1. **State the problem:** We have rectangle EFGH with diagonals intersecting at point I. Given that $GI = 84$ and $HI = -x - 2$, we need to solve for $x$. 2. **Recall properties of rectangles and diagonals:** In a rectangle, the diagonals are equal in length and bisect each other. This means point I is the midpoint of both diagonals. 3. **Use the midpoint property:** Since I is the midpoint of diagonal GH, the segments GI and HI are equal in length. 4. **Set up the equation:** $$GI = HI$$ $$84 = |-x - 2|$$ 5. **Solve for $x$ considering absolute value:** Case 1: $-x - 2 = 84$ $$-x = 84 + 2$$ $$-x = 86$$ $$x = -86$$ Case 2: $-x - 2 = -84$ $$-x = -84 + 2$$ $$-x = -82$$ $$x = 82$$ 6. **Final answer:** $$x = -86 \text{ or } x = 82$$