$\begin{array}{*{20}{l}} {A = \left\{ {x \in R|\dfrac{1}{{\left| {x - 1} \right|}} > 2} \right\}}\\ {{\rm{ \;}} + \dfrac{1}{{\left| {x - 2} \right|}} > 2 \Leftrightarrow \left| {x - 2} \right| < \dfrac{1}{2}}\\ {{\rm{ \;}} \Leftrightarrow {\rm{ \;}} - \dfrac{1}{2} < x - 2 < \dfrac{1}{2}}\\ {{\rm{ \;}} \Leftrightarrow \dfrac{3}{2} < x < \dfrac{5}{2}}\\ {{\rm{ \;}} \Rightarrow A = \left( {\dfrac{3}{2};\dfrac{5}{2}} \right)}\\ {B = \left\{ {x \in R|\left| {x - 1} \right| < 1} \right\}}\\ {{\rm{ \;}} + \left| {x - 1} \right| < 1 \Leftrightarrow {\rm{ \;}} - 1 < x - 1 < 1 \Leftrightarrow 0 < x < 2}\\ {{\rm{ \;}} \Rightarrow B = \left( {0;2} \right)}\\ { \Rightarrow A \cap B = A = \left( {\dfrac{3}{2};2} \right)}\\ {{\rm{ \;}} \Rightarrow A \cup B = \left( {0;\dfrac{3}{2}} \right] \cup \left( {\dfrac{3}{2};\dfrac{5}{2}} \right)} \end{array}$
