## Task 3 - 240823

Determine all natural numbers $x$ that satisfy the inequality

$$
\frac{11}{15}<\frac{7}{x}<\frac{15}{11} \quad \text { ! }
$$
The required inequality is satisfied if and only if the two inequalities

$$
\frac{11}{15}0$ and the inequality $15 x>77$ is satisfied. This is true if and only if $x>\frac{77}{15}$, which, since $\frac{77}{15}=5 \frac{2}{15}$, is satisfied precisely by the natural numbers $x$ with $x \geq 6$.

Thus, the validity of (1) and (2) for natural numbers $x$ is equivalent to $6 \leq x \leq 9$ (i.e., the numbers sought are exactly the numbers 6, 7, 8, and 9).