Simplify this expression:

Question

(3/y-1) - (3/y²-1)

 

The answers can be any of the following (I am prepping for a placement exam at a university):

 

A) 3y/(y+1)(y-1)

 

B) 5y/(y+1)(y-1)

 

C) 3y/(y+2)(y-1)

 

D) 3y/(y+1)(y+2)

 

E) 4y/(y+1)(y-1)

Answer 1

3/(y-1) - 3/(y²-1)

 

You need to recognize y²-1 as a difference of squares (y²-1²).  It factors to (y+1)(y-1):

 

3/(y-1) - 3/((y-1)(y+1))

 

To add or subtract fractions, you need a common denominator, which is (y+1)(y-1).  Put the first fraction over the common denominator by multiplying it by (y+1)/(y+1):

 

3(y+1)/((y-1)(y+1)) - 3/((y-1)(y+1))

(3(y+1) - 3)/((y-1)(y+1))

(3y - 3 + 3)/((y-1)(y+1))

3y/((y-1)(y+1))