How to solve this (2x+5)/(x+1) > (x+1)/(x-1)

Question

and graph?

I tried this and got (x+3)(x-2)

and got -3<x>2 which isn't right

the right answer is x<-3 or -1<x<1 or x>2

How would I get this?

I asked this before.

Answer 1

I was sure that I answered this question before, but I can't find it in the postings.
I did find the sheet of paper with the solution on it

Here is what I did ...
(2x+5)/(x+1) > (x+1)/(x-1)
(2x+5)/(x+1) - (x+1)/(x-1) > 0
[2x^2 + 3x - 5 - x^2 - 2x - 1]/[(x+1)(x-1)] > 0
(x^2 + x - 6)/(x^2 - 1) > 0
(x+3)(x-2)/[(x+1)(x-1] > 0

clearly x ≠ ±1
so we have "critical" values at x= -3,-1,+1, and 2

Mark these values on a number line, I then pick any number in each of these 5 regions bordered by the above points. You don't have to get the actual numercal value, just the sign of the answer.

1. let x = -5
(-)(-)/(-)(-) which is > 0 . Good one!

2. let x = -2
(+)(-)/(-)(-) < 0 , no good

3. let x = 0
(+)(-)/(+)(-) > ) , Good one

4. let x = 1.5
(+)(-)/(+)(+) < 0 , no good

5. let x = 10
(+)(+)/(+)(+) > 0 Good one!

so x < -3 OR -1 < x < 1 OR x > 2

Answer 2

but for (-)(-)/(-)(-)
wouldn't it become +

does the answer have to be a negative for it to be true ?

Answer 3

Or would it having to be negative only for polynomial inequalities and for rational inequalities they have t be positive?

Answer 4

It is positive or negative according to the stated inequality.
In your case after I moved the term to the left I was left with > 0, which means positive.
So we were looking for cases where the result is positive

That is why for case 1.
(-)(-)/(-)(-)
which is > 0 , or positive, I said "Good one"