Answer
$\left( -\infty, -\dfrac{11}{2} \right]
\cup
\left[ 5,\infty \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|4x+1|\ge21
,$ use the definition of absolute value inequalities. Then use the properties of inequality to isolate the variable.
For the interval notation, use a parenthesis for the symbols $\lt$ or $\gt.$ Use a bracket for the symbols $\le$ or $\ge.$
For graphing inequalities, use a hollowed dot for the symbols $\lt$ or $\gt.$ Use a solid dot for the symbols $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
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}
4x+1\ge21
\\\\\text{OR}\\\\
4x+1\le-21
.\end{array}
Using the properties of inequality to isolate the variable results to
\begin{array}{l}\require{cancel}
4x+1\ge21
\\\\
4x\ge21-1
\\\\
4x\ge20
\\\\
x\ge\dfrac{20}{4}
\\\\
x\ge5
\\\\\text{OR}\\\\
4x+1\le-21
\\\\
4x\le-21-1
\\\\
4x\le-22
\\\\
x\le-\dfrac{22}{4}
\\\\
x\le-\dfrac{11}{2}
.\end{array}
In interval notation, the solution set is $
\left( -\infty, -\dfrac{11}{2} \right]
\cup
\left[ 5,\infty \right)
.$
The colored graph is the graph of the solution set.