Unable to figure out why my approach gives the wrong answer to this Power Plus question.

Q: x > 1 : Quantitative Comparison Questions

Ans:

Cross multiply so we get

x(1-x) *#*#*#*# -x(x+1)

x-x^2 *#*#*#*# -x^2 -x

Add x^2 on both sides

x *#*#*#*# -x

The answer is A as it is just "X’ and x > 1

however, the correct answer is B. I was able to understand the other approach in the comments. But, I want to know why my approach is wrong.