Answer
$\left( -\infty,-7 \right)\cup\left( -7,\infty \right)$
Work Step by Step
Solving the inequalities separately, then,
\begin{array}{l}
-x\lt7\\
x\gt-7
,\\\\\text{OR}\\\\
3x+1\lt-20\\
3x\lt-21\\
x\lt-7
.\end{array}
Using the conjunction "OR", the solution set is the combined solution of both inequalities. Hence, in interval notation form, the solution is $
\left( -\infty,-7 \right)\cup\left( -7,\infty \right)
$.