The inequality is given to be:
[tex]\mleft|x-6\mright|\ge2[/tex]
Recall the Absolute Rule:
[tex]\mathrm{If}\: |u|\: \ge\: a,\: a\: >\: 0\: \mathrm{then}\: u\: \le\: -a\: \quad \mathrm{or}\quad \: u\: \ge\: a[/tex]
Applying this to the inequality, we have:
[tex]\begin{gathered} x-6\ge2 \\ x\ge2+6 \\ x\ge8 \end{gathered}[/tex]
or
[tex]\begin{gathered} x-6\le-2 \\ x\le-2+6 \\ x\le4 \end{gathered}[/tex]
Therefore, the solution to the inequality is:
[tex]x\le\: 4\quad \mathrm{or}\quad \: x\ge\: 8[/tex]
