4. Sketch the set of points $(x, y)$ in the coordinate plane such that

$$
y \geq|2| x|+| x-2||, \quad y \leq 8
$$

Calculate the area of the resulting geometric figure.
4. We must consider these three cases:

$1^{\circ} \quad x \geq 2 ; \quad 2^{\circ} \quad 0 \leq x \leq 2 ; \quad 3^{\circ} \quad x \leq 0$.

In the first case, $y \geq 3x - 2$, in the second $y \geq x + 2$, and in the third $y \geq -3x + 2$.

(10 points).

The geometric figure obtained is a quadrilateral whose vertices are $A(0,2)$,

$B(2,4), \quad C\left(\frac{10}{3}, 8\right), \quad D(-2,8)$.

(10 points)

![](https://cdn.mathpix.com/cropped/2024_06_03_843fcc95a1719fe2319dg-03.jpg?height=551&width=537&top_left_y=935&top_left_x=614)

The area of this quadrilateral is equal to the sum of the areas of triangle $ABE$ and $AFD$ and trapezoid $EBCF$, where $E(0,4)$ and $F(0,8)$. We get

$$
P=\frac{1}{2} \cdot 2 \cdot 2+\frac{1}{2} \cdot 2 \cdot 6+\frac{2+\frac{10}{3}}{2} \cdot 4=\frac{56}{3}
$$

(5 points)

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## CROATIAN MATHEMATICAL SOCIETY

## MATHEMATICS