## Task B-2.3.

The quadratic function $f$ is positive only on the interval $\langle-2,3\rangle$, and the quadratic function $g$ is positive only on $\langle-\infty, 0\rangle \cup\langle 5,+\infty\rangle$. The number 1 is a root of the increasing linear function $h$. Determine all real numbers $x$ for which $\frac{f(x) \cdot g(x)}{h(x)}>0$.
## First Solution.

The increasing linear function $h$ is positive on the interval $\langle 1,+\infty\rangle$, and negative on the interval $\langle-\infty, 1\rangle$.

Let's determine the signs of the functions $f, g$, and $h$ on each individual interval.

|  | $f$ | $g$ | $h$ | $\frac{f \cdot g}{h}$ |
| :---: | :---: | :---: | :---: | :---: |
| $\langle-\infty,-2\rangle$ | - | + | - | + |
| $\langle-2,0\rangle$ | + | + | - | - |
| $\langle 0,1\rangle$ | + | - | - | + |
| $\langle 1,3\rangle$ | + | - | + | - |
| $\langle 3,5\rangle$ | - | - | + | + |
| $\langle 5,+\infty\rangle$ | - | + | + | - |

We conclude that $\frac{f(x) \cdot g(x)}{h(x)}>0$ for $x \in\langle-\infty,-2\rangle \cup\langle 0,1\rangle \cup\langle 3,5\rangle$.