Answer
$x=\left\{ -\dfrac{3}{2}, 2 \right\}
$
Work Step by Step
Using the properties of equality, the given equation, $
\dfrac{2}{2x-1}=\dfrac{x}{3}
,$ is equivalent to
\begin{align*}\require{cancel}
2(3)&=(2x-1)x
&\text{(cross multiply)}
\\
2(3)&=2x(x)-1(x)
&\text{(use Distributive Property)}
\\
6&=2x^2-x
\\
0&=2x^2-x-6
.\end{align*}
Using the factoring of trinomials in the form $ax^2+bx+c,$ the expression
\begin{align*}
2x^2-x-6
\end{align*} has $ac=
2(-6)=-12
$ and $b=
-1
.$
The two numbers with a product of $c$ and a sum of $b$ are $\left\{
-4,3
\right\}.$ Using these $2$ numbers to decompose the middle term of the trinomial expression above results to
\begin{align*}
2x^2-4x+3x-6=0
.\end{align*}
Grouping the first and second terms and the third and fourth terms, the expression above is equivalent to
\begin{align*}
(2x^2-4x)+(3x-6)=0
.\end{align*}
Factoring the $GCF$ in each group results to
\begin{align*}
2x(x-2)+3(x-2)=0
.\end{align*}
Factoring the $GCF=
(x-2)
$ of the entire expression above results to
\begin{align*}
(x-2)(2x+3)=0
.\end{align*}
The factored form of the given equation is $(x-2)(2x+3)=0.$
Equating each factor to zero (Zero Product Property) and then solving for the variable, then
\begin{array}{lcl}
x-2=0 &\text{ OR }& 2x+3=0
\\
x=2 && 2x=-3
\\
&& x=-\dfrac{3}{2}
\end{array}
Hence, the solutions are $
x=\left\{ -\dfrac{3}{2}, 2 \right\}
.$
