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1.
[tex]\it x\in\mathbb{Z},\ \ \dfrac{11}{|2x-1|}\in \mathbb{Z} \Rightarrow |2x-1|\in\{1,\ \ 11\} \Rightarrow 2x-1\in\{\pm1,\ \ \pm11\} \Rightarrow \\ \\ \\ \Rightarrow 2x-1\in\{-11,\ \ -1,\ \ 1,\ \ 11\}\Big|_{+1} \Rightarrow 2x\in\{-10,\ \ 0,\ \ 2,\ \ 12\}\Big|_{:2} \Rightarrow \\ \\ \\ \Rightarrow x\in\{-5,\ \ 0,\ \ 1,\ \ 6\}[/tex]