Trigonometric inequality solving How to solve this inequality $\left|\dfrac{\cos 2x + 3}{\cos x}\right|\geq 4$ ?
I tried to consider 2 cases:
1) When $\cos 2x \geq 0$ and $0<\cos x<1$
2) $\cos 2x\leq 0$ and $-1 < \cos x < 0$. 
But I think that's wrong.

As $\cos2x=2\cos^2x-1$
$$\frac{\cos2x+3}{\cos x}=\frac{2\cos^2x-1+3}{\cos x}=2\left(\cos x+\frac1{\cos x}\right)$$
If $\displaystyle \cos x>0, \cos x+\frac1{\cos x} \ge 2\sqrt{\cos x\cdot\frac1{\cos x}}=2$ (using A.M.$\ge$ G.M.)
I leave the case $\cos x<0$ for you
