Given the inequality:
$$\cot{\left(x \right)} < 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(x \right)} = 2$$
Solve:
Given the equation
$$\cot{\left(x \right)} = 2$$
transform
$$\cot{\left(x \right)} - 2 = 0$$
$$\cot{\left(x \right)} - 2 = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
Move free summands (without w)
from left part to right part, we given:
$$w = 2$$
We get the answer: w = 2
do backward replacement
$$\cot{\left(x \right)} = w$$
substitute w:
$$x_{1} = \operatorname{acot}{\left(2 \right)}$$
$$x_{1} = \operatorname{acot}{\left(2 \right)}$$
This roots
$$x_{1} = \operatorname{acot}{\left(2 \right)}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \operatorname{acot}{\left(2 \right)}$$
=
$$- \frac{1}{10} + \operatorname{acot}{\left(2 \right)}$$
substitute to the expression
$$\cot{\left(x \right)} < 2$$
$$\cot{\left(- \frac{1}{10} + \operatorname{acot}{\left(2 \right)} \right)} < 2$$
-cot(1/10 - acot(2)) < 2
but
-cot(1/10 - acot(2)) > 2
Then
$$x < \operatorname{acot}{\left(2 \right)}$$
no execute
the solution of our inequality is:
$$x > \operatorname{acot}{\left(2 \right)}$$
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