Minimum of $\left| \sin x- 1\right| + \left|\sin x- 2\right| + \left| \sin x -3\right| + \left| \sin x+1\right|$ I would like to know the minimum value of $$\left| \sin x- 1\right| + \left|\sin x- 2\right| + \left| \sin x -3\right| + \left| \sin x+1\right|$$ 
for $x \in \mathbb{R}$.

$\sin x-1$ is always non-positive.
$\sin x - 2$ is always negative.
$\sin x -3$ is always negative.
$\sin x + 1$ is always non-negative.
Therefore the whole expression reduces to
$$
-(\sin x-1) -(\sin x-2) -(\sin x-3) + (\sin x+1).
$$
(But if you meant $\sin(x-1)$ where you wrote $\sin x-1$, then none of the above will help you.)
You get $7-2\sin x$.  Since the sine oscillates between $1$ and $-1$, the smallest this function ever gets is $7-2=5$.
