Solve for: (x+1)/(x-5)=(x+4)/(x-4)

Expression: $\frac{ x+1 }{ x-5 }=\frac{ x+4 }{ x-4 }$

Determine the defined range

$\begin{array} { l }\frac{ x+1 }{ x-5 }=\frac{ x+4 }{ x-4 },& \begin{array} { l }x≠5,& x≠4\end{array}\end{array}$

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

$x=8$