1. **State the problem:** Simplify the expression $$\frac{1}{x-2} - \frac{2}{x-1} + \frac{1}{x-3}$$. 2. **Find a common denominator:** The denominators are $x-2$, $x-1$, and $x-3$. The common denominator is their product: $$(x-2)(x-1)(x-3)$$. 3. **Rewrite each fraction with the common denominator:** $$\frac{1}{x-2} = \frac{(x-1)(x-3)}{(x-2)(x-1)(x-3)}$$ $$\frac{2}{x-1} = \frac{2(x-2)(x-3)}{(x-2)(x-1)(x-3)}$$ $$\frac{1}{x-3} = \frac{(x-2)(x-1)}{(x-2)(x-1)(x-3)}$$ 4. **Combine the fractions:** $$\frac{(x-1)(x-3) - 2(x-2)(x-3) + (x-2)(x-1)}{(x-2)(x-1)(x-3)}$$ 5. **Expand the numerators:** - $(x-1)(x-3) = x^2 - 4x + 3$ - $2(x-2)(x-3) = 2(x^2 - 5x + 6) = 2x^2 - 10x + 12$ - $(x-2)(x-1) = x^2 - 3x + 2$ 6. **Substitute back and simplify numerator:** $$x^2 - 4x + 3 - (2x^2 - 10x + 12) + x^2 - 3x + 2$$ $$= x^2 - 4x + 3 - 2x^2 + 10x - 12 + x^2 - 3x + 2$$ 7. **Combine like terms:** $$ (x^2 - 2x^2 + x^2) + (-4x + 10x - 3x) + (3 - 12 + 2)$$ $$= 0x^2 + 3x - 7$$ 8. **Final simplified expression:** $$\frac{3x - 7}{(x-2)(x-1)(x-3)}$$ This is the simplified form of the original expression.