1. **State the problem:** We are given a rectangle with sides \( \frac{2}{x-2} \) and \( \frac{3}{x-1} \), and the perimeter is 10 cm. We need to write an equation for \( x \) and solve it algebraically for \( x > 2 \). 2. **Formula for perimeter of a rectangle:** \[ P = 2(\text{length} + \text{width}) \] Here, \( P = 10 \), length = \( \frac{2}{x-2} \), and width = \( \frac{3}{x-1} \). 3. **Write the equation:** \[ 10 = 2\left( \frac{2}{x-2} + \frac{3}{x-1} \right) \] 4. **Divide both sides by 2:** \[ \cancel{2} \times 5 = \cancel{2} \left( \frac{2}{x-2} + \frac{3}{x-1} \right) \implies 5 = \frac{2}{x-2} + \frac{3}{x-1} \] 5. **Find common denominator and combine:** \[ 5 = \frac{2(x-1)}{(x-2)(x-1)} + \frac{3(x-2)}{(x-1)(x-2)} = \frac{2(x-1) + 3(x-2)}{(x-2)(x-1)} \] 6. **Simplify numerator:** \[ 2(x-1) + 3(x-2) = 2x - 2 + 3x - 6 = 5x - 8 \] 7. **Rewrite equation:** \[ 5 = \frac{5x - 8}{(x-2)(x-1)} \] 8. **Multiply both sides by \( (x-2)(x-1) \):** \[ 5(x-2)(x-1) = 5x - 8 \] 9. **Expand left side:** \[ 5(x^2 - 3x + 2) = 5x - 8 \] \[ 5x^2 - 15x + 10 = 5x - 8 \] 10. **Bring all terms to one side:** \[ 5x^2 - 15x + 10 - 5x + 8 = 0 \implies 5x^2 - 20x + 18 = 0 \] 11. **Divide entire equation by 5:** \[ \cancel{5}x^2 - \cancel{5} \times 4x + \cancel{5} \times \frac{18}{5} = 0 \implies x^2 - 4x + \frac{18}{5} = 0 \] 12. **Use quadratic formula:** \[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \times 1 \times \frac{18}{5}}}{2} = \frac{4 \pm \sqrt{16 - \frac{72}{5}}}{2} = \frac{4 \pm \sqrt{\frac{80}{5} - \frac{72}{5}}}{2} = \frac{4 \pm \sqrt{\frac{8}{5}}}{2} \] 13. **Simplify square root:** \[ \sqrt{\frac{8}{5}} = \frac{\sqrt{8}}{\sqrt{5}} = \frac{2\sqrt{2}}{\sqrt{5}} = \frac{2\sqrt{2} \sqrt{5}}{5} = \frac{2\sqrt{10}}{5} \] 14. **Final exact solutions:** \[ x = \frac{4 \pm \frac{2\sqrt{10}}{5}}{2} = 2 \pm \frac{\sqrt{10}}{5} \] 15. **Since \( x > 2 \), choose:** \[ x = 2 + \frac{\sqrt{10}}{5} \] 16. **Approximate value:** \[ \sqrt{10} \approx 3.1623 \implies x \approx 2 + \frac{3.1623}{5} = 2 + 0.6325 = 2.6 \] **Answer:** Exact: \( x = 2 + \frac{\sqrt{10}}{5} \) Rounded to nearest tenth: \( x \approx 2.6 \)