|x-15|>10 inequation

Given the inequality:
$$\left|{x - 15}\right| > 10$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{x - 15}\right| = 10$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$x - 15 \geq 0$$
or
$$15 \leq x \wedge x < \infty$$
we get the equation
$$\left(x - 15\right) - 10 = 0$$
after simplifying we get
$$x - 25 = 0$$
the solution in this interval:
$$x_{1} = 25$$

2.
$$x - 15 < 0$$
or
$$-\infty < x \wedge x < 15$$
we get the equation
$$\left(15 - x\right) - 10 = 0$$
after simplifying we get
$$5 - x = 0$$
the solution in this interval:
$$x_{2} = 5$$


$$x_{1} = 25$$
$$x_{2} = 5$$
$$x_{1} = 25$$
$$x_{2} = 5$$
This roots
$$x_{2} = 5$$
$$x_{1} = 25$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + 5$$
=
$$\frac{49}{10}$$
substitute to the expression
$$\left|{x - 15}\right| > 10$$
$$\left|{-15 + \frac{49}{10}}\right| > 10$$

101     
--- > 10
 10     

one of the solutions of our inequality is:
$$x < 5$$

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

Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 5$$
$$x > 25$$