1. **State the problem:** We have points L, M, and N on a line segment in that order, with M between L and N. 2. **Given:** - Length LM = $2x$ - Length MN = $3x - 1$ - Length LN = 24 3. **Formula used:** Since M lies between L and N, the sum of LM and MN equals LN: $$LM + MN = LN$$ 4. **Substitute the given expressions:** $$2x + (3x - 1) = 24$$ 5. **Simplify the equation:** $$2x + 3x - 1 = 24$$ $$5x - 1 = 24$$ 6. **Add 1 to both sides:** $$5x - 1 + 1 = 24 + 1$$ $$5x = 25$$ 7. **Divide both sides by 5:** $$\cancel{5}x = \cancel{5}5$$ $$x = 5$$ 8. **Find MN:** $$MN = 3x - 1 = 3(5) - 1 = 15 - 1 = 14$$ **Final answer:** $$\boxed{14}$$