Question

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

Answer 1

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

Explanation:

Find the lowest common factor which is `x(x+1)`

`(x+1)/x = x/(x+1)`

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

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

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

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

Answer 2

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

Explanation:

`"we require the fractions to have a "color(blue)"common denominator"`

`"to obtain this"`

`"multiply the numerator/denominator of "(x+1)/x`
`"by "(x+1)`

`"and multiply numerator/denominator of "x/(x+1)" by "x`

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

`"the fractions now have a common denominator so expand"`
`"and subtract the numerator leaving the denominator"`

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

`=(2x+1)/(x(x+1))`

Answer 3

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

Explanation:

I am not sure what is meant by simplified here, but we can find:

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

`color(white)((x+1)/x - x/(x+1)) = (1+1/x)-(1-1/(x+1))`

`color(white)((x+1)/x - x/(x+1)) = 1/x+1/(x+1)`