$\begin{array} { l }\frac{ 6 }{ x+1 }-\frac{ 3 }{ 2 }=\frac{ 8 }{ 3\left( x+1 \right) },& x≠-1\end{array}$
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$\begin{align*}&x=\frac{ 11 }{ 9 } \\&\begin{array} { l }x=1 \frac{ 2 }{ 9 },& x=1.\overset{ \cdot }{ 2 } \end{array}\end{align*}$