Solve for: (\frac{8)/(9)-3}{(4)/(6)}/(3)/(2)+(3)/(4)

Expression: $\frac{ \frac{ 8 }{ 9 }-3 }{ \frac{ 4 }{ 6 } }\div\frac{ 3 }{ 2 }+\frac{ 3 }{ 4 }$

Calculate the difference

$\frac{ -\frac{ 19 }{ 9 } }{ \frac{ 4 }{ 6 } }\div\frac{ 3 }{ 2 }+\frac{ 3 }{ 4 }$

To divide by a fraction, multiply by the reciprocal of that fraction

$\frac{ -\frac{ 19 }{ 9 } }{ \frac{ 4 }{ 6 } } \times \frac{ 2 }{ 3 }+\frac{ 3 }{ 4 }$

Use $\frac{ -a }{ b }=\frac{ a }{ -b }=-\frac{ a }{ b }$ to rewrite the fraction

$-\frac{ \frac{ 19 }{ 9 } }{ \frac{ 4 }{ 6 } } \times \frac{ 2 }{ 3 }+\frac{ 3 }{ 4 }$

Simplify the complex fraction

$-\frac{ 19 }{ 6 } \times \frac{ 2 }{ 3 }+\frac{ 3 }{ 4 }$

Cancel out the greatest common factor $2$

$-\frac{ 19 }{ 3 } \times \frac{ 1 }{ 3 }+\frac{ 3 }{ 4 }$

Multiply the fractions

$-\frac{ 19 }{ 9 }+\frac{ 3 }{ 4 }$

Calculate the sum

$\begin{align*}&-\frac{ 49 }{ 36 } \\&\begin{array} { l }\begin{array} { l }-1 \frac{ 13 }{ 36 },& -1.36\overset{ \cdot }{ 1 } \end{array},& -{\left( \frac{ 7 }{ 6 } \right)}^{2}\end{array}\end{align*}$