## Intermediate Algebra: Connecting Concepts through Application

$m \ge-\dfrac{68}{5}$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $\dfrac{1}{4}m+\dfrac{2}{3}(m-5)\le\dfrac{1}{3}(4m+7) ,$ use the Distributive Property and the properties of inequality to isolate the variable. $\bf{\text{Solution Details:}}$ Using the Distributive Property, which is given by $a(b+c)=ab+ac,$ the inequality above is equivalent to \begin{array}{l}\require{cancel} \dfrac{1}{4}m+\dfrac{2}{3}(m-5)\le\dfrac{1}{3}(4m+7) \\\\ \dfrac{1}{4}m+\dfrac{2}{3}(m)+\dfrac{2}{3}(-5)\le\dfrac{1}{3}(4m)+\dfrac{1}{3}(7) \\\\ \dfrac{1}{4}m+\dfrac{2}{3}m-\dfrac{10}{3}\le\dfrac{4}{3}m+\dfrac{7}{3} .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} \dfrac{1}{4}m+\dfrac{2}{3}m-\dfrac{10}{3}\le\dfrac{4}{3}m+\dfrac{7}{3} \\\\ 12\left( \dfrac{1}{4}m+\dfrac{2}{3}m-\dfrac{10}{3} \right)\le12\left( \dfrac{4}{3}m+\dfrac{7}{3} \right) \\\\ 3m+8m-40 \le16m+28 \\\\ 3m+8m-16m \le28+40 \\\\ -5m \le68 .\end{array} Dividing both sides by a negative number (and consequently reversing the inequality symbol) results to \begin{array}{l}\require{cancel} -5m \le68 \\\\ m \ge\dfrac{68}{-5} \\\\ m \ge-\dfrac{68}{5} .\end{array}