Answer
$x=\left\{ \dfrac{11}{27},\dfrac{25}{27} \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\left| \dfrac{7}{2-3x} \right|-9=0
,$ use the properties of equality to isolate the absolute value expression. Then use the definition of absolute value equality.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
\left| \dfrac{7}{2-3x} \right|=9
.\end{array}
Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{7}{2-3x}=9
\\\\\text{OR}\\\\
\dfrac{7}{2-3x}=-9
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
\dfrac{7}{2-3x}=9
\\\\
(2-3x)\left( \dfrac{7}{2-3x}\right)=(9)(2-3x)
\\\\
1(7)=(9)(2-3x)
\\\\
7=18-27x
\\\\
27x=18-7
\\\\
27x=11
\\\\
x=\dfrac{11}{27}
\\\\\text{OR}\\\\
\dfrac{7}{2-3x}=-9
\\\\
(2-3x)\left( \dfrac{7}{2-3x} \right)=(-9)(2-3x)
\\\\
1(7)=(-9)(2-3x)
\\\\
7=-18+27x
\\\\
-27x=-18-7
\\\\
-27x=-25
\\\\
x=\dfrac{-25}{-27}
\\\\
x=\dfrac{25}{27}
.\end{array}
Hence, the solutions are $
x=\left\{ \dfrac{11}{27},\dfrac{25}{27} \right\}
.$
