#### Answer

no solution

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inequality, $ |3x-7|+1\lt-2 ,$ use the properties of inequality to isolate the absolute value expression. Then analyze the resulting inequality.
$\bf{\text{Solution Details:}}$
Using the properties of inequality, the statement above is equivalent to \begin{array}{l}\require{cancel} |3x-7|\lt-2-1 \\\\ |3x-7|\lt-3 .\end{array}
The absolute value of $x,$ written as $|x|,$ is the distance of $x$ from $0,$ and hence is always a nonnegative number. In the same way, for any $x$, the expression at the left side is always a nonnegative number. This will never be $\text{ less than }$ the expression at the right. Hence, there is $\text{
no solution
.}$