How do you solve algebraic fractions with quadratics?

First you need to remove the fractions from each side,

take the equation: (x+1)/(x+3) = (2x-1)/(3x-1)

now multiply by x+3 to give x+1 = (2x-1)(x+3)/(3x-1)

now multiply by (3x-1) to give (3x-1)(x+1)=(2x-1)(x+3)

Then expand out the brackets

3x² +2x-1 = 2x² +5x-3

as x²  appears on both sides, subtract (2x² +5x-3) from both sides to give x² -3x +2=0  which can now be solved to give values of x

factorise to give (x-2)(x-1)=0

therfore x= 2 or x=1