1. **State the problem:** We need to find the value of $x$ such that the ratio $\frac{x - 6}{2x}$ is equivalent to the ratio $\frac{1}{3}$. 2. **Write the equation from the ratio:** Since the two ratios are equivalent, we set them equal: $$\frac{x - 6}{2x} = \frac{1}{3}$$ 3. **Cross multiply to solve for $x$:** $$3(x - 6) = 1 \times 2x$$ $$3x - 18 = 2x$$ 4. **Isolate $x$ on one side:** $$3x - 2x = 18$$ $$x = 18$$ 5. **Check the solution:** Substitute $x = 18$ back into the original ratio: $$\frac{18 - 6}{2 \times 18} = \frac{12}{36} = \frac{1}{3}$$ which matches the given ratio. **Final answer:** $$x = 18$$