** $$(x-3)(x+1) = x^2 + x - 3x - 3 = x^2 - 2x - 3$$ 9. **Substitute back:** $$\frac{x - (x^2 - 2x - 3)}{(x-1)(x+1)} = \frac{x - x^2 + 2x + 3}{(x-1)(x+1)}$$ 10. **Simplify the numerator:** $$x - x^2 + 2x + 3 = -x^2 + 3x + 3$$ 11. **Rewrite numerator:** $$-x^2 + 3x + 3 = -(x^2 - 3x - 3)$$ 12. **Final simplified expression:** $$\frac{-(x^2 - 3x - 3)}{(x-1)(x+1)}$$ This is the simplified form of the original expression. **Answer:** $$\boxed{\frac{-(x^2 - 3x - 3)}{(x-1)(x+1)}}$$