Answer
$x=\dfrac{25}{7}$
Work Step by Step
For easier work, it is better to get rid of the fractions.
This can be achieved by multiplying the LCD of $30$ on both sides of the equation to obtain:
$30(\frac{3x}{5} - \frac{x-3}{2}) = 30(\frac{x+2}{3})
\\\frac{90x}{5} - \frac{30(x-3)}{2} =\frac{30(x+2)}{3}
\\18x - 15(x-3) = 10(x+2)
\\18x - 15x +45 = 10x+20
\\3x+45 =10x+20$
Subtract $3x$ and $20$ on both sides to obtain:
$45-20=10x-3x
\\25=7x
\\\frac{25}{7} = \frac{7x}{7}
\\\frac{25}{7}=x$
Check:
$\begin{array}{ccc}
&\dfrac{3(\frac{25}{7})}{5} - \dfrac{\frac{25}{7}-3}{2} &= &\dfrac{\frac{25}{7}+2}{3}
\\&\dfrac{(\frac{75}{7})}{5} - \dfrac{\frac{25}{7}-\frac{21}{7}}{2} &= &\dfrac{\frac{25}{7}+\frac{14}{7}}{3}
\\&\frac{75}{7(5)} - \dfrac{\frac{4}{7}}{2} &= &\dfrac{\frac{39}{7}}{3}
\\&\frac{15}{7} - \frac{4}{7(2)}&= &\frac{39}{7(3)}
\\&\frac{15}{7} - \frac{2}{7} &= &\frac{13}{7}
\\&\frac{13}{7} &= &\frac{13}{7}\end{array}$
