(abs(x-(5/2)))>(9/2) inequation

Given the inequality:
$$\left|{x - \frac{5}{2}}\right| > \frac{9}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{x - \frac{5}{2}}\right| = \frac{9}{2}$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$x - \frac{5}{2} \geq 0$$
or
$$\frac{5}{2} \leq x \wedge x < \infty$$
we get the equation
$$\left(x - \frac{5}{2}\right) - \frac{9}{2} = 0$$
after simplifying we get
$$x - 7 = 0$$
the solution in this interval:
$$x_{1} = 7$$

2.
$$x - \frac{5}{2} < 0$$
or
$$-\infty < x \wedge x < \frac{5}{2}$$
we get the equation
$$\left(\frac{5}{2} - x\right) - \frac{9}{2} = 0$$
after simplifying we get
$$- x - 2 = 0$$
the solution in this interval:
$$x_{2} = -2$$


$$x_{1} = 7$$
$$x_{2} = -2$$
$$x_{1} = 7$$
$$x_{2} = -2$$
This roots
$$x_{2} = -2$$
$$x_{1} = 7$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-2 + - \frac{1}{10}$$
=
$$- \frac{21}{10}$$
substitute to the expression
$$\left|{x - \frac{5}{2}}\right| > \frac{9}{2}$$
$$\left|{- \frac{5}{2} + - \frac{21}{10}}\right| > \frac{9}{2}$$

23/5 > 9/2

one of the solutions of our inequality is:
$$x < -2$$

 _____           _____          
      \         /
-------ο-------ο-------
       x2      x1

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < -2$$
$$x > 7$$