If f(x) = -x/4x-1 and g(x) = 2/x-9, algebraically determine when f(x)>g(x)

Question 1

Question 2

$\: -11) In this question, to determine when f(x) > g(x) we need to plug into this inequality the given values:[tex]\begin{gathered} f(x)>g(x) \\ (-x)/(4x-1)>(2)/(x-9) \\ Take\: the\: LCM\: for\: the\: denominators \\ (-x(x-9))/((4x-1)(x-9))>(2(4x-1))/((4x-1)(x-9)) \\ (-x^2+x+2)/((4x-1)(x-9))>0 \\ Factor\: the\: numerator \\ (-\mleft(x+1\mright)\mleft(x-2\mright))/((4x-1)(x-9))>0 \\ (\left(x+1\right)\left(x-2\right))/(\left(x-9\right)\left(4x-1\right))<0 \end{gathered}$

2) Let's identify the valid intervals for this inequality:

$\begin{gathered} x+1<0,x<-1 \\ x-2<0,x<2 \\ x-9<0,x<9 \\ 4x-1<0,x<(1)/(4) \\ \\ \end{gathered}$

So the answer is:

[tex]\begin{gathered} \: -1

If f(x) = -x/4x-1 and g(x) = 2/x-9, algebraically determine when f(x)>g(x)

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