(|4*x+3|)<-9 inequation

Given the inequality:
$$\left|{4 x + 3}\right| < -9$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{4 x + 3}\right| = -9$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$4 x + 3 \geq 0$$
or
$$- \frac{3}{4} \leq x \wedge x < \infty$$
we get the equation
$$\left(4 x + 3\right) + 9 = 0$$
after simplifying we get
$$4 x + 12 = 0$$
the solution in this interval:
$$x_{1} = -3$$
but x1 not in the inequality interval

2.
$$4 x + 3 < 0$$
or
$$-\infty < x \wedge x < - \frac{3}{4}$$
we get the equation
$$\left(- 4 x - 3\right) + 9 = 0$$
after simplifying we get
$$6 - 4 x = 0$$
the solution in this interval:
$$x_{2} = \frac{3}{2}$$
but x2 not in the inequality interval


This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example

x0 = 0

$$\left|{0 \cdot 4 + 3}\right| < -9$$

3 < -9

but

3 > -9

so the inequality has no solutions