1. **State the problem:** We are given an arc length expression $6 + 12x$ and an angle of $45^\circ$ at point $X$ on a circle. We need to solve for $x$. 2. **Relevant formula:** The measure of an inscribed angle in a circle is half the measure of its intercepted arc. That is, $$\text{Angle} = \frac{1}{2} \times \text{Arc measure}$$ 3. **Apply the formula:** The angle at $X$ is $45^\circ$, so the intercepted arc measure is $$\text{Arc} = 2 \times 45^\circ = 90^\circ$$ 4. **Set up the equation:** The arc length expression given is $6 + 12x$, which corresponds to the arc measure in degrees. So, $$6 + 12x = 90$$ 5. **Solve for $x$:** $$12x = 90 - 6$$ $$12x = 84$$ $$x = \frac{84}{12}$$ $$x = 7$$ 6. **Final answer:** $$\boxed{7}$$