Question

Solve 

\[\left| \frac{2x - 1}{x - 1} \right| > 2\] 

Answer 1

\[\text{ As }, \left| \frac{2x - 1}{x - 1} \right| > 2\]

\[ \Rightarrow \frac{2x - 1}{x - 1} < - 2 \text{ or } \frac{2x - 1}{x - 1} > 2 \left( \text{ As }, \left| x \right| > 2 \Rightarrow x < - 2 \text{ or } x > 2 \right)\]

\[ \Rightarrow \frac{2x - 1}{x - 1} + 2 < 0 \text{ or } \frac{2x - 1}{x - 1} - 2 > 0\]

\[ \Rightarrow \frac{2x - 1 + 2x - 2}{x - 1} < 0 \text{ or } \frac{2x - 1 - 2x + 2}{x - 1} > 0\]

\[ \Rightarrow \frac{4x - 3}{x - 1} < 0 \text{ or } \frac{1}{x - 1} > 0\]

\[ \Rightarrow \frac{4x - 3}{x - 1} < 0 \text{ or } x - 1 > 0\]

\[ \Rightarrow \left[ \left( 4x - 3 > 0 \text{ and } x - 1 < 0 \right) or \left( 4x - 2 < 0 \text{ and } x - 1 > 0 \right) \right] or \left[ x - 1 > 0 \right]\]

\[ \Rightarrow \left[ \left( x > \frac{3}{4} \text{ and } x < 1 \right) or \left( x < \frac{3}{4} \text{ and } x > 1 \right) \right] or \left[ x > 1 \right]\]

\[ \Rightarrow \left[ \left( \frac{3}{4} < x < 1 \right) \text{ or } \phi \right] or \left[ x > 1 \right]\]

\[ \Rightarrow \left[ \frac{3}{4} < x < 1 \right] \text{ or } \left[ x < 1 \right]\]

\[ \Rightarrow \frac{3}{4} < x < 1 \text{ or } x > 1\]

\[ \therefore x \in \left( \frac{3}{4}, 1 \right) \cup \left( 1, \infty \right)\]