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?
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Best Answer
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.