#### Answer

$x=-\dfrac{7}{3}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|3x+7|\le0
,$ use the definition of absolute value to analyze the given inequality.
$\bf{\text{Solution Details:}}$
The absolute value of $x$, given by $|x|,$ is the distance of $x$ from $0,$ and hence is always a nonnegative number. Therefore, for any $x,$ the expression at the left, $
|3x+7|
,$ is nonnegative. The given inequality is satisfied only when $3x+7=0
.$ Using the properties of equality, then
\begin{array}{l}\require{cancel}
3x+7=0
\\\\
3x=-7
\\\\
x=-\dfrac{7}{3}
.\end{array}