Add 4 to both sides:
[tex]\begin{gathered} 3|x\text{ + 6| - 4 +4 }\ge\text{ 8 + 4} \\ 3|x\text{ + 6| }\ge\text{ 1}2 \end{gathered}[/tex]Divide both sides by 3:
[tex]\begin{gathered} \frac{3|x\text{ + 6|}}{3}\text{ }\ge\text{ }\frac{\text{1}2}{3} \\ |x\text{ + 6| }\ge\text{ 4} \\ |x\text{ + 6| = (x + 6) or -(x + 6)} \end{gathered}[/tex][tex]\begin{gathered} \text{ x + 6 }\ge\text{ 4} \\ x\text{ }\ge\text{ 4 - 6} \\ x\text{ }\ge\text{ -2} \\ OR \\ \text{-(x + 6) }\ge\text{ 4} \\ x\text{ + 6 }\le\text{ -4} \\ x\text{ }\le\text{ - 4 - 6} \\ x\text{ }\le\text{ -10} \end{gathered}[/tex]