Area of bounded quadrilateral

The region is bounded between $x = 0$ (Y-axis) and $x = 5$.

For $0 \le x \le 5$, we have $|x - 5| = 5 - x$.

Substitute this into the given equation: $|x - y| - (5 - x) = 2 \implies |x - y| = 7 - x$

This gives two linear boundary functions for $y$:

1. $x - y = 7 - x \implies y = 2x - 7$
2. $x - y = -(7 - x) \implies y = 7$

Thus, the region is bounded by $x = 0, x = 5, y = 7,$ and $y = 2x - 7$.

At $x = 0$, $y$ ranges from $-7$ to $7$ (vertical length $= 14$). At $x = 5$, $y$ ranges from $3$ to $7$ (vertical length $= 4$).

This forms a trapezoid with parallel vertical sides of lengths $14$ and $4$, and width $5$. $\text{Area} = \frac{14 + 4}{2} \times 5 = 9 \times 5 = 45$