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I moved one part of the inequality to the other to create the following:

$\frac{2|x-1|}{x}-3-|x-9|>0$

Eventually, I get to a case where I have 2 inequalities after opening 1 of the absolute values, but then get confused on what do I do with the second one. Am I supposed to get 4 inequalities? Isn't there a simpler method which I am missing?

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Unfortunately, there is no easier way than splitting this into different cases. You can restrict yourself to the cases $0 < x \leq 1$, $1 < x < 9$ and $x \geq 9$. However, two of these cases are immediately impossible, so you only end up with one case. Can you finish?

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