Why does the sign have to be flipped in this inequality?

We are learning about inequalities. I originally assumed it would be the same as equations, except with a different sign. And so far, it has been - except for this.

Take the simple inequality: $-5m>25$ To solve it, we divide by $-5$ on both sides, as expected. $m>-5$.

But, I have been told that now we have to flip the inequality sign because we divided by a negative (and this also applies to multiplying negatives).

$m<-5$

And this does work. Plug in any value less than $-5$ and it does turn out to be more than 25, but why?

Mathematically, why do we flip the sign here?

Surely you believe that we can add/subtract from inequalities without a problem. I show you why using this.

If you have that $x>y$, then subtract $y$ to get $x-y>0$ and subtract $x$ to get $-y>-x$. That is, multiplying by $-1$ flips the inequality.

The act of multiplying by a positive scalar is to stretch the number line outward from the origin (or shrink inward if the scaling factor is less than one). If one point on the number line is to the left of another, that fact remains true after stretching. Multiplying by a negative not only stretches/shrinks it but also flips it across the origin - think of it as a $180^\circ$ rotation. If you do that to two points, then that will flip what order they were in. If point A was to the left of point B to begin with, then after flipping, point B will be to the left of A afterwards.

In symbols, $a<b\implies ra<rb$ if $r>0$ and $a<b\implies rb<ra$ if $r<0$.

One may prove the above axiomatically, using nothing but addition and subtraction as avid19 says.