Given the inequality:
$$\left|{x - 1}\right| - \frac{6}{\left|{x - 1}\right|} < -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{x - 1}\right| - \frac{6}{\left|{x - 1}\right|} = -1$$
Solve:
$$x_{1} = -1$$
$$x_{2} = 3$$
$$x_{1} = -1$$
$$x_{2} = 3$$
This roots
$$x_{1} = -1$$
$$x_{2} = 3$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$-1 + - \frac{1}{10}$$
=
$$-1.1$$
substitute to the expression
$$\left|{x - 1}\right| - \frac{6}{\left|{x - 1}\right|} < -1$$
$$- \frac{6}{\left|{-1.1 - 1}\right|} + \left|{-1.1 - 1}\right| < -1$$
-0.757142857142857 < -1
but
-0.757142857142857 > -1
Then
$$x < -1$$
no execute
one of the solutions of our inequality is:
$$x > -1 \wedge x < 3$$
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