Question

How do you solve: [ (x-2) / (x+3) ] < [ (x+1) / (x) ]?

Answer 1

`x in (-1/2, + oo) "\" {0}`

Explanation:

The inequality given to you looks like this

`(x-2)/(x+3) < (x+1)/x`

Right from the start, you know that any solution interval must not contain the values of `x` that will make the two denominators equal to zero.

More specifically, you need to have

`x+3 != 0 implies x != -3" "` and `" "x != 0`

With that in mind, use the common denominator of the two fractions, which is equal to `(x+3) * x`, to get rid of the denominators.

More specifically, multiply the first fraction by `1 = x/x` and the second fraction by `1 = (x+3)/(x+3)`.

This will get you

`(x-2)/(x+3) * x/x < (x+1)/x * (x+3)/(x+3)`

`(x(x-2))/(x(x+3)) < ((x+1)(x+3))/(x(x+3))`

This is equivalent to

`x(x-2) < (x+1)(x+3)`

Expand the parantheses to get

`color(red)(cancel(color(black)(x^2))) - 2x < color(red)(cancel(color(black)(x^2))) + x + 3x + 3`

Rearrange the inequality to isolate `x` on one side

`-6x < 3 implies x > 3/((-6)) <=> x > -1/2`

This means that any value of `x` that is greater than `-1/2`, except `x = 0`, will be a solution to the original inequality.

Therefore, the solution interval will be `x in (-1/2, + oo) "\" {0}`