Answer
$a\le-\dfrac{10}{3} \text{ or } a\ge\dfrac{2}{3}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Solve the given inequality, $
|3a+4|+2\ge8
,$ by isolating first the absolute value expression. Then use the definition of a greater than absolute value inequality. Finally, graph the solution set.
In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Using the properties of inequality, the given expression is equivalent to
\begin{array}{l}\require{cancel}
|3a+4|+2\ge8
\\\\
|3a+4|\ge8-2
\\\\
|3a+4|\ge6
.\end{array}
Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the given inequality is equivalent to
\begin{array}{l}\require{cancel}
3a+4\ge6
\\\\\text{OR}\\\\
3a+4\le-6
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
3a+4\ge6
\\\\
3a\ge6-4
\\\\
3a\ge2
\\\\
a\ge\dfrac{2}{3}
\\\\\text{OR}\\\\
3a+4\le-6
\\\\
3a\le-6-4
\\\\
3a\le-10
\\\\
a\le-\dfrac{10}{3}
.\end{array}
Hence, the solution set is $
a\le-\dfrac{10}{3} \text{ or } a\ge\dfrac{2}{3}
.$