Task 4. (Option 1)

For which values of a does the inequality $x^{2}+|x+a|<2$ have at least one positive solution?
Solution: On the xOa plane, we will plot the set of points whose coordinates satisfy the inequality $x^{2}+|x+a|x^{2}-x-2$. In the shaded area, points with positive abscissas exist when $a \in\left[-\frac{9}{4} ; 2\right)$.

![](https://cdn.mathpix.com/cropped/2024_05_06_96a99fe03936e779e014g-05.jpg?height=708&width=653&top_left_y=280&top_left_x=1244)

Answer: $a \in\left[-\frac{9}{4} ; 2\right)$