## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$x\le-\dfrac{5}{2}$
$\bf{\text{Solution Outline:}}$ Use the Distributive Property and the properties of inequality to solve the given, $-2(x-5)\ge6(x+7)-12 .$ $\bf{\text{Solution Details:}}$ Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the inequality above is equivalent to \begin{array}{l}\require{cancel} -2(x-5)\ge6(x+7)-12 \\\\ -2(x)-2(-5)\ge6(x)+6(7)-12 \\\\ -2x+10\ge6x+42-12 .\end{array} Using the properties of equality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -2x+10\ge6x+42-12 \\\\ -2x-6x\ge42-12-10 \\\\ -8x\ge20 .\end{array} Dividing both sides by a negative number (and consequently reversing the inequality symbol), the inequality above is equivalent to \begin{array}{l}\require{cancel} -8x\ge20 \\\\ x\le\dfrac{20}{-8} \\\\ x\le-\dfrac{5}{2} .\end{array}