∣x∣<10 inequation

Given the inequality:
$$\left|{x}\right| < 10$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{x}\right| = 10$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$x \geq 0$$
or
$$0 \leq x \wedge x < \infty$$
we get the equation
$$x - 10 = 0$$
after simplifying we get
$$x - 10 = 0$$
the solution in this interval:
$$x_{1} = 10$$

2.
$$x < 0$$
or
$$-\infty < x \wedge x < 0$$
we get the equation
$$- x - 10 = 0$$
after simplifying we get
$$- x - 10 = 0$$
the solution in this interval:
$$x_{2} = -10$$


$$x_{1} = 10$$
$$x_{2} = -10$$
$$x_{1} = 10$$
$$x_{2} = -10$$
This roots
$$x_{2} = -10$$
$$x_{1} = 10$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-10 + - \frac{1}{10}$$
=
$$- \frac{101}{10}$$
substitute to the expression
$$\left|{x}\right| < 10$$
$$\left|{- \frac{101}{10}}\right| < 10$$

101     
--- < 10
 10     

but

101     
--- > 10
 10     

Then
$$x < -10$$
no execute
one of the solutions of our inequality is:
$$x > -10 \wedge x < 10$$

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        /     \  
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       x2      x1