Step-by-Step Solving?

Question

i REALLY need help simplifying this
(1+(1/x))/(1-(1/x^2)) i know the answer is x/(x-1) but i have no idea how to get there

Answer 1

(1+(1/x))/(1-(1/x^2))

First, let's simplify each of the pieces of the fraction:

top:
1+(1/x)
= (x/x) + (1/x)
= (x + 1) / x

bottom:
1-(1/x^2)
= (x^2 / x^2) - (1 / x^2)
= (x^2 - 1) / x^2
= [(x - 1)(x + 1)] / x^2

Now you have two fractions like this:
(a/b) / (c/d)
You can simplify that by multiplying the first by the reciprocal of the second:
(a/b) * (d/c) = (ad) / (bc)

Right now...
a = x + 1
b = x
c = (x - 1)(x + 1)
d = x^2

So...
[(x + 1)(x^2)] / [x(x - 1)(x + 1)]

You can cancel out x(x + 1) from the top and bottom:

You're left with:
x / (x - 1)

Answer 2

(1+(1/x))/(1-(1/x^2))
Look at the numerator first:
1 + 1/x
Put these terms over a common denominator x. That gives:
(x + 1) / x .........(1)
Now look at the denominator:
1 - 1/x^2
Put these terms over a common denominator x^2. That gives:
( x^2 - 1 ) / x^2 .........(2)
The original fraction is (1) / (2). That is equivalent to inverting (2) and then multiplying:
[ (x + 1) x^2 ] / [ x(x^2 - 1) ]
Factorise the x^2 - 1 as the difference of two squares. That gives:
[ (x + 1) x^2 ] / [ x(x - 1)(x + 1) ]
Now cancel factors x and (x + 1) as these both occur in the numerator and in the denominator:
x / (x - 1).

Answer 3

ok, well first you need to find common denominators. So (1)+ (1/x) will turn to (x + 1)/(x). Then (1) - (1/x^2) is (x^2-1)(x^2). Then flip the bottom fraction and multiply by the top. (x+1)(x^2) / (x)(x^2-1). x^2-1 is the same as (x-1)(x+1).

(x+1)(x^2)
(x)(x-1)(x+1)

x+1 cancels. then take an x out of the bottom and top (from x^2 and x). that leaves you with x over x-1!!!

hope it helps

Answer 4

= [ (x + 1) / x ] / [ (x² - 1) / x² ]
= [ (x + 1) / x ] . [ x² / (x - 1).(x + 1) ]
= x / (x - 1)

Answer 5

(1+(1/x)) /(1-(1/x^2))

[(x+1) /x ] / [(x^2-1) /x^2]

[(x+1)/x ] * [x^2 /( x^2-1)]

[(x+1)/x ] * [x^2 / (x+1)(x-1)]

x /(x-1)

Answer 6

replace 1 in the numerator with x/x and 1 in the denominator with x^2/x^2, giving us

((x/x) + 1)/ x
-------------------
(x^2/x^2) - (1/x^2)

which simplifies to:

((x+1)/x)
-------------
(x^2 -1)/ x^2

but by factoring, x^2 -1 = (x+1) (x-1), so we get:

(x+1)/x
----------
(x+1)(x-1)/ x^2

which equals

x^2 (x+1)
-------------
x (x+1) (x-1)

which after cancelling brings us (x/x-1)