2*(|x|)-1<13 inequation

Given the inequality:
$$2 \left|{x}\right| - 1 < 13$$
To solve this inequality, we must first solve the corresponding equation:
$$2 \left|{x}\right| - 1 = 13$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$x \geq 0$$
or
$$0 \leq x \wedge x < \infty$$
we get the equation
$$2 x - 14 = 0$$
after simplifying we get
$$2 x - 14 = 0$$
the solution in this interval:
$$x_{1} = 7$$

2.
$$x < 0$$
or
$$-\infty < x \wedge x < 0$$
we get the equation
$$2 \left(- x\right) - 14 = 0$$
after simplifying we get
$$- 2 x - 14 = 0$$
the solution in this interval:
$$x_{2} = -7$$


$$x_{1} = 7$$
$$x_{2} = -7$$
$$x_{1} = 7$$
$$x_{2} = -7$$
This roots
$$x_{2} = -7$$
$$x_{1} = 7$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-7 + - \frac{1}{10}$$
=
$$- \frac{71}{10}$$
substitute to the expression
$$2 \left|{x}\right| - 1 < 13$$
$$-1 + 2 \left|{- \frac{71}{10}}\right| < 13$$

66/5 < 13

but

66/5 > 13

Then
$$x < -7$$
no execute
one of the solutions of our inequality is:
$$x > -7 \wedge x < 7$$

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