Hi !
Can you help me how to solve inequations of this form:
$\displaystyle \frac{a}{b}<\frac{c}{d}<\frac{e}{f}$
a,b,c,d,e and f are polynomials
Sure. What you've got there are really two inequalities that are logically AND'ed together. That is you're solving the following simultaneous inequalities:
$\displaystyle \frac{a}{b}<\frac{c}{d}$ AND
$\displaystyle \frac{c}{d}<\frac{e}{f}.$
Can you write up a specific example to work on?
Once you have the problem reduced to $\displaystyle \frac{a}{b}< \frac{c}{d}$, where a, b, c, and d are polynomials, you need to realize that "<" can change to ">" only where (1) they are "=" or (2) where at least one of the denominators is 0.
So: first solve the equation $\displaystyle \frac{a}{b}= \frac{c}{d}$ as well as b= 0 and d= 0. Those point separate "<" from ">". You can choose one point in each interval to see whether "<" or ">" is correct in that interval.
Another way is to rewrite the problem as $\displaystyle \frac{a}{b}- \frac{c}{d}= \frac{ac- bd}{bd}< 0$, and then factor ac- bd as well as bd. For each interval between zeros of ac- bd, b, and d, you can count the number of negative factors to see if "> 0 " or "< 0" is correct.