1. **State the problem:** Solve the equation $$\frac{|D^2 - 3|}{|D^2 - 1|} = \frac{|D^2 - 2|}{|D^2 - 1|}$$ for $D$. 2. **Simplify the equation:** Since the denominators on both sides are the same and nonzero (assuming $D^2 \neq 1$), we can multiply both sides by $|D^2 - 1|$ to get: $$|D^2 - 3| = |D^2 - 2|$$ 3. **Solve the absolute value equation:** The equation $|A| = |B|$ implies either $A = B$ or $A = -B$. So, we have two cases: **Case 1:** $$D^2 - 3 = D^2 - 2$$ Simplify: $$D^2 - 3 = D^2 - 2 \implies -3 = -2$$ This is false, so no solution from case 1. **Case 2:** $$D^2 - 3 = -(D^2 - 2)$$ Simplify: $$D^2 - 3 = -D^2 + 2$$ Bring all terms to one side: $$D^2 + D^2 = 2 + 3$$ $$2D^2 = 5$$ $$D^2 = \frac{5}{2}$$ 4. **Find $D$:** $$D = \pm \sqrt{\frac{5}{2}} = \pm \frac{\sqrt{10}}{2}$$ 5. **Check for restrictions:** We must ensure the denominator $|D^2 - 1| \neq 0$, so $D^2 \neq 1$. Since $\frac{5}{2} = 2.5 \neq 1$, these solutions are valid. **Final answer:** $$D = \pm \frac{\sqrt{10}}{2}$$