|3x-4|+<=5 inequation

Given the inequality:
$$\left|{3 x - 4}\right| \leq 5$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{3 x - 4}\right| = 5$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$3 x - 4 \geq 0$$
or
$$\frac{4}{3} \leq x \wedge x < \infty$$
we get the equation
$$\left(3 x - 4\right) - 5 = 0$$
after simplifying we get
$$3 x - 9 = 0$$
the solution in this interval:
$$x_{1} = 3$$

2.
$$3 x - 4 < 0$$
or
$$-\infty < x \wedge x < \frac{4}{3}$$
we get the equation
$$\left(4 - 3 x\right) - 5 = 0$$
after simplifying we get
$$- 3 x - 1 = 0$$
the solution in this interval:
$$x_{2} = - \frac{1}{3}$$


$$x_{1} = 3$$
$$x_{2} = - \frac{1}{3}$$
$$x_{1} = 3$$
$$x_{2} = - \frac{1}{3}$$
This roots
$$x_{2} = - \frac{1}{3}$$
$$x_{1} = 3$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{3} + - \frac{1}{10}$$
=
$$- \frac{13}{30}$$
substitute to the expression
$$\left|{3 x - 4}\right| \leq 5$$
$$\left|{-4 + \frac{\left(-13\right) 3}{30}}\right| \leq 5$$

53     
-- <= 5
10     

but

53     
-- >= 5
10     

Then
$$x \leq - \frac{1}{3}$$
no execute
one of the solutions of our inequality is:
$$x \geq - \frac{1}{3} \wedge x \leq 3$$

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       x2      x1