Answer
$x=-\dfrac{3}{7}$
Work Step by Step
The given inequality, $
|7x+3|\lt0
,$ is equivalent to
\begin{array}{l}\require{cancel}
7x+3=0
.\end{array}
Using the properties of equality, the solution to the equation above is
\begin{array}{l}\require{cancel}
7x=-3
\\\\
x=-\dfrac{3}{7}
.\end{array}