Question

How do you solve and write the following in interval notation: `(x + 5)/(x + 9) < 3`?

Answer 1

`x in (-oo, -11) uu (-9, oo)`

Explanation:

Given:

`(x+5)/(x+9) < 3`

I would like first to note what you can't do, which is to simply multiply both sides by `(x+9)`. The problem is that if `x < -9` then `(x+9) < 0`, so you would have to reverse the inequality too, while if `x > -9` then you would leave the inequality as `<`. So you would need to split into cases - messy.

So what can we do?

Noting first that `x != -9`, we can multiply both sides of the inequality by `(x+9)^2` which is positive, to get:

`(x+5)(x+9) < 3(x+9)^2`

which multiplies out to:

`x^2+14x+45 < 3x^2+54x+243`

Then subtracting `x^2+14x+45` from both sides we get:

`0 < 2x^2+40x+198 = 2(x^2+20x+99) = 2(x+9)(x+11)`

The right hand side is a quadratic with positive leading coefficient, which intersects the `x` axis at `x=-11` and `x=-9`.

Therefore it is positive when:

`x < -11" "` or `" "x > -9`

That is:

`x in (-oo, -11) uu (-9, oo)`