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

Answer 1

Assuming you meant:
(2x-1)/(x-5) > (x+1)/(x+5)

taking a look where the boundary curves meet:
(2x-1)/(x-5) = (x+1)/(x+5)
2x^2 + 9x - 5 = x^2 - 4x - 5
x^2 + 13x = 0
x(x+13) = 0
so they intersect at x = 0 and x = -13
other critical values are x = -5 and x = 5, we have asymptotes there

so let's look at 5 different segments on the x-axis, picking any value in each segment
1. for x < -13, let x = -15
-31/-20 > -14/-10 ---> 1.55 > 1.4 ? YES <----- winner
2. for -13 < x < -5 , let x = -10
-21/-15 > -9/-5 ---> 1.4 > 1.8 ? NO
3. for -5 < x < 0 , let x = -2
-5/-7 > -1/3 ---> .714.. > -.333, YES <---- another winner
4. for 0 to 5, let x = 2
3/-3 > 3/7, NO
5. for x > 5 , let x = 10
19/5 > 11/15 ---> 3.8 > .733 , YES

so :
x < -13 OR -5 < x < 0 OR x > 5

I graphed the regions using DESMOS and it shows it as correct