Question

Write the set of values of x satisfying |x − 1| ≤ 3 and |x − 1| ≥ 1.

Answer 1

\[\text{ We have }, \]
\[\left| x - 1 \right| \leq 3 \text{ and } \left| x - 1 \right| \geq 1\]
\[\text{ We know }: \]
\[\left| x - a \right| \leq r \Rightarrow a - r \leq x \leq a + r\]
\[\text{ And }, \left| x - a \right| \geq r \Rightarrow x \leq a - r \text{ or } x \geq a + r\]
\[ \therefore 1 - 3 \leq x \leq 1 + 3 \text{ and } x \leq 1 - 1 \text{ or } x \geq 1 + 1\]
\[ \Rightarrow - 2 \leq x \leq 4 \text{ and } x \leq 0 \text{ or } x \geq 2\]
\[ \Rightarrow x \in [ - 2, 4] \text{ and } x \in ( - \infty , 0] \cup [2, \infty )\]
\[ \Rightarrow x \in \left[ - 2, 0 \right] U \left[ 2, 4 \right]\]