Appendix A - Appendix A Exercises - Page 1158: 32

$$\left[ { - 10, - 1} \right] \cup \left[ {1,10} \right]$$

$$\eqalign{ & 1 \leqslant \left| x \right| \leqslant 10 \cr & {\text{The solution of the inequality is any number that is a solution }} \cr & {\text{of both of these inequalities}}: \cr & 1 \leqslant \left| x \right|{\text{ and }}\left| x \right| \leqslant 10. \cr & \cr & {\text{Solving }}1 \leqslant \left| x \right|,{\text{use the property }}\left| x \right| \geqslant a \Leftrightarrow x \leqslant - a{\text{ or }}x \geqslant a \cr & x \leqslant - 1{\text{ or }}x \geqslant 1 \cr & {\text{Express in form of intervals}} \cr & \left( { - \infty , - 1} \right] \cup \left[ {1,\infty } \right) \cr & \cr & {\text{Solving }}10 \leqslant \left| x \right|,{\text{use the property }}\left| x \right| \leqslant a \Leftrightarrow - a \leqslant x \leqslant a \cr & - 10 \leqslant x \leqslant 10 \cr & {\text{Express in form of intervals}} \cr & \left[ { - 10,10} \right] \cr & {\text{Intersecting both solutions}} \cr & \left[ {\left( { - \infty , - 1} \right] \cup \left[ {1,\infty } \right)} \right] \cap \left[ { - 10,10} \right] \cr & {\text{We obtain}} \cr & \left[ { - 10, - 1} \right] \cup \left[ {1,10} \right] \cr & {\text{The solution set is }} \cr & - 10 \leqslant x \leqslant - 1{\text{ or }}1 \leqslant x \leqslant 10 \cr} $$