How can i add algebraic fractions, such as 1/(1-x) + 2/x

The trick to working out how to do this, is to remember how to add normal fractions! We know 1/8 + 3/8 = 4/8 . The point here is that IF THE DENOMINATORS ARE THE SAME, we can add the numerators. So what about 1/3 + 1/4 = ??? Its not straight away obvious what to do but if we could make the denominators the same, adding the fractions would be easy. We look for numbers in the times tables of both 3 and 4 , for example 12 = 3x4. Why do this?? Well because we know we can write each fraction as (something)/12. For example 1/3 =(4x1)/(4x3) = 4/12 and 1/4 = (3x1)/(3x4) = 3/12. Finally we are ready to add the fractions, 1/3 + 1/4 = 4/12 + 3/12 = 7/12.

Lets apply this thinking to the algebraic fractions. The two denominators are (1-x) and (x) so we can write both fractions as (something)/x(1-x). For example 1/(1-x) = x/x(1-x) and 2/x = 2(1-x)/x(1-x) .... so 1/(1-x) + 2/x = x/x(1-x) + 2(1-x)/x(1-x) = (x + 2 - 2x )/x(1-x) = (2-x)/x(1-x). Done! Again the point is that once both fractions had the same denominator, we could add the numerators.