Answer
$\left(-\infty,\dfrac{5}{6} \right]$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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]
.$
