1. **State the problem:** Solve the equation $$\frac{X - 1}{X} - 2 = \frac{-X - 1}{X^2 - 2X}$$ for $X$. 2. **Rewrite the equation:** Note that $X^2 - 2X = X(X - 2)$, so the equation becomes: $$\frac{X - 1}{X} - 2 = \frac{-X - 1}{X(X - 2)}$$ 3. **Find a common denominator:** The denominators are $X$ and $X(X - 2)$. The least common denominator (LCD) is $X(X - 2)$. 4. **Express all terms with the LCD:** - Left side first term: $$\frac{X - 1}{X} = \frac{(X - 1)(X - 2)}{X(X - 2)}$$ - Left side second term: $$2 = \frac{2X(X - 2)}{X(X - 2)}$$ 5. **Rewrite the equation with common denominator:** $$\frac{(X - 1)(X - 2)}{X(X - 2)} - \frac{2X(X - 2)}{X(X - 2)} = \frac{-X - 1}{X(X - 2)}$$ 6. **Combine the left side numerator:** $$(X - 1)(X - 2) - 2X(X - 2) = -X - 1$$ 7. **Expand the terms:** - Expand $(X - 1)(X - 2)$: $$X^2 - 2X - X + 2 = X^2 - 3X + 2$$ - Expand $2X(X - 2)$: $$2X^2 - 4X$$ 8. **Substitute expansions:** $$X^2 - 3X + 2 - (2X^2 - 4X) = -X - 1$$ 9. **Simplify the left side:** $$X^2 - 3X + 2 - 2X^2 + 4X = -X - 1$$ $$-X^2 + X + 2 = -X - 1$$ 10. **Bring all terms to one side:** $$-X^2 + X + 2 + X + 1 = 0$$ $$-X^2 + 2X + 3 = 0$$ 11. **Multiply both sides by -1 to simplify:** $$X^2 - 2X - 3 = 0$$ 12. **Factor the quadratic:** $$(X - 3)(X + 1) = 0$$ 13. **Solve for $X$:** $$X - 3 = 0 \Rightarrow X = 3$$ $$X + 1 = 0 \Rightarrow X = -1$$ 14. **Check for restrictions:** - Denominators cannot be zero. - $X \neq 0$ and $X \neq 2$ (from $X(X - 2)$ in denominator). - Both $3$ and $-1$ are allowed. **Final answer:** $$X = 3 \text{ or } X = -1$$