## Task B-2.4.

Determine all natural numbers $x$ that satisfy the system of inequalities

$$
x-\frac{6}{x} \geq 1, \quad \frac{1}{x-6} \leq 0
$$
## Solution.

We seek the solution in the set of natural numbers, so we can multiply the first inequality by the denominator. We get the inequality $x^{2}-x-6 \geq 0$ whose solution in the set $\mathbb{N}$ is $x \in\{3,4,5, \ldots\}$.

The solution to the inequality $x-6<0$ in the set $\mathbb{N}$ is $x \in\{1,2,3,4,5\}$.

The final solution to the problem is $x \in\{3,4,5\}$.