## Intermediate Algebra: Connecting Concepts through Application

$-\dfrac{5}{2}\lt x \lt-\dfrac{3}{2}$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $4-5|4x+8|\gt-6 ,$ isolate first the absolute value expression. Then use the definition of absolute value inequality. Use the properties of inequality to isolate the variable. Finally, graph the solution set. In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$ $\bf{\text{Solution Details:}}$ Using the properties of inequality, the given is equivalent to \begin{array}{l}\require{cancel} 4-5|4x+8|\gt-6 \\\\ -5|4x+8|\gt-6-4 \\\\ -5|4x+8|\gt-10 .\end{array} Dividing both sides by a negative number (and consequently reversing the inequality symbol) results to \begin{array}{l}\require{cancel} -5|4x+8|\gt-10 \\\\ |4x+8|\lt\dfrac{-10}{-5} \\\\ |4x+8|\lt2 .\end{array} Since for any $c\gt0$, $|x|\lt c$ implies $-c\lt x\lt c$ (or $|x|\le c$ implies $-c\le x\le c$), the inequality above is equivalent to \begin{array}{l}\require{cancel} -2\lt 4x+8 \lt2 .\end{array} Using the properties of inequality to isolate the variable results to \begin{array}{l}\require{cancel} -2\lt 4x+8 \lt2 \\\\ -2-8\lt 4x+8-8 \lt2-8 \\\\ -10\lt 4x \lt-6 \\\\ -\dfrac{10}{4}\lt \dfrac{4x}{4} \lt-\dfrac{6}{4} \\\\ -\dfrac{5}{2}\lt x \lt-\dfrac{3}{2} .\end{array} Upon checking, any value of the variable in the solution set satisfies the original inequality and any value not in the solution set does not satisfy the original inequality. Hence, the solution set is $-\dfrac{5}{2}\lt x \lt-\dfrac{3}{2} .$