|2x-5|<=7 inequation

Given the inequality:
$$\left|{2 x - 5}\right| \leq 7$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{2 x - 5}\right| = 7$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$2 x - 5 \geq 0$$
or
$$\frac{5}{2} \leq x \wedge x < \infty$$
we get the equation
$$\left(2 x - 5\right) - 7 = 0$$
after simplifying we get
$$2 x - 12 = 0$$
the solution in this interval:
$$x_{1} = 6$$

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


$$x_{1} = 6$$
$$x_{2} = -1$$
$$x_{1} = 6$$
$$x_{2} = -1$$
This roots
$$x_{2} = -1$$
$$x_{1} = 6$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-1 + - \frac{1}{10}$$
=
$$- \frac{11}{10}$$
substitute to the expression
$$\left|{2 x - 5}\right| \leq 7$$
$$\left|{-5 + \frac{\left(-11\right) 2}{10}}\right| \leq 7$$

36/5 <= 7

but

36/5 >= 7

Then
$$x \leq -1$$
no execute
one of the solutions of our inequality is:
$$x \geq -1 \wedge x \leq 6$$

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        /     \  
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       x2      x1