Answer:I won't be able to vote on answers until the "SE day" ends, so: the right idea is to reduce your inequality to the either of the forms stuff>0 or stuff≥0. Multiplication by −1 could be treated as reflection about the number line. –
J. M. ain't a mathematician
Dec 28 '11 at 23:34
I already found the answer which I wanted to rehash as answer to my own question but I must wait for 5 more hours. Instead I give link on Inequalities by Lawrence Spector which answers it in thorough detail introducing theorem of inequalities and their proofs. –
Sniper Clown
Dec 28 '11 at 23:53
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Dividing by a negative number is the same as dividing by a positive number and then multiplying by −1. Dividing an inequality by a positive number retains the same inequality. But, multiplying by −1 is the same as switching the signs of the numbers on both sides of the inequality, which reverses the inequality:
a<b⟺−a>−b.(1)
You should be able to convince yourself why the above is true by looking at the number line and considering the various cases involved.
Seeing why (1) is true is not too hard.
Here is the hand waving approach I suggested above:
Consider, for example, in (1), the case when a is negative and b is positive. We have a<b. Then −a is positive and −b is negative. Thus, we have −b<−a.
As another case, suppose a and b are both negative with a<b. Switching the signs here makes the resulting numbers both positive with −a>−b (you can see this by drawing the points on the number line and noting that with the given conditions, b is closer to the origin than a):
enter image description here ).
The other cases can be handled similarly.
Step-by-step explanation:
