$\begin{array} { l }\frac{ x+1 }{ x-5 }=\frac{ x+4 }{ x-4 },& \begin{array} { l }x≠5,& x≠4\end{array}\end{array}$
Simplify the equation using cross-multiplication$\left( x+1 \right) \times \left( x-4 \right)=\left( x+4 \right) \times \left( x-5 \right)$
Simplify the expression${x}^{2}-4x+x-4=\left( x+4 \right) \times \left( x-5 \right)$
Simplify the expression${x}^{2}-4x+x-4={x}^{2}-5x+4x-20$
Cancel equal terms on both sides of the equation$-4x+x-4=-5x+4x-20$
Collect like terms$-3x-4=-5x+4x-20$
Collect like terms$-3x-4=-x-20$
Move the variable to the left-hand side and change its sign$-3x-4+x=-20$
Move the constant to the right-hand side and change its sign$-3x+x=-20+4$
Collect like terms$-2x=-20+4$
Calculate the sum$-2x=-16$
Divide both sides of the equation by $-2$$\begin{array} { l }x=8,& \begin{array} { l }x≠5,& x≠4\end{array}\end{array}$
Check if the solution is in the defined range$x=8$