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

$\left( -\dfrac{27}{19},\infty \right)$
$\bf{\text{Solution Outline:}}$ Use the order of operations and the properties of inequality to solve the given inequality, $6[4-2(6+3t)]\gt5[3(7-t)-4(8+2t)]-20 .$ $\bf{\text{Solution Details:}}$ Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the inequality above is equivalent to \begin{array}{l}\require{cancel} 6[4-2(6+3t)]\gt5[3(7-t)-4(8+2t)]-20 \\\\ 6[4-2(6)-2(3t)]\gt5[3(7)+3(-t)-4(8)-4(2t)]-20 \\\\ 6[4-12-6t]\gt5[21-3t-32-8t]-20 \\\\ 6[-8-6t]\gt5[-11-11t]-20 \\\\ 6[-8]+6[-6t]\gt5[-11]+5[-11t]-20 \\\\ -48-36t\gt-55-55t-20 \\\\ -48-36t\gt-75-55t .\end{array} Using the properties of inequality, the given is equivalent to \begin{array}{l}\require{cancel} -48-36t\gt-75-55t \\\\ -36t+55t\gt-75+48 \\\\ 19t\gt-27 \\\\ t\gt-\dfrac{27}{19} .\end{array} Hence, the solution set is $\left( -\dfrac{27}{19},\infty \right) .$