1. **Problem statement:** Given that line $n$ bisects segment $CE$ at point $D$, and the lengths $CD = x + 6$ and $DE = 4x - 21$, find the length $CD$. 2. **Key concept:** When a point $D$ bisects segment $CE$, it means $D$ is the midpoint of $CE$. Therefore, the two segments $CD$ and $DE$ are equal in length. 3. **Set up the equation:** Since $CD = DE$, we have: $$x + 6 = 4x - 21$$ 4. **Solve for $x$:** Subtract $x$ from both sides: $$\cancel{x} + 6 = 4\cancel{x} - 21 - x$$ $$6 = 3x - 21$$ Add 21 to both sides: $$6 + 21 = 3x - 21 + 21$$ $$27 = 3x$$ Divide both sides by 3: $$\frac{27}{\cancel{3}} = \frac{3x}{\cancel{3}}$$ $$9 = x$$ 5. **Find $CD$:** Substitute $x=9$ into $CD = x + 6$: $$CD = 9 + 6 = 15$$ **Final answer:** $$\boxed{15}$$