1. **State the problem:** We have a rectangle ABCD with diagonals intersecting. The diagonal segments are given as $5x - 6$ and $3x + 2$. We need to find the value of $x$. 2. **Recall the property of rectangle diagonals:** In a rectangle, the diagonals are equal in length. Therefore, the expressions for the diagonals must be equal. 3. **Set up the equation:** $$5x - 6 = 3x + 2$$ 4. **Solve for $x$:** Subtract $3x$ from both sides: $$5x - \cancel{3x} - 6 = \cancel{3x} + 2 - 3x$$ $$2x - 6 = 2$$ Add 6 to both sides: $$2x - 6 + 6 = 2 + 6$$ $$2x = 8$$ Divide both sides by 2: $$\frac{2x}{\cancel{2}} = \frac{8}{\cancel{2}}$$ $$x = 4$$ 5. **Final answer:** $$\boxed{4}$$