Questioning Inequalities Why do you reverse inequality signs when you multiply or divide by a negative number? Why is it only with negative numbers and only multiplication and division?
 A: Let $$a<b$$
Subtract $b$ from both sides and you get $$a-b<0$$
Subtract $a$ from both sides of this new inequality and you get $$ -b<-a$$
Therefore $$a<b \implies -b<-a$$
Similarly you can show that   $$ -b<-a \implies a<b$$ 
Therefore multiplying by negatives change the orientation.
When you divide by a negative number it is the same as multiplying by the inverse of the number which is also negative so you have to change the orientation.
A: Rather than multiply by a negative instead consider that if we have $a > b$ then by adding $-a-b$ to both sides we see that $-b > -a$. This leads immediately the rule you've given.
A: I will add an answer which might be able to help you visualize the situation geometrically.
Consider a real number line with zero in the center, with the negative values extending towards the left and the positive values extending to the right, as per usual. Every real number can be thought of as a point on this line. 
Then, let $n$ be some real number, which we can treat as a point on this real number line. Multiplying $n$ by $-1$ is equivalent to keeping the distance of $n$ from $0$ the same, but flipping it to over to the other side of $0$. In other words, multiplying $n$ by $-1$ is equivalent to reflecting its position across the vertical line at $0$. 
Hence, multiplying (or dividing) $n$ by any negative number, say $-m$, can be thought of as first reflecting $n$ across the vertical line at $0$, and then appropriately scaling it by the magnitude of the $-m$. 
Now, we return to your question. Suppose we have some inequality, $a<b$, which we know to be true. Then, this can be visualized as $b$ being on the right of $a$ and likewise $a$ being on the left of $b$ on the real number line. For simplicity, let $a$ and $b$ both be positive; that is, they are both to the right of $0$. Then, multiplying both $a$ and $b$ by $-1$, a.k.a multiplying both sides of the inequality by $-1$, is equivalent to reflecting both points across the vertical line at $0$. At this point, since $a$ was originally to the left of $b$, looking at the mirror image, we can see that $-b$ is now to the left of $-a$. In other words, $-b>-a$.
Now, multiplying or dividing both sides of any inequality by any negative number is the same by first multiplying by $-1$ and then scaling by the appropriate magnitude. So we can see that this geometric intuition mandates that we obey the rule of flipping the equality sign in all such cases. 
We can see that such reasoning only applies to multiplication and division. Returning to points on our number line, adding or subtracting a number to both sides of an inequality like $a<b$ is like shifting both $a$ and $b$ in the same direction by the same distance. Hence, there is no reason why the order of the two points would change: if $a$ is to the left of $b$ at the beginning, then shifting both points in any direction by any amount is not going to change that. 
A: Pretend you don't know the rule, but you do know to add and subtract the same amount from both sides. Any time you have something like
$$-x < -10,$$
you can add $x$ to both sides
$$0 < -10 + x,$$
and add 10 to both sides 
$$ 10 < x.$$
So it's not so much that the inequality sign has changed, but that the $x$ and 10 terms are now on opposite sides to where they started. This is, of course, equivalent to the rule you stated.
