<s>[INST] 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 $x \in \mathbb{R}$.
 [/INST] Note that $-1\le\sin x\le1$ for all real $x$. This allows you to rewrite each of the $4$ summands without the absolute value bars. For example: $\sin x-3\leq 1-3=-2<0$, so $|\sin x-3|=3-\sin x$; and $\sin x+1\geq -1+1=0$, so $|\sin x+1|=\sin x+1$. Simplify the resulting expression, and then use the fact that $\sin x$ ranges from $-1$ to $1$ (inclusive) to minimize it.
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