Task: Solve the inequality: $$\cot x < \sqrt3\ \mbox{ for }\ x\in\mathbb{R}.$$ Marie solved the task in the following steps:
(1) She identified the points where the cotangent function is undefined: $$x=k\cdot\pi,\ \mbox{ where }\ k\in\mathbb{Z}$$
(2) She set up and solved the equation $\cot x=\sqrt3$: $$x=\frac{\pi}{6}+k\cdot\pi,\ \mbox{ where }\ k\in\mathbb{Z}$$ (3) She stated that the cotangent function is decreasing on every open interval bounded by two consecutive points where it is undefined, i.e., on the intervals: $$\left(0+k\cdot\pi;\pi+k\cdot\pi\right),\ \mbox{ where }\ k\in\mathbb{Z}$$ (4) Marie further claimed that the previous two steps imply that for each $k\in\mathbb{Z}$: $$\cot x<\sqrt3\Leftrightarrow 0+k\cdot\pi< x < \frac{\pi}{6}+k\cdot\pi$$ (5) Finally, she wrote the solution of the inequality obtained in the previous step in the form: $$K=\bigcup_{k\in\mathbb{Z}}\left(0+k\cdot\pi;\frac{\pi}{6}+k\cdot\pi\right)$$ The solution is incorrect. In which step did Marie make a mistake?
The mistake is in step (1). Marie incorrectly identified the points where the cotangent function is undefined.
The mistake is in step (2). Marie incorrectly solved the equation $\cot x=\sqrt3$.
The mistake is in step (3). The cotangent function is decreasing over its domain.
The mistake is in step (4). Marie incorrectly applied the property of the cotangent function described in step (3) to solve the inequality $\cot x<\sqrt3$.
The cotangent function is decreasing on every open interval bounded by two consecutive points where the function is undefined, i.e., on the intervals $$\left(0+k\cdot\pi; \pi+k\cdot\pi\right),\mbox{ where } k\in\mathbb{Z}.$$ Therefore, in each of these intervals (i.e. for each $k\in\mathbb{Z}$), it holds (see the figure): $$\cot x< \sqrt3\Leftrightarrow\frac{\pi}{6}+k\cdot\pi< x< \pi+k\cdot\pi$$
We can write the solution of the inequality $\cot x< \sqrt3$ as: $$K=\bigcup_{k \in \mathbb{Z}}\left(\frac{\pi}{6}+k\cdot\pi;\pi+k\cdot\pi\right)$$
