Extraneous solution from substituting in equations

If we call $A(x)=x^2+x+1$ and $B(x)=x+1+\frac1x$, we can schematize your passages as such: $$A(x)=0\Leftrightarrow \begin{cases}x\ne 0\\ B(x)=0\end{cases}\Leftrightarrow \begin{cases}x\ne 0\\ A(x)=0 \\B(x)=0\end{cases}\stackrel{!!!}\Rightarrow \begin{cases}x\ne 0\\ B(x)-A(x)=0\end{cases}$$

Whereas to preserve equivalence you should have kept $A(x)=0$ in $\begin{cases}x\ne0\\ B(x)-A(x)=0\\ A(x)=0\end{cases}$

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Algebra Precalculus

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