Answer
$\left\{ t|t \ge -\dfrac{8}{33} \right\}
\text{ or }
\left[ -\dfrac{8}{33},\infty \right)
$
Work Step by Step
Using the properties of inequality, the given inequality, $
\dfrac{2}{3}t+\dfrac{8}{9}\ge\dfrac{4}{6}-\dfrac{1}{4}t
,$ is equivalent to
\begin{array}{l}\require{cancel}
36\left( \dfrac{2}{3}t+\dfrac{8}{9} \right) \ge \left(\dfrac{4}{6}-\dfrac{1}{4}t \right)36
\\\\
12(2t)+4(8) \ge 4(6)-t(9)
\\\\
24t+32 \ge 24-9t
\\\\
24t+9t \ge 24-32
\\\\
33t \ge -8
\\\\
t \ge -\dfrac{8}{33}
.\end{array}
Hence, the solution is $
\left\{ t|t \ge -\dfrac{8}{33} \right\}
\text{ or }
\left[ -\dfrac{8}{33},\infty \right)
.$
