<s>[INST] Domain of $\frac{x^2-3x-10}{x+2}\div\frac{x^2-25}{x}$ I saw a question asked to simplify
$\dfrac{x^2-3x-10}{x+2}\div\dfrac{x^2-25}{x}$
It is not so hard to do :   $$\dfrac{(x-5)(x+2)}{x+2}\times\dfrac{x}{(x-5)(x+5)}=\dfrac x{x+5}$$
But my question is: should we have necessary all the conditions $x\ne-2$ , $x\ne5$ , $x\ne-5$ and also $x\ne0$ (because in former fraction we had $x$ in denominator) ? or we should have some of them?
 [/INST] Yes, you have to state all of the conditions $x\neq2$, $x\neq5$, $x\neq-5$ and $x\neq0$. This is because when we write
$$
\frac{x^2-3x-10}{x+2} \div \frac{x^2-25}{x}=\frac{x}{x+5} \, ,
$$
we are asserting that for every value of $x$ for which the LHS and RHS are defined, the LHS represents the same number as the RHS. When $x=5$, for instance, the LHS does not make sense, even though the RHS does. Hence, equality does not hold.
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