|v-6|>3 inequation

Given the inequality:
$$\left|{v - 6}\right| > 3$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{v - 6}\right| = 3$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$v - 6 \geq 0$$
or
$$6 \leq v \wedge v < \infty$$
we get the equation
$$\left(v - 6\right) - 3 = 0$$
after simplifying we get
$$v - 9 = 0$$
the solution in this interval:
$$v_{1} = 9$$

2.
$$v - 6 < 0$$
or
$$-\infty < v \wedge v < 6$$
we get the equation
$$\left(6 - v\right) - 3 = 0$$
after simplifying we get
$$3 - v = 0$$
the solution in this interval:
$$v_{2} = 3$$


$$v_{1} = 9$$
$$v_{2} = 3$$
$$v_{1} = 9$$
$$v_{2} = 3$$
This roots
$$v_{2} = 3$$
$$v_{1} = 9$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$v_{0} < v_{2}$$
For example, let's take the point
$$v_{0} = v_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 3$$
=
$$\frac{29}{10}$$
substitute to the expression
$$\left|{v - 6}\right| > 3$$
$$\left|{-6 + \frac{29}{10}}\right| > 3$$

31    
-- > 3
10    

one of the solutions of our inequality is:
$$v < 3$$

 _____           _____          
      \         /
-------ο-------ο-------
       v2      v1

Other solutions will get with the changeover to the next point
etc.
The answer:
$$v < 3$$
$$v > 9$$