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