1. **State the problem:** In rectangle ABCD, we are given that DE = 3x + 6 and BE = 5x - 6. We need to find the value of $x$. 2. **Understand the properties of a rectangle:** In a rectangle, opposite sides are equal and all angles are right angles. Points D, E, and B are on the rectangle, and since DE and BE are segments meeting at E, we can use the Pythagorean theorem if E is the right angle vertex. 3. **Assuming E is the vertex where DE and BE meet at a right angle:** Then by the Pythagorean theorem, $$AB^2 = DE^2 + BE^2$$ 4. **Since ABCD is a rectangle, AB is a side, but we don't have AB directly. However, if E lies on the diagonal or side, we need more info.** 5. **Alternatively, if DE and BE are segments from point E on the rectangle such that DE and BE are equal (e.g., if E is midpoint or similar), then set DE = BE:** $$3x + 6 = 5x - 6$$ 6. **Solve for $x$:** $$3x + 6 = 5x - 6$$ $$6 + 6 = 5x - 3x$$ $$12 = 2x$$ $$x = \frac{12}{2}$$ $$x = 6$$ 7. **Check the value:** $$DE = 3(6) + 6 = 18 + 6 = 24$$ $$BE = 5(6) - 6 = 30 - 6 = 24$$ Both are equal, which is consistent with the assumption. **Final answer:** $$x = 6$$