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

$\left(-\infty,\dfrac{5}{6} \right]$
$\bf{\text{Solution Outline:}}$ Use the order of operations and the properties of inequality to solve the given inequality, $27-4[2(4x-3)+7]\ge2[4-2(3-x)]-3 .$ $\bf{\text{Solution Details:}}$ Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the inequality above is equivalent to \begin{array}{l}\require{cancel} 27-4[2(4x-3)+7]\ge2[4-2(3-x)]-3 \\\\ 27-4[2(4x)+2(-3)+7]\ge2[4-2(3)-2(-x)]-3 \\\\ 27-4[8x-6+7]\ge2[4-6+2x]-3 \\\\ 27-4[8x+1]\ge2[-2+2x]-3 \\\\ 27-4[8x]-4[1]\ge2[-2]+2[2x]-3 \\\\ 27-32x-4\ge-4+4x-3 \\\\ 23-32x\ge4x-7 .\end{array} Using the properties of inequality, the given is equivalent to \begin{array}{l}\require{cancel} 23-32x\ge4x-7 \\\\ -32x-4x\ge-7-23 \\\\ -36x\ge-30 .\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} -36x\ge-30 \\\\ x\le\dfrac{-30}{-36} \\\\ x\le\dfrac{\cancel{-6}(5)}{\cancel{-6}(6)} \\\\ x\le\dfrac{5}{6} .\end{array} Hence, the solution set is $\left(-\infty,\dfrac{5}{6} \right] .$