1. **Problem statement:** We are given a triangle $\triangle ABC$ with sides $AB = 7.5x$, $BC = 6x + 3$, and $AC = 10x - 5$. The triangle is equilateral, meaning all sides are equal. 2. **Formula and rule:** In an equilateral triangle, all three sides are equal. So, we set the sides equal to each other: $$7.5x = 6x + 3 = 10x - 5$$ 3. **Step 1: Equate $AB$ and $BC$:** $$7.5x = 6x + 3$$ Subtract $6x$ from both sides: $$7.5x - 6x = 3$$ $$1.5x = 3$$ Divide both sides by 1.5: $$x = \frac{3}{1.5}$$ Use $\cancel{1.5}$ to simplify: $$x = 2$$ 4. **Step 2: Verify with $AB$ and $AC$:** Set $7.5x = 10x - 5$: $$7.5x = 10x - 5$$ Subtract $7.5x$ from both sides: $$0 = 10x - 7.5x - 5$$ $$0 = 2.5x - 5$$ Add 5 to both sides: $$5 = 2.5x$$ Divide both sides by 2.5: $$x = \frac{5}{2.5}$$ Simplify: $$x = 2$$ 5. **Conclusion:** Both equations give $x = 2$, which matches option D. **Final answer:** $\boxed{2}$