[tex]\: -1
1) In this question, to determine when f(x) > g(x) we need to plug into this inequality the given values:
[tex]\begin{gathered} f(x)>g(x) \\ \frac{-x}{4x-1}>\frac{2}{x-9} \\ Take\: the\: LCM\: for\: the\: denominators \\ \frac{-x(x-9)}{(4x-1)(x-9)}>\frac{2(4x-1)}{(4x-1)(x-9)} \\ \frac{-x^2+x+2}{(4x-1)(x-9)}>0 \\ Factor\: the\: numerator \\ \frac{-\mleft(x+1\mright)\mleft(x-2\mright)}{(4x-1)(x-9)}>0 \\ \frac{\left(x+1\right)\left(x-2\right)}{\left(x-9\right)\left(4x-1\right)}<0 \end{gathered}[/tex]
2) Let's identify the valid intervals for this inequality:
[tex]\begin{gathered} x+1<0,x<-1 \\ x-2<0,x<2 \\ x-9<0,x<9 \\ 4x-1<0,x<\frac{1}{4} \\ \\ \end{gathered}[/tex]
So the answer is:
[tex]\begin{gathered} \: -1
