|x^2-7x|>-3 inequation

Given the inequality:
$$\left|{x^{2} - 7 x}\right| > -3$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{x^{2} - 7 x}\right| = -3$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$x^{2} - 7 x \geq 0$$
or
$$\left(7 \leq x \wedge x < \infty\right) \vee \left(x \leq 0 \wedge -\infty < x\right)$$
we get the equation
$$\left(x^{2} - 7 x\right) + 3 = 0$$
after simplifying we get
$$x^{2} - 7 x + 3 = 0$$
the solution in this interval:
$$x_{1} = \frac{7}{2} - \frac{\sqrt{37}}{2}$$
but x1 not in the inequality interval
$$x_{2} = \frac{\sqrt{37}}{2} + \frac{7}{2}$$
but x2 not in the inequality interval

2.
$$x^{2} - 7 x < 0$$
or
$$0 < x \wedge x < 7$$
we get the equation
$$\left(- x^{2} + 7 x\right) + 3 = 0$$
after simplifying we get
$$- x^{2} + 7 x + 3 = 0$$
the solution in this interval:
$$x_{3} = \frac{7}{2} - \frac{\sqrt{61}}{2}$$
but x3 not in the inequality interval
$$x_{4} = \frac{7}{2} + \frac{\sqrt{61}}{2}$$
but x4 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

$$\left|{0^{2} - 0 \cdot 7}\right| > -3$$

0 > -3

so the inequality is always executed