Simplifying 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

I'll do #1 and then leave you to do the others.
1 / x + 3 = 1 / x + 3x / x = (1 + 3x) / x
1 / x - 5 = 1 / x - 5x / x = (1 - 5x) / x
Therefore original = (1 + 3x) / x divided by (1 - 5x) / x
= (1 + 3x) / x * x / (1 - 5x) = (1 + 3x) / (1 - 5x)

The second also involves common denominator. the third involves factorising x^2 - y^2.

Answer 2

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

2_((2/y)+(1/2y))/(y+y/2)
=(4y+y/2y^2)/((2y+1)/2)
=2y+1((5y)/y^2)
3_(x/(x-y))/((x^2)/(x^2-y^2)
since x^2-y^2=( x+y)(x-y)
by formula
thenthe above expression becomes aas
x/x-y * x^2-y^2/x^2
=(x+y)(x-y)/x-y = x+y