Question

Solve \[1 \leq \left| x - 2 \right| \leq 3\] 

Answer 1

\[\text{ As }, 1 \leq \left| x - 2 \right| \leq 3\]
\[ \Rightarrow \left| x - 2 \right| \geq 1 \text{ and } \left| x - 2 \right| \leq 3\]
\[ \Rightarrow \left( \left( x - 2 \right) \leq - 1 \text{ or } \left( x - 2 \right) \geq 1 \right) \text{ and } \left( - 3 \leq \left( x - 2 \right) \leq 3 \right) \left( As, \left| x \right| \geq a \Rightarrow x \leq - a or x \geq a; \text{ and } \left| x \right| \leq a \Rightarrow - a \leq x \leq a \right)\]
\[ \Rightarrow \left( x \leq 1 \text{ or } x \geq 3 \right) \text{ and } \left( - 3 + 2 \leq x \leq 3 + 2 \right)\]
\[ \Rightarrow \left( x \leq 1 or x \geq 3 \right) \text{ and } \left( - 1 \leq x \leq 5 \right)\]
\[ \Rightarrow x \in ( - \infty , 1] \cup [3, \infty ) \text{ and } x \in \left[ - 1, 5 \right]\]
\[ \therefore x \in \left[ - 1, 1 \right] \cup \left[ 3, 5 \right]\]