## College Algebra (11th Edition)

$x=-\dfrac{7}{24}$
$\bf{\text{Solution Outline:}}$ 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 $x=-\dfrac{7}{24} .$