Does changing a sign in an inequality relate to decreasing functions in any way? Edited (not by original poster): When we multiply both sides on an inequality by a negative number, we change the sign of the inequality (we flip it around). I have been told that this has something to do with functions. More specifically, I think that we can use decreasing functions to show that multiplying both sides of an inequality by a negative number leads to changing the sign of the inequality. Can anybody elaborate or explain to me, in an easy, simple, and clear way, how (or even whether) we can use decreasing functions to show that multiplying both sides of an inequality by a negative number leads to changing the sign?
Original: I got the point of changing the sign, but I also heard that it has something to do with functions. It's like Decreasing Functions causes us to change the sign in an inequality. Can anybody broaden this idea in an easy, simple and clear way?
 A: Suppose $a<0$ and consider the linear function $L(x)=ax.$ The connection you seek can be explained by noting that the graph of $L$ is strictly decreasing, meaning that if $x_1<x_2$ then $L(x_1)>L(x_2).$ So if we have some inequality $u<v$ and multiply both sides by the negative $a$ the above would say that $L(u)>L(v),$ i.e. that $au>av$ and we see the reversal of the inequality.
It also works if we start with $u>v$ since another way to write that is $v<u,$ so using the above we get $av>au,$ which written another way is $au<av.$ we see again that from $u>v$ the direction of the inequality has reversed to $au<av.$
It may be argued this is all redundant, and one does not need the linear function, just remember the rule. However if you draw some examples of negatively sloped lines passing through $(0,0)$ it does give a somewhat geometrical reason why inequality directions get switched on multiplying by a negative.
A: Yes, it does in a way relate to decreasing functions. I'm sure that you know that
$$x = y \implies f(x) = f(y).$$
However, it is not quite as simple for inequalities as it is for equalities. Now assume that $f$ is an increasing function, then we have that
$$x\leq y \implies f(x) \leq f(y). $$
However, if $g$ is a decreasing function then we have
$$x\leq y \implies g(x)\geq g(y),$$
that is we change the direction of the inequality.
A: I think this will be more clearer to you when you look at the graph of a decreasing function,
for simplicity let's take a strictly decreasing straight line function. Its graph would look something like "\"
Now take 2 points: x1 and x2 on the  $x$-axis such that $x_1<x_2$
looking at the graph we see that the $y$-value associated with $x_1$ [a.k.a. $f(x_1)$]is Greater than the $y$-value associated with x2[aka f(x2)]
=> when $x_1<x_2$
$f(x_1)>f(x_2)$
that is, the more your inputs increase in value, the lesser your outputs turn out to be.
I hope this was helpful in some way or the other.
