Answer
$(-\infty, \infty)$
Work Step by Step
Using the properties of inequality, the given expression, $
7|x+2|+5\gt4
,$ is equivalent to
\begin{array}{l}\require{cancel}
7|x+2|\gt4-5
\\\\
7|x+2|\gt-1
\\\\
|x+2|\gt-\dfrac{1}{7}
.\end{array}
The left side of the inequality is always a nonnegative number. This is always greater than the negative value at the right. Hence, the solution set is $
(-\infty, \infty)
.$
