Question 247175
There are many problems with the way you present this problem.
It is not an equation as you wrote it. Nothing is set equal to it. It is an expression.
let's begin by simplifying
x-7+5=x-2 and x-3+1=x-2
Should I assume that (x-1) is the denominator and not just x?

(x-2)/(x-1))/((x-2)/(x-1)) equals one because not only do (x-1) cancel out but also ((x-2) and part of the denominator (x-2)
Notice the difference
((x-2)/(x-1))/(x-2)/(x-1)

here the numerator (x-2) and part of the denominator (x-2) cancel out leaving  1/(x-1)^2

It is hard to tell what your expression was meant to be.