About range inequalities

Question

I'm confused to the range of the inequality question. Let say there is a question about a line intersecting a curve to which I managed to derive to the discriminant as below

     m^2 + 6m + 5 > 0
     (m+5)(m+1)>0

My answer would be m>-5 and m>-1, but the solution states otherwise m<-5 and m>-1...

Another question is to find the range of x such that the area of rectangle is greater than 27cm^2, which I have the inequality as below

    12x - x^2 > 27
    x^2-12x+27 < 0
    (x-3)(x-9) < 0

My answer would be x<3 and x<9...

So my question is why the inequality for the first question is not m>-5 and m>-1...How do you know which greater or lesser symbol should be facing?

Answer 1

We have that

$$(x-3)(x-9)<0 \iff \left(x-3<0 \quad\land \quad x-9>0\right) \lor \left(x-3>0 \quad\land\quad x-9<0\right) $$

and since the first one is impossible, the solution is

$$x-3>0 \quad \land \quad x-9<0$$

that is $3<x<9$.

Answer 2

Use the interval's method.

Take on $m$-axis numbers $-5$ and $-1$.

We got three intervals: $m<-5$, $-5<m<-1$ and $m>-1$.

The right and the left intervals give the answer: $$(-\infty,-5)\cup(-1,+\infty).$$ By the same way $$(x-3)(x-9)<0$$ gives $$3<x<9.$$