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Let's say I have

$$a > b$$

Now we know that the inequality sign will switch if:

  1. We take the reciprocal of both sides

  2. Multiply $-1$ to both sides.

But, are there more? What if we do more "complicated" things to it?

Will it change if I apply inverse trig on it? For instance is it still true then if

$$\arccos(a) > \arccos(b)$$

Note that $a,b \in\mathbb R$.

So what I really want to ask is, are there any other operations that would make you change the inequality signs?

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    Look up the definitions of *increasing* and *decreasing* functions. When you apply an increasing function to both sides of an inequality, the order of numbers stays the same. When you apply a decreasing function, the order is reversed. Care needs to be taken, when the function is increasing/decreasin only on an interval.2012-01-01

2 Answers 2