Rational Expressions...?

Question

Can anyone help me with simplifying these problems?

1) ((1/x)+3)
-----------
((1/x)-5)

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

3) (x/(x-y))
----------------
((x^2)/(x^2-y^2))

Thanks for any help

Answer 1

You have to come up with common denominators.
1) (1/x) + 3 = (1/x) + (3x/x) = (1+ 3x)/x
Similarly, (1/x) - 5 = (1 - 5x)/x
Remembering that the fraction bar is the same as a division sign, our problem now is
(1+3x)/x divided by (1-5x)/x
Invert the second fraction and multiply
(1+3x)/x * x/(1-5x)
the x on the bottom of the first fraction and the x on the top of the second fraction simplify to one or "cancel out", and the fraction simplifies to (1+3x)/(1-5x)

2) (2/y) + (1/2y) = (4/2y) + (1/2y) = 5/2y
y + (y/2) = (2y/2) + (y/2) = 3y/2

(5/2y)/(3y/2) = (5/2y) * (2/3y) = 10/6y = 5/3y

3/[x/(x-y)] / [(x^2)/(x^2 - y^2)
Factor x^2 - y^2 == (x-y)(x+y)
Invert and multiply
[x/(x-y)] * [(x-y)(x+y)/(x + 2)]
cancel ( make a 1) with the (x-y) terms
x * (x+y)/(x+1) = x(x+y)/(x+1)

Hope I didnt' make any mistakes!

Answer 2

Look back to previous time you asked this question. Don't expect answers too quickly.