1. **State the problem:** Solve for $x$ given the segments on parallel lines cut by transversals with lengths $7x - 11$, $4x - 2$, $20$, and $35$. 2. **Use the Segment Proportionality Theorem:** When two parallel lines are cut by transversals, corresponding segments are proportional. So, $$\frac{7x - 11}{20} = \frac{4x - 2}{35}$$ 3. **Cross multiply:** $$(7x - 11) \times 35 = (4x - 2) \times 20$$ 4. **Expand both sides:** $$245x - 385 = 80x - 40$$ 5. **Bring like terms together:** $$245x - 80x = -40 + 385$$ $$165x = 345$$ 6. **Solve for $x$:** $$x = \frac{345}{165}$$ 7. **Simplify the fraction:** $$x = \frac{\cancel{345}^{\div 15}}{\cancel{165}^{\div 15}} = \frac{23}{11}$$ **Final answer:** $$x = \frac{23}{11}$$