Solve for: (6)/(x+1)-(3)/(2)=(8)/(3(x+1))

Expression: $\frac{ 6 }{ x+1 }-\frac{ 3 }{ 2 }=\frac{ 8 }{ 3\left( x+1 \right) }$

Determine the defined range

$\begin{array} { l }\frac{ 6 }{ x+1 }-\frac{ 3 }{ 2 }=\frac{ 8 }{ 3\left( x+1 \right) },& x≠-1\end{array}$

Move the constant to the right-hand side and change its sign

$\frac{ 6 }{ x+1 }=\frac{ 8 }{ 3\left( x+1 \right) }+\frac{ 3 }{ 2 }$

Move the expression to the left-hand side and change its sign

$\frac{ 6 }{ x+1 }-\frac{ 8 }{ 3\left( x+1 \right) }=\frac{ 3 }{ 2 }$

Write all numerators above the least common denominator $3\left( x+1 \right)$

$\frac{ 10 }{ 3\left( x+1 \right) }=\frac{ 3 }{ 2 }$

Simplify the equation using cross-multiplication

$20=9\left( x+1 \right)$

Distribute $9$ through the parentheses

$20=9x+9$

Move the variable to the left-hand side and change its sign

$20-9x=9$

Move the constant to the right-hand side and change its sign

$-9x=9-20$

Calculate the difference

$-9x=-11$

Divide both sides of the equation by $-9$

$\begin{array} { l }x=\frac{ 11 }{ 9 },& x≠-1\end{array}$

Check if the solution is in the defined range

$\begin{align*}&x=\frac{ 11 }{ 9 } \\&\begin{array} { l }x=1 \frac{ 2 }{ 9 },& x=1.\overset{ \cdot }{ 2 } \end{array}\end{align*}$