Solve the following equation. Show all your work

Question

x/x-2+x-1/x+1=-1

Do you first do (x-2)(x+1)? I need help. I can't seem to get the idea of what to do.

Thanks

Answer 1

Yes, just as when you add 2/3 and 5/7, you need to find the common denominator. In this case, that is indeed (x-2)(x+1)

So, multiplying by (x-2)(x+1), the common factors cancel, and you are left with
x(x+1) + (x-1)(x+2) = -1(x-2)(x+1)
You can now ignore the denominator, because all the terms have the same denominator. The fractions add up if the new numerators add up.

The only thing you have to watch out for is extraneous roots. Because the original fractions had (x-2) and (x+1) in the denominator, those fractions are undefined if x=2 or x = -1. So if one of your solutions is -1 or 2, it must be discarded.

Answer 2

Thank you for helping!