Answer
$\left( -\infty, -\dfrac{12}{5} \right)
\cup
\left( \dfrac{8}{5},\infty \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|5x+2|\gt10
,$ 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}
5x+2\gt10
\\\\\text{OR}\\\\
5x+2\lt-10
.\end{array}
Using the properties of inequality to isolate the variable results to
\begin{array}{l}\require{cancel}
5x+2\gt10
\\\\
5x\gt10-2
\\\\
5x\gt8
\\\\
x\gt\dfrac{8}{5}
\\\\\text{OR}\\\\
5x+2\lt-10
\\\\
5x\lt-10-2
\\\\
5x\lt-12
\\\\
x\lt-\dfrac{12}{5}
.\end{array}
In interval notation, the solution set is $
\left( -\infty, -\dfrac{12}{5} \right)
\cup
\left( \dfrac{8}{5},\infty \right)
.$
The colored graph is the graph of the solution set.