Given the inequality:
$$\left|{\cos{\left(x \right)}}\right| < - \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{\cos{\left(x \right)}}\right| = - \frac{1}{2}$$
Solve:
Given the equation
$$\left|{\cos{\left(x \right)}}\right| = - \frac{1}{2}$$
transform
$$\left|{\cos{\left(x \right)}}\right| + \frac{1}{2} = 0$$
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
$$\left|{\cos{\left(0 \right)}}\right| < - \frac{1}{2}$$
1 < -1/2
but
1 > -1/2
so the inequality has no solutions