Answer
$x=-\dfrac{7}{24}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\dfrac{2}{x}-\dfrac{4}{3x}=8+\dfrac{3}{x}
,$ multiply both sides by the $LCD.$ Then use the properties of equality to isolate the variable. Finally, do checking and ensure that the denominator does not become $0.$
$\bf{\text{Solution Details:}}$
The $LCD$ of the denominators, $x,3x,1,$ and $x$ is $3x$ since it is the lowest expression which can be divided exactly by all the given denominators.
Multiplying both sides by the $LCD=
3x
,$ the given expression is equivalent to
\begin{array}{l}\require{cancel}
3x\left( \dfrac{2}{x}-\dfrac{4}{3x} \right) =\left(8+\dfrac{3}{x}\right)3x
\\\\
3(2)-1(4)=8(3x)+3(3)
\\\\
6-4=24x+9
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
-24x=9-6+4
\\\\
-24x=7
\\\\
x=\dfrac{7}{-24}
\\\\
x=-\dfrac{7}{24}
.\end{array}
Upon checking, the solution does not produce an expression with a $0$ in the denominator. Hence, the solution is $
x=-\dfrac{7}{24}
.$
