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?
 A: 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.
A: For those who benefit from imagery, consider this number line:
$<----- (-5) ---- 0 ------ (7) --->$
You can see $7$ is farther to the right than $-5$, so $7 > -5$.
Multiply both of those values by $-1$, and you flip the number line:
$<--- (-7) ------ 0 ---- (5) ----->$
Now you can see $5$ is farther to the right than $-7$, so $5 > -7$, or $-7 < 5$.
You can extend that logic to an equation with variables, like the example in the question:

$-5m>25$

is represented on the number line as follows:
$<------------- 0 ----- (25) ----- (-5m) --->$
Divide both sides by $-5$ to get $m$ by itself. As before, this flips the number line. It also scales the whole thing down by a factor of 5. The result:
$<------- (m) - (-5) - 0 ------------------>$
That number line can be represented as $-5>m$ or, as the question points out:

$m < -5$

A: Imagine two points on a number axis, say $1$ and $3$. Certainly $1$ is to the left of $3$, which we write $$1<3$$ Now, let's multiply both sides by $2$. That means we scale the situation to a twice bigger: $1$ becomes $2$ and $3$ lands at $6$. Of course what was on the left in the pair, is still on the left: $$2<6$$
Multiplying by a negative value, however, is not just scaling, it is also flipping the whole image with respect to $0$ (zero). It is like you have rotated the line $180^\circ$ around a pivot at the origin. That way what was previously on the left side now appears on the right side: $$(-6)<(-2)$$
The values actually get swapped, but we can equivalently keep them on their previous sides and flip the inequality direction instead: $$(-2)>(-6)$$
A: It's just a matter of equation reforming.
If you have $x < y$ this is equivalent to $f(x) < f(y)$ if and only if $f$ is a strictly monotonically increasing function.  If $f$ is a strictly monotonically decreasing function (like multiplying with a negative number is), it flips the inequality.  If the function is only non-decreasing, you get $f(x) \le f(y)$ and cannot get back to the original equation any more.  If $f$ has both increasing and decreasing parts (like $f(x)=x^2$), you need to split the (in)equality into monotonically increasing and decreasing parts and combine the results.
A: 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.
A: It is just the fact that the product of a negative number by a positive is negative.
You have $25-5m>0$. If you multiply by the negative number $-1/5$, the result is negative:
$$
(-1/5)(25-5m)<0.
$$
After expanding,
$$
m-5 <0.
$$
A: One way to look at this problem, is to change the problem to one with a positive coefficient.
\begin{array}{lrcl}
   \text{We start with : }                    & -5m &\gt &25\\
   \text{Add $5m$ to both sides : }           & 0   &\gt &5m + 25\\
   \text{Add $-25$ to both sides : }          & -25 &\gt &5m\\
   \text{Make $5m$ the subject : }            & 5m  &\lt &-25\\
   \text{Multiply both sides by $\frac 15$ :} & m   &\gt &-5\\
\end{array}
Once you have worked out the logic of this solution, it makes more sense that the direction of the inequality changes when you multiply both sides by a negative number.
