Answer
$a.\quad \left\{\begin{array}{l}
x\neq-3\\
x\neq 2
\end{array}\right. $
$ b.\quad$ Solution set = $\{-8\}$
Work Step by Step
Factor the denominators.
$\displaystyle \frac{3}{x+3}=\frac{5}{2(x+3)}+\frac{1}{x-2}$
$a.$
The denominators in the equation must not be zero:
$\left\{\begin{array}{l}
x+3\neq 0\\
2(x+3)\neq 0\\
x-2\neq 0
\end{array}\right\}\Rightarrow\left\{\begin{array}{l}
x\neq-3\\
x\neq 2
\end{array}\right. \qquad(*)$
$b.$
Multiply both sides with the LCD,$\quad 2(x+3)$
$ 2(x+3) \displaystyle \left[ \frac{3}{x+3} \right]= 2(x+3) \left[\frac{5}{2(x+3)}+\frac{1}{x-2} \right]\quad$ ... distribute and simplify
$ 6(x-2)=5(x-2)+2(x+3)\quad$ ... distribute and simplify
$6x-12=5x-10+2x+6$
$ 6x-12=7x-4\quad$ ... add $-7x+12$ to both sides
$-x=8$
$ x=-8 \quad$ ... satisfies (*), a valid solution.
Solution set = $\{-8\}$
