Solve 2x-1/x-5 > x+1/x+5

Answer 1

Assuming you mean

(2x-1)/(x-5) > (x+1)/(x+5)

if (x+5)(x-5)<0 then

(2x-1)(x+5)< (x+1)(x-5)

2x^2 + 9x - 5 < x^2 -4x - 5

x^2 + 13x < 0

x(x+13) < 0

which has solution per sign table (-13,0)

on the other hand if (x+5)(x-5)>0 then

(2x-1)(x+5)>(x+5)(x-5)

2x^2 + 9x - 5 > x^2 -4x - 5

x^2 + 13x > 0

x(x+13)>0

which has solutions per sign table (-inf,-13) U (0,inf)

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Without the parenthesis:

2x - 1/x - 5 > x + 1/x + 5

Subtracts x:

x - 1/x - 5 > 1/x + 5

adds 1/x:

x - 5 > 2/x + 5

adds 5:

x > 2/x + 10

If x>0 then

x^2 > 2 + 10x

x^2 - 10x - 2 > 0

which per quadratic formula has solutions x<-0.196 and x>10.196

But if x<0 then the solution is (-0.196, 10.196)