find all values of x which satisfy the equation?

Question

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

if there is more than one solution write them all

Answer 1

The first important rule when finding x for this problem is that the denominator cannot be zero, so x≠1 or 5.

Multiply both sides by (x-1)(x-5):
(x+2)(x-5)+(x-1)(x-5)=(x-2)(x-1)
x^2-3x-10+x^2-6x+5=x^2-3x+2
2x^2-9x-5=x^2-3x+2

Subtract x^2, add 3x, and subtract 2 from both sides:
x^2-6x-7=0

Factor the equation:
(x+1)(x-7)=0
x=-1 and 7

Check:
(-1+2)/(-1-1)+1=(-1-2)/(-1-5)
-1/2+2/2=3/6
1/2=1/2

(7+2)/(7-1)+1=(7-2)/(7-5)
9/6+1=5/2
3/2+2/2=5/2
5/2=5/2

I hope this helps!

Answer 2

(x + 2/x - 1)+1 = x-2/x-5
we put (x-1)/(x-1) instead of 1 , so :
(x + 2/x - 1)+(x-1/x-1) = x-2/x-5
2x+1/x-1 = x-2/x-5
2x^2-9x-5 = x^2 -3x+2
x^2 -6x-7= 0
x has two answers :
x= -1 , x= 7