Question

For what x is `1 / ( x + 3 ) > 2/(x-3)`?

Answer 1

`x in (-oo, -9) uu (-3, 3)`

Explanation:

Note that `x = -3` and `x = 3` are not solutions since one or the other side of the inequality is undefined for these values of `x`.

Split the remaining possibilities into cases:

Case `bb (x in (-oo, -3) uu (3, oo))`

`x+3` and `x-3` are both negative or both positive.

In either case `(x+3)(x-3) > 0` so multiply both sides of the inequality by that to get:

`x-3 > 2x+6`

Subtract `x+6` from both sides to get:

`-9 > x`

Combining with the conditions of the case, that gives us the solution:

`x in (-oo, -9)`

Case `bb (x in (-3, 3))`

`x+3 > 0` and `x-3 < 0`, so `(x+3)(x-3) < 0`, so multiply both sides of the inequality by this and reverse the inequality to get:

`x-3 < 2x+6`

Subtract `x+6` from both sides to get:

`-9 < x`

Combining with the conditions of the case, that gives us the solution:

`x in (-3, 3)`

Conclusion

Putting the two cases together, the original inequality holds for all

`x in (-oo, -9) uu (-3, 3)`