Answer
$\left( -\infty,\dfrac{14}{17} \right)$
Work Step by Step
Using the properties of inequality, the given inequality, $
\dfrac{3}{4}(x-2)-\dfrac{1}{3}(5-2x)\lt-2
,$ simplifies to
\begin{array}{l}\require{cancel}
12\left[ \dfrac{3}{4}(x-2)-\dfrac{1}{3}(5-2x)\right]\lt(-2)12
\\\\
9(x-2)-4(5-2x)\lt-24
\\\\
9x-18-20+8x\lt-24
\\\\
9x+8x\lt-24+18+20
\\\\
17x\lt14
\\\\
x\lt\dfrac{14}{17}
.\end{array}
Hence, the solution is the interval $
\left( -\infty,\dfrac{14}{17} \right)
.$
