[tex]\frac{x+1}{x-1}\ \textgreater \ \frac{x+3}{x+2} \\ \\
\frac{x+1}{x-1}-\frac{x+3}{x+2}\ \textgreater \ \frac{x+3}{x+2}-\frac{x+3}{x+2} \\ \\
\frac{x+1}{x-1}-\frac{x+3}{x+2}\ \textgreater \ 0 \\ \\
\frac{x+1}{x-1}-\frac{x+3}{x+2} \\ \\
=\frac{\left(x+1\right)\left(x+2\right)}{\left(x+2\right)\left(x-1\right)}-\frac{\left(x+3\right)\left(x-1\right)}{\left(x+2\right)\left(x-1\right)} \\ \\
=\frac{\left(x+1\right)\left(x+2\right)-\left(x+3\right)\left(x-1\right)}{\left(x+2\right)\left(x-1\right)} \\ \\
[/tex][tex]\left(x+1\right)\left(x+2\right)-\left(x+3\right)\left(x-1\right) \\ \\
=x^2+3x+2-\left(x+3\right)\left(x-1\right) \\ \\
=x^2+3x+2-x^2-2x+3 \\ \\
=x^2-x^2+3x-2x+2+3 \\ \\
=x^2-x^2+x+2+3 \\ \\
=x+2+3 \\ \\
=x+5 \\ \\
=\frac{x+5}{\left(x+2\right)\left(x-1\right)} \\ \\
\frac{x+5}{\left(x+2\right)\left(x-1\right)}\ \textgreater \ 0 \\ \\
-5\ \textless \ x\ \textless \ -2\quad \mathrm{sau}\quad \:x\ \textgreater \ 1[/tex]
