#### Answer

$\left( \dfrac{59}{31},\infty \right)$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
\dfrac{5}{3}(x-2)+\dfrac{2}{5}(x+1)\gt1
,$ use the properties of inequality.
For the interval notation, use a parenthesis for the symbols $\lt$ or $\gt.$ Use a bracket for the symbols $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Multiplying both sides by the $LCD=15,$ the inequality above is equivalent to
\begin{array}{l}\require{cancel}
15\left( \dfrac{5}{3}(x-2)+\dfrac{2}{5}(x+1) \right) \gt 15(1)
\\\\
25(x-2)+6(x+1)\gt 15(1)
.\end{array}
Using the Distributive Property and the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
25(x)+25(-2)+6(x)+6(1)\gt 15(1)
\\\\
25x-50+6x+6\gt 15
\\\\
25x+6x\gt 15+50-6
\\\\
31x\gt 59
\\\\
x\gt \dfrac{59}{31}
.\end{array}
In interval notation, the solution set is $
\left( \dfrac{59}{31},\infty \right)
.$