Answer
$\left( -\infty,-6
\right]\cup\left[ 5,\infty \right)
$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|2x+1|-11\ge0
,$ isolate first the absolute value expression. Then use the definition of absolute value inequalities.
$\bf{\text{Solution Details:}}$
Using the properties of inequality, the given inequality is equivalent to
\begin{array}{l}\require{cancel}
|2x+1|\ge0+11
\\\\
|2x+1|\ge11
.\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 inequality above is equivalent to
\begin{array}{l}\require{cancel}
2x+1\ge11
\\\\\text{OR}\\\\
2x+1\le-11
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
2x+1\ge11
\\\\
2x\ge11-1
\\\\
2x\ge10
\\\\
x\ge\dfrac{10}{2}
\\\\
x\ge5
\\\\\text{OR}\\\\
2x+1\le-11
\\\\
2x\le-11-1
\\\\
2x\le-12
\\\\
x\le-\dfrac{12}{2}
\\\\
x\le-6
.\end{array}
Hence, the solution set is the interval $
\left( -\infty,-6
\right]\cup\left[ 5,\infty \right)
.$
