Question

How do you simplify `(x/9 - 1/x) / (1+ 3/x)`?

Answer 1

`(x/9 - 1/x)/(1/1+3/x) = (x-3)/9`

Explanation:

For this particular problem, we'd have to take note of some algebraic properties such as

`a/b - c/d = (ad - bc) / (bd)`

In this case, we follow this property to start simplifying:

`(x/9 - 1/x)/(1/1+3/x) = ((x^2-9)/(9x)) /((x+3)/(x) )`

We can simplify further by multiplying top and bottom (numerator and denominator) by the reciprocal of `(x+3)/x`, namely `x/(x+3)`.

`(x/9 - 1/x)/(1/1+3/x) = (((x^2-9)/(9x)) * (x/(x+3)))/((cancel((x+3)/(x))) * (cancel(x/(x+3)))) = (cancel(x)(x^2-9))/(9cancel(x)(x+3)`

We've now simplified our expression, although we're not done yet.
Note that we can still factor the top (numerator), that is, `x^2 - 9`.
In order to factor this expression, we proceed as follows:

`x^2 - 9 = 0`

`x^2 = 9`

Solving for `x` gives us

`x = ± sqrt(9)`

`x = ± 3`

Since the solutions are `x = 3` and `x = -3`, we now have our factors and can rewrite the numerator in the following way:

`(cancel(x)(x^2-9))/(9cancel(x)(x+3)) = ((x-3)cancel((x+3)))/(cancel((x+3))*9) = (x-3)/9`