Answer
$\left[ -\dfrac{1}{2},\dfrac{3}{2} \right]$
Work Step by Step
Using the properties of inequality, the given expression, $
8-|2x-1|\ge6
,$ is equivalent to
\begin{array}{l}\require{cancel}
-|2x-1|\ge6-8
\\\\
-|2x-1|\ge-2
\\\\
|2x-1|\le2
.\end{array}
For any $a\gt0$, $|x|\lt a$ implies $-a\lt x\lt a$. (The symbol $\lt$ may be replaced with $\le$.)
Using the concept above, the solutions to the given inequality, $
-2\le 2x-1\le2
,$ is
\begin{array}{l}\require{cancel}
-2+1\le 2x\le2+1
\\\\
-1\le 2x\le3
\\\\
-\dfrac{1}{2}\le x\le\dfrac{3}{2}
.\end{array}
In interval notation, the solution set is $
\left[ -\dfrac{1}{2},\dfrac{3}{2} \right]
.$
