1. **State the problem:** We are given a trapezoid with two parallel sides and need to find the value of $x$ given the lengths of the sides: left vertical side $10x - 2$, right vertical side $4x$, and the longer base $13$. 2. **Identify the relationship:** In a trapezoid, the sum of the lengths of the non-parallel sides (legs) is equal to the sum of the lengths of the parallel sides if it is an isosceles trapezoid or if the problem implies equality between these sides. Here, since only one base length is given, we assume the legs are equal or set an equation based on the problem context. 3. **Set up the equation:** Since the trapezoid shape and labels suggest the legs are equal, we set: $$10x - 2 = 4x$$ 4. **Solve for $x$:** $$10x - 2 = 4x$$ Subtract $4x$ from both sides: $$10x - 4x - 2 = 0$$ $$6x - 2 = 0$$ Add 2 to both sides: $$6x = 2$$ Divide both sides by 6: $$\cancel{6}x = \frac{2}{\cancel{6}}$$ $$x = \frac{1}{3}$$ 5. **Conclusion:** The value of $x$ is $\frac{1}{3}$.