1. (17 points) Find the area of the triangle cut off by the line $y=2x+2$ from the figure defined by the inequality $|x-2|+|y-3| \leq 3$.
Answer: 3.

Solution. The figure defined by the given inequality is a square.

![](https://cdn.mathpix.com/cropped/2024_05_06_0785b6f179aec901e215g-3.jpg?height=417&width=486&top_left_y=805&top_left_x=882)

The side of the square is $3 \sqrt{2}$ (this value can be found using the Pythagorean theorem). The given line passes through the vertex of the square and cuts off a triangle with the smaller leg equal to $\sqrt{2}$, and the larger leg - the side of the square. Therefore, the area of the cut-off right triangle is $S=\frac{1}{2} \cdot \sqrt{2} \cdot 3 \sqrt{2}=3$ square units.