Initial program 29.4
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
- Using strategy
rm Applied *-un-lft-identity29.4
\[\leadsto \frac{x}{x + 1} - \frac{x + 1}{x - \color{blue}{1 \cdot 1}}\]
Applied *-un-lft-identity29.4
\[\leadsto \frac{x}{x + 1} - \frac{x + 1}{\color{blue}{1 \cdot x} - 1 \cdot 1}\]
Applied distribute-lft-out--29.4
\[\leadsto \frac{x}{x + 1} - \frac{x + 1}{\color{blue}{1 \cdot \left(x - 1\right)}}\]
Applied associate-/r*29.4
\[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{\frac{x + 1}{1}}{x - 1}}\]
Applied *-un-lft-identity29.4
\[\leadsto \frac{\color{blue}{1 \cdot x}}{x + 1} - \frac{\frac{x + 1}{1}}{x - 1}\]
Applied associate-/l*29.4
\[\leadsto \color{blue}{\frac{1}{\frac{x + 1}{x}}} - \frac{\frac{x + 1}{1}}{x - 1}\]
Applied frac-sub29.1
\[\leadsto \color{blue}{\frac{1 \cdot \left(x - 1\right) - \frac{x + 1}{x} \cdot \frac{x + 1}{1}}{\frac{x + 1}{x} \cdot \left(x - 1\right)}}\]
Simplified25.4
\[\leadsto \frac{\color{blue}{\left(x - \left(x + 1\right) \cdot \frac{x + 1}{x}\right) - 1}}{\frac{x + 1}{x} \cdot \left(x - 1\right)}\]
Taylor expanded around inf 0.1
\[\leadsto \frac{\color{blue}{\left(-\left(\frac{1}{x} + 2\right)\right)} - 1}{\frac{x + 1}{x} \cdot \left(x - 1\right)}\]
Simplified0.1
\[\leadsto \frac{\color{blue}{\left(\frac{-1}{x} + -2\right)} - 1}{\frac{x + 1}{x} \cdot \left(x - 1\right)}\]
- Using strategy
rm Applied *-un-lft-identity0.1
\[\leadsto \frac{\left(\frac{-1}{x} + -2\right) - 1}{\color{blue}{1 \cdot \left(\frac{x + 1}{x} \cdot \left(x - 1\right)\right)}}\]
Applied associate-/r*0.1
\[\leadsto \color{blue}{\frac{\frac{\left(\frac{-1}{x} + -2\right) - 1}{1}}{\frac{x + 1}{x} \cdot \left(x - 1\right)}}\]
Simplified0.0
\[\leadsto \frac{\frac{\left(\frac{-1}{x} + -2\right) - 1}{1}}{\color{blue}{\left(1 + x\right) - \frac{1 + x}{x}}}\]
Final simplification0.0
\[\leadsto \frac{\left(-2 + \frac{-1}{x}\right) - 1}{\left(x + 1\right) - \frac{x + 1}{x}}\]
