Respuesta :
The given inequality is:
[tex]|x|>2[/tex]Since this is an absolute value inequality, separate the inequality into possible cases using the definition of an absolute value:
[tex]\begin{gathered} \text{case I}\colon x>2,x\geqslant0 \\ \text{case II}\colon\; -x>2,x<0 \end{gathered}[/tex]For the case I, find the intersection:
[tex]x\in(2,+\infty)[/tex]For case II, solve the first inequality:
[tex]\begin{gathered} -x>2\Rightarrow x<-2 \\ (\text{the inequality was divided by -1 and the inequality sign changed)} \end{gathered}[/tex]Hence, the set of inequalities for case II becomes:
[tex]x<-2,x<0[/tex]Find the intersection:
[tex]x\in(-\infty,-2)[/tex]Find the union of the two solutions to get the solution of the inequality:
[tex]x\in(-\infty,-2)\cup(2,+\infty)[/tex]