## Intermediate Algebra: Connecting Concepts through Application

$\bf{\text{Solution Outline:}}$ 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 $\text{ all real numbers .}$