Answer
$\left( -\infty,\dfrac{31}{5} \right] \cup \left[ \dfrac{39}{5},\infty \right)$
Work Step by Step
Using the properties of inequality, the given statement, $
|0.5x-3.5|+0.2\ge0.6
,$ is equivalent to
\begin{array}{l}\require{cancel}
|0.5x-3.5|\ge0.6-0.2
\\\\
|0.5x-3.5|\ge0.4
.\end{array}
Since for any $a\gt0$, $|x|\gt a$ implies $x\gt a$ OR $x\lt-a$, then the inequality above is equivalent to
\begin{array}{l}\require{cancel}
0.5x-3.5\ge0.4 \text{ OR } 0.5x-3.5\le-0.4
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
0.5x-3.5\ge0.4
\\\\
10(0.5x-3.5)\ge(0.4)10
\\\\
5x-35\ge4
\\\\
5x\ge4+35
\\\\
5x\ge39
\\\\
x\ge\dfrac{39}{5}
\\\\\text{ OR }\\\\
0.5x-3.5\le-0.4
\\\\
10(0.5x-3.5)\le(-0.4)10
\\\\
5x-35\le-4
\\\\
5x\le-4+35
\\\\
5x\le31
\\\\
x\le\dfrac{31}{5}
.\end{array}
Hence, the solution set is the interval $
\left( -\infty,\dfrac{31}{5} \right] \cup \left[ \dfrac{39}{5},\infty \right)
.$
