|x-4|<=2 inequation

Given the inequality:
$$\left|{x - 4}\right| \leq 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{x - 4}\right| = 2$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$x - 4 \geq 0$$
or
$$4 \leq x \wedge x < \infty$$
we get the equation
$$\left(x - 4\right) - 2 = 0$$
after simplifying we get
$$x - 6 = 0$$
the solution in this interval:
$$x_{1} = 6$$

2.
$$x - 4 < 0$$
or
$$-\infty < x \wedge x < 4$$
we get the equation
$$\left(4 - x\right) - 2 = 0$$
after simplifying we get
$$2 - x = 0$$
the solution in this interval:
$$x_{2} = 2$$


$$x_{1} = 6$$
$$x_{2} = 2$$
$$x_{1} = 6$$
$$x_{2} = 2$$
This roots
$$x_{2} = 2$$
$$x_{1} = 6$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 2$$
=
$$\frac{19}{10}$$
substitute to the expression
$$\left|{x - 4}\right| \leq 2$$
$$\left|{-4 + \frac{19}{10}}\right| \leq 2$$

21     
-- <= 2
10     

but

21     
-- >= 2
10     

Then
$$x \leq 2$$
no execute
one of the solutions of our inequality is:
$$x \geq 2 \wedge x \leq 6$$

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       x2      x1