Calculate: (x+1)/(x)+(x)/(x+1)=(25)/(12)

Expression: $\frac{ x+1 }{ x }+\frac{ x }{ x+1 }=\frac{ 25 }{ 12 }$

Determine the defined range

$\begin{array} { l }\frac{ x+1 }{ x }+\frac{ x }{ x+1 }=\frac{ 25 }{ 12 },& \begin{array} { l }x≠0,& x≠-1\end{array}\end{array}$

To get an equation that is easier to solve, substitute $t$ for $\frac{ x+1 }{ x }$

$t+\frac{ 1 }{ t }=\frac{ 25 }{ 12 }$

Solve the equation for $t$

$\begin{array} { l }t=\frac{ 3 }{ 4 },\\t=\frac{ 4 }{ 3 }\end{array}$

Substitute back $t=\frac{ x+1 }{ x }$

$\begin{array} { l }\frac{ x+1 }{ x }=\frac{ 3 }{ 4 },\\\frac{ x+1 }{ x }=\frac{ 4 }{ 3 }\end{array}$

Solve the equation for $x$

$\begin{array} { l }x=-4,\\\frac{ x+1 }{ x }=\frac{ 4 }{ 3 }\end{array}$

Solve the equation for $x$

$\begin{array} { l }\begin{array} { l }x=-4,\\x=3\end{array},& \begin{array} { l }x≠0,& x≠-1\end{array}\end{array}$

Check if the solution is in the defined range

$\begin{array} { l }x=-4,\\x=3\end{array}$

The equation has $2$ solutions

$\begin{array} { l }x_1=-4,& x_2=3\end{array}$