To rewrite the inequality:
[tex]\lvert{x-8}\rvert\ge4[/tex]
we need to remember that:
[tex]\lvert{x}\rvert\ge a\text{ is equivalent to }x\ge a\text{ or }x\leq-a[/tex]
Then in this case we have:
[tex]\begin{gathered} \lvert{x-8}\rvert\ge4 \\ \text{ Is equivalent to:} \\ x-8\ge4\text{ or }x-8\leq-4 \end{gathered}[/tex]
Therefore, we can rewrite the inequality as:
[tex]x-8\leq-4\text{ or }x-8\ge4[/tex]
Once we have it written in this form we can solve it:
[tex]\begin{gathered} x-8\leq-4\text{ or }x-8\ge4 \\ x\leq-4+8\text{ or }x\ge4+8 \\ x\leq4\text{ or }x\ge12 \end{gathered}[/tex]
Therefore, the solution set of the inequality is:
[tex](-\infty,4\rbrack\cup\lbrack12,\infty)[/tex]
