Answer
$m \ge-\dfrac{68}{5}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
\dfrac{1}{4}m+\dfrac{2}{3}(m-5)\le\dfrac{1}{3}(4m+7)
,$ use the Distributive Property and the properties of inequality to isolate the variable.
$\bf{\text{Solution Details:}}$
Using the Distributive Property, which is given by $a(b+c)=ab+ac,$ the inequality above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{4}m+\dfrac{2}{3}(m-5)\le\dfrac{1}{3}(4m+7)
\\\\
\dfrac{1}{4}m+\dfrac{2}{3}(m)+\dfrac{2}{3}(-5)\le\dfrac{1}{3}(4m)+\dfrac{1}{3}(7)
\\\\
\dfrac{1}{4}m+\dfrac{2}{3}m-\dfrac{10}{3}\le\dfrac{4}{3}m+\dfrac{7}{3}
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{4}m+\dfrac{2}{3}m-\dfrac{10}{3}\le\dfrac{4}{3}m+\dfrac{7}{3}
\\\\
12\left( \dfrac{1}{4}m+\dfrac{2}{3}m-\dfrac{10}{3} \right)\le12\left( \dfrac{4}{3}m+\dfrac{7}{3} \right)
\\\\
3m+8m-40 \le16m+28
\\\\
3m+8m-16m \le28+40
\\\\
-5m \le68
.\end{array}
Dividing both sides by a negative number (and consequently reversing the inequality symbol) results to
\begin{array}{l}\require{cancel}
-5m \le68
\\\\
m \ge\dfrac{68}{-5}
\\\\
m \ge-\dfrac{68}{5}
.\end{array}