Question

How do you add and simplify `(1)/(1+x) + (1-x)/x`?

Answer 1

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

Explanation:

When fractions are adding, we can't combine them unless they have a common denominator.

To make `1/(1+x)+(1-x)/x` have the same denominator, we need to "give" `1/(1+x)` an `x` and `(1-x)/x` an `(1+x)`.

We'll "give" the fractions those denominators by multiplying both top and bottom that value. We do that because `x/x` and `(1-x)/(1-x)` are both equal to `1`, and we can multiply anything by `1` without changing it's actual value.

`x/x*1/(1+x)+(1-x)/x*(1+x)/(1+x)`
`x/(x^2+x)+(1-x^2)/(x^2+x)`
`(x+1-x^2)/(x^2+x)`
`(-x^2+x+1)/(x^2+x)`
Lets factor out that negative
`(-1(x^2-x-1))/(x^2+x)`
An now we can factor the denominator
`(-1(x^2-x-1))/((x)(x+1))`
`x^2-x-1` can't be factored, so this is as simplified as we can get

Answer 2

Enlarge to reach a common base first. Then you will get `(1-x)/x`

Explanation:

Your equation:
`(1/(1+x))+((1-x)/x)`
will be multiplied with x (for the first half) and (1+x) for the second half:

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

Now add terms:

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

You can simplify:

`[(-x+1)*(x+1)]/(x(x+1))`

Simplfying `(x+1)`s will provide your solution:

`(-x+1)/x` or `-1+1/x`