1. **Problem (الف): Solve the equation** $$\frac{2x}{x^2 - 1} + \frac{2}{x + 1} = \frac{2 - x}{x^2 - x}$$ 2. **Identify the denominators and restrictions:** - $x^2 - 1 = (x - 1)(x + 1)$ - $x^2 - x = x(x - 1)$ Restrictions: $x \neq 1$, $x \neq -1$, $x \neq 0$ to avoid division by zero. 3. **Find a common denominator:** $$\text{LCD} = x(x - 1)(x + 1)$$ 4. **Multiply both sides by the LCD to clear denominators:** $$2x \cdot x + 2 \cdot x(x - 1) = (2 - x)(x + 1)$$ 5. **Simplify each term:** - Left side: $$2x \cdot x = 2x^2$$ $$2 \cdot x(x - 1) = 2x^2 - 2x$$ - Right side: $$(2 - x)(x + 1) = 2x + 2 - x^2 - x = -x^2 + x + 2$$ 6. **Combine left side terms:** $$2x^2 + 2x^2 - 2x = 4x^2 - 2x$$ 7. **Set equation:** $$4x^2 - 2x = -x^2 + x + 2$$ 8. **Bring all terms to one side:** $$4x^2 - 2x + x^2 - x - 2 = 0$$ $$5x^2 - 3x - 2 = 0$$ 9. **Solve quadratic equation:** Use quadratic formula: $$x = \frac{3 \pm \sqrt{(-3)^2 - 4 \cdot 5 \cdot (-2)}}{2 \cdot 5} = \frac{3 \pm \sqrt{9 + 40}}{10} = \frac{3 \pm \sqrt{49}}{10}$$ 10. **Calculate roots:** $$x = \frac{3 + 7}{10} = 1$$ $$x = \frac{3 - 7}{10} = -\frac{4}{10} = -0.4$$ 11. **Check restrictions:** $x = 1$ is excluded (denominator zero), so discard. 12. **Final solution for (الف):** $$x = -0.4$$ --- 13. **Problem (ب): Solve the equation** $$\sqrt{x + 1} - \sqrt{2x - 5} = 1$$ 14. **Domain restrictions:** - $x + 1 \geq 0 \Rightarrow x \geq -1$ - $2x - 5 \geq 0 \Rightarrow x \geq \frac{5}{2} = 2.5$ So domain is $x \geq 2.5$ 15. **Isolate one square root:** $$\sqrt{x + 1} = 1 + \sqrt{2x - 5}$$ 16. **Square both sides:** $$x + 1 = (1 + \sqrt{2x - 5})^2 = 1 + 2\sqrt{2x - 5} + (2x - 5)$$ 17. **Simplify:** $$x + 1 = 1 + 2\sqrt{2x - 5} + 2x - 5$$ $$x + 1 = 2x - 4 + 2\sqrt{2x - 5}$$ 18. **Bring terms to one side:** $$x + 1 - 2x + 4 = 2\sqrt{2x - 5}$$ $$-x + 5 = 2\sqrt{2x - 5}$$ 19. **Isolate the square root:** $$2\sqrt{2x - 5} = 5 - x$$ 20. **Square both sides again:** $$4(2x - 5) = (5 - x)^2$$ $$8x - 20 = 25 - 10x + x^2$$ 21. **Bring all terms to one side:** $$0 = 25 - 10x + x^2 - 8x + 20$$ $$0 = x^2 - 18x + 45$$ 22. **Solve quadratic:** $$x = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot 45}}{2} = \frac{18 \pm \sqrt{324 - 180}}{2} = \frac{18 \pm \sqrt{144}}{2}$$ 23. **Calculate roots:** $$x = \frac{18 + 12}{2} = 15$$ $$x = \frac{18 - 12}{2} = 3$$ 24. **Check domain and original equation:** - For $x=15$: $$\sqrt{15 + 1} - \sqrt{2 \cdot 15 - 5} = \sqrt{16} - \sqrt{25} = 4 - 5 = -1 \neq 1$$ - For $x=3$: $$\sqrt{3 + 1} - \sqrt{6 - 5} = \sqrt{4} - \sqrt{1} = 2 - 1 = 1$$ 25. **Final solution for (ب):** $$x = 3$$