<s>[INST] Solution for $\frac{3}{x-9} \gt \frac{2}{x+2}$ What I did:
$\frac{3}{x-9} \gt \frac{2}{x+2}$
$3(x+2) \gt 2(x-9)$
$3x+6 \gt 2x-18$
$x \gt -24$
When typing this in in symbolab, it showed me that the solution is $-24 \lt x \lt -2$ or $x \gt 9$
What did i do wrong ?
How come i didnt get the correct solution ?
 [/INST] HINT
The first step
$$\frac{3}{x-9} \gt \frac{2}{x+2}\iff 3(x+2) \gt 2(x-9)$$
is wrong since $x-9$ and $x+2$ are not both positive (or both negative) in general.
To solve properly use that
$$\frac{3}{x-9} \gt \frac{2}{x+2}\iff \frac{3}{x-9} - \frac{2}{x+2}>0\iff\frac{3(x+2)-2(x-9)}{(x-9)(x+2)}>0$$$$\iff\frac{x+24}{(x-9)(x+2)}>0$$
and study separetely the sign for numerator and denominator.
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