## Intermediate Algebra: Connecting Concepts through Application

$k\le\dfrac{4416}{17}$
$\bf{\text{Solution Outline:}}$ To solve the given inequality, $1.85+1.34(2.4k-5.7)\ge3.25k-14.62 ,$ 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} 1.85+1.34(2.4k-5.7)\ge3.25k-14.62 \\\\ 1.85+1.34(2.4k)+1.34(-5.7)\ge3.25k-14.62 \\\\ 1.85+3.216k-7.638\ge3.25k-14.62 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} 1.85+3.216k-7.638\ge3.25k-14.62 \\\\ 3.216k-3.25k\ge-14.62-1.85+7.638 \\\\ -0.034k\ge-8.832 \\\\ 1000(-0.034)k\ge1000(-8.832) \\\\ -34k\ge-8832 .\end{array} Dividing both sides by a negative number (and consequently reversing the inequality symbol) results to \begin{array}{l}\require{cancel} -34k\ge-8832 \\\\ \dfrac{-34k}{-34}\le\dfrac{-8832}{{-34}} \\\\ k\le\dfrac{\cancel{2}(4416)}{\cancel{2}(17)} \\\\ k\le\dfrac{4416}{17} .\end{array}