Answer
$\left( -\dfrac{27}{19},\infty \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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)
.$
