<s>[INST] Inequalities with absolute values My question is:

Show that for all $|x-1|+|x-2|+\dots+|x-10| > 23$

I have solved above problem as below,
If $x-1 > 0$ and $x-2 > 0$ and ......$x-10 > 0$ then
LHS $= x-1+x-2+x-3+\dots+x-10 = 10x-55 > 23$ (because $x>10$)
If $x-1<0$ and $x-2 < 0$ and ......$x-10 < 0$ then
LHS $= -x+1-x+2-x+3.....-x+10 = -10x+55 > 23 $ (because $x<1$)
If the above solution is wrong, please give me the correct method
 [/INST] Just use triangle inequality:
$$|x-1|+|x-2|+\dots+|x-10| = |1-x|+|2-x|+\dots+|5-x|+ |x-6|+|x-7|\dots+|x-10| >$$ $$|1-x+2-x+...+x-10| = |15-6-7-8-9-10| = |-25|$$
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