<s>[INST] Solving the equation $\frac{2x+3}{x+1} = \frac{2x+2}{x-1}$ I'm a college student and I've got a problem solving this:
$$\frac{2x+3}{x+1} = \frac{2x+2}{x-1}.$$
Since neither the numerators nor the denominators are equal, I figured that $(2x+3)(x-1) = (x+1)(2x+2)$ would provide me with a suitable $x$.
However calculating this I found that $x = -\frac13$ while my answer sheet states that $x = -\frac23$. What am I doing wrong?
 [/INST] There’s nothing wrong with your basic approach, but you seem to have solved the equation $(2x+3)(x-1)=(x+1)(2x+2)$ incorrectly. After multiplying out both sides, you should have $$2x^2+x-3=2x^2+4x+2\;,$$ which simplifies progressively to $$\begin{align*}
x-3&=4x+2\;,\\
-3&=3x+2\;,\\
-5&=3x\;,\text{ and}\\
x&=-\frac53.
\end{align*}$$
Substituting this into the original equation yields $$\begin{align*}
\frac{2\left(-\frac53\right)+3}{-\frac53+1} &\stackrel{?}= \frac{2\left(-\frac53\right)+2}{-\frac53-1}\\
\frac{-\frac13}{-\frac23} &\stackrel{?}= \frac{-\frac43}{-\frac83}\\
\frac12&=\frac12.
\end{align*}$$ 
In other words, both you and the answer sheet are wrong.
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