Simplify 1 over quantity x minus 3 plus 4 over x all over 4 over x minus 1 over quantity x minus 3

We can begin this problem by creating a common denominator. our fraction is:

(1 / (x-3) + 4/x) / (x-1) / (x-3), giving us the common denominator of x (x-3). We then take each individual fraction in the complex fraction, making it so that each one has the denominator x (x-3). 1 / (x-3) becomes x/x (x-3), 4/x becomes 4 (x-3) / x (x-3), and (x-1) / (x-3) becomes x (x-1) / x (x-3).

Now, our fraction is (x/x (x-3) + 4 (x-3) / x (x-3)) / x (x-1) / x (x-3), or simply ((x+4 (x-3) / x (x-3)) / x (x-1) / x (x-3)). If we multiply the numerator and denominator by x (x-3), we get the fraction (x+4 (x+3)) / (x (x-1)).

Distributing the 4, we get (x+4x+12) / (x (x-1)), or (3x+12) / (x (x-1)). (I don't know if you want the denominator factored or not, but if you want it expanded then it's (3x+12) / (x^2-x).)

I hope this was easy enough to follow! I haven't written anything like this before so I'm sorry if it wasn't very good.