<s>[INST] Solving $|x^2-2x|+|x-4|>|x^2-3x+4|$ 
How do I solve $|x^2-2x|+|x-4|>|x^2-3x+4|?$

I can see the difference of the two terms on the left gives the term on the right. Now,what should I do? Is there any general method for solving $$|a|+|b|>|a-b|?$$
Thanks for any help!!
 [/INST] Note that $|x^2-2x|+|x-4|=|x^2-2x|+|-x+4|\ge|x^2-3x+4|$ with the equality holds if and only if $x^2-2x$ and $-x+4$ are of the same sign. 
Therefore, $|x^2-2x|+|x-4|>|x^2-3x+4|$ if and only if $x^2-2x$ and $x-4$ are of the same sign. So we have
\begin{align*}
(x^2-2x)(x-4)&>0\\
x(x-2)(x-4)&>0\\
0<x<2 \quad\textrm{or}\quad x&>4
\end{align*}
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