1. **State the problem:** We are given two angles formed by a transversal intersecting two parallel lines. The angles are $(52 - 2x)^\circ$ and $(22x + 88)^\circ$. We need to solve for $x$. 2. **Identify the relationship:** Since the lines are parallel and the transversal intersects them, the given angles are corresponding angles and therefore equal. 3. **Set up the equation:** $$52 - 2x = 22x + 88$$ 4. **Solve for $x$:** Subtract 22x from both sides: $$52 - 2x - 22x = 88$$ $$52 - 24x = 88$$ Subtract 52 from both sides: $$-24x = 88 - 52$$ $$-24x = 36$$ Divide both sides by -24: $$x = \frac{36}{-24} = -\frac{3}{2} = -1.5$$ 5. **Final answer:** $$x = -1.5$$