#### Answer

all real numbers

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
-4|x+3|+9\lt20
,$ isolate first the absolute value expression. Then use the definition of absolute value to analyze the solution.
$\bf{\text{Solution Details:}}$
Using the properties of inequality, the given is equivalent to
\begin{array}{l}\require{cancel}
-4|x+3|+9\lt20
\\\\
-4|x+3|\lt20-9
\\\\
-4|x+3|\lt11
.\end{array}
Dividing both sides by a negative number (and consequently reversing the inequality symbol) results to
\begin{array}{l}\require{cancel}
-4|x+3|\lt11
\\\\
\dfrac{-4|x+3|}{-4}\gt\dfrac{11}{-4}
\\\\
|x+3|\gt-\dfrac{11}{4}
.\end{array}
The absolute value of $x,$ written as $|x|,$ is the distance of $x$ from zero. Hence, it is always a nonnegative number. In the same way, the left side of the inequality above is a nonnegative number. This is always $\text{
greater than
}$ the negative number at the right. Hence, the solution is the set of $\text{
all real numbers
.}$