Answer
$\left( -\infty,20 \right] \cup \left[ 30,\infty \right)$
Work Step by Step
Using the properties of inequality, the given statement, $
|0.1x-2.5|+0.3\ge0.8
,$ is equivalent to
\begin{array}{l}\require{cancel}
|0.1x-2.5|\ge0.8-0.3
\\\\
|0.1x-2.5|\ge0.5
.\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}
0.1x-2.5\ge0.5 \text{ OR } 0.1x-2.5\le-0.5
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
0.1x-2.5\ge0.5
\\\\
10(0.1x-2.5)\ge(0.5)10
\\\\
x-25\ge5
\\\\
x\ge5+25
\\\\
x\ge30
\\\\\text{ OR }\\\\
0.1x-2.5\le-0.5
\\\\
10(0.1x-2.5)\le(-0.5)10
\\\\
x-25\le-5
\\\\
x\le-5+25
\\\\
x\le20
.\end{array}
Hence, the solution set is the interval $
\left( -\infty,20 \right] \cup \left[ 30,\infty \right)
.$
