Inequation: quadratic difference equations Given:
$$\frac{(x - 3)}{(x-4)} > \frac{(x + 4)}{(x + 3)}$$
Step 1:
$$(x + 3)(x - 3) > (x + 4)(x - 4)$$
Step2 : Solving step 1:
$$x^2 - 3^2 > x^2 - 4^2$$
*Step 3: 
$ 0 > -16 + 9$ ???
As you see, I can delete the $x^2$, but there is no point in doing that.
What should be the next step?

I think people ought to draw graphs of things as long as these are simple enough to do so. In this case I drew the two hyperbolas indicated in the question. As in Mark Bennet's answer, we see that the red curve is higher between the two vertical asymptotes, that is $-3 < x < 4.$ However, for $x > 4$ and again for $x < -3,$ it appears the green curve is very slightly higher.
 
Also, we see how the picture suggests that the red and green curves are translates of each other. This turns out to be true: if 
$$ f(x) = \frac{x-3}{x-4}, $$ then
$$ f(x + 7) = \frac{x+ 4}{x+3}. $$
