Let's solve the given problem step by step.

Given Information:

• Parallelogram $ABCD$ with area $2$.
• Given points: $A(0,2)$ and $B(3,0)$.
• The intersection of the diagonals $I$ lies on the line $y = -x$.
• The $x$-coordinate of $D$ is greater than $-14$.

Step 1: Find the Coordinates of $I$

Since diagonals of a parallelogram bisect each other, we have:

$I = \frac{A + C}{2} = \frac{B + D}{2}$

Let $C(x_C, y_C)$ and $D(x_D, y_D)$ be the unknown points.

From the given information, $I$ satisfies $y = -x$, so let $I(a, -a)$.

$I = \frac{(0,2) + (x_C, y_C)}{2} = \frac{(3,0) + (x_D, y_D)}{2}$

Equating components:

$\left( \frac{0 + x_C}{2}, \frac{2 + y_C}{2} \right) = (a, -a)$

$\frac{0 + x_C}{2} = a \Rightarrow x_C = 2a$

$\frac{2 + y_C}{2} = -a \Rightarrow y_C = -2a - 2$

Similarly, for $D$:

$\left( \frac{3 + x_D}{2}, \frac{0 + y_D}{2} \right) = (a, -a)$

$\frac{3 + x_D}{2} = a \Rightarrow x_D = 2a - 3$

$\frac{0 + y_D}{2} = -a \Rightarrow y_D = -2a$

Step 2: Use the Area Condition

The area of parallelogram $ABCD$ is given by:

$S = \left| \overrightarrow{AB} \times \overrightarrow{AD} \right|$

Vectors:

$\overrightarrow{AB} = (3 - 0, 0 - 2) = (3, -2)$

$\overrightarrow{AD} = (x_D - 0, y_D - 2) = (2a - 3, -2a - 2)$

Determinant formula:

$S = \left| 3(-2a - 2) - (-2)(2a - 3) \right| = 2$

$\left| -6a - 6 + 4a - 6 \right| = 2$

$\left| -2a - 12 \right| = 2$

Solving:

$-2a - 12 = 2 \quad \text{or} \quad -2a - 12 = -2$

$-2a = 14 \Rightarrow a = -7$

$-2a = 10 \Rightarrow a = -5$

Step 3: Find $x_D$ and Choose the Correct One

For $a = -7$:

x_D = 2(-7) - 3 = -14 - 3 = -17 \quad \text{(Invalid, since x_D > -14)}

For $a = -5$:

$x_D = 2(-5) - 3 = -10 - 3 = -13 \quad \text{(Valid)}$

Thus, the final answer is:

$\boxed{-13}$

Would you like further clarifications? Here are some related questions:

1. How do you determine the intersection of two diagonals in a parallelogram?
2. Why does the determinant formula help in finding the area of a parallelogram?
3. Can this problem be solved using vector cross-product methods?
4. What happens if the given area was negative instead of positive?
5. How do you verify the correctness of your calculated coordinates?

Tip: Always check conditions carefully when solving for unknowns, especially when given constraints like $x_D > -14$.