abs(x-2)<=7 inequation

Given the inequality:
$$\left|{x - 2}\right| \leq 7$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{x - 2}\right| = 7$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$x - 2 \geq 0$$
or
$$2 \leq x \wedge x < \infty$$
we get the equation
$$\left(x - 2\right) - 7 = 0$$
after simplifying we get
$$x - 9 = 0$$
the solution in this interval:
$$x_{1} = 9$$

2.
$$x - 2 < 0$$
or
$$-\infty < x \wedge x < 2$$
we get the equation
$$\left(2 - x\right) - 7 = 0$$
after simplifying we get
$$- x - 5 = 0$$
the solution in this interval:
$$x_{2} = -5$$


$$x_{1} = 9$$
$$x_{2} = -5$$
$$x_{1} = 9$$
$$x_{2} = -5$$
This roots
$$x_{2} = -5$$
$$x_{1} = 9$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-5 + - \frac{1}{10}$$
=
$$- \frac{51}{10}$$
substitute to the expression
$$\left|{x - 2}\right| \leq 7$$
$$\left|{- \frac{51}{10} - 2}\right| \leq 7$$

71     
-- <= 7
10     

but

71     
-- >= 7
10     

Then
$$x \leq -5$$
no execute
one of the solutions of our inequality is:
$$x \geq -5 \wedge x \leq 9$$

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       x2      x1