I'm studying inequalities as part of a course on Numbers, Proofs and Mathematical Induction. There is one type of question that I don't understand, primarily because there's only one example in the notes and two questions. I need more examples/questions. Note that I don't need help with these specific questions, but I'd like to know how to do questions like this in general and to be pointed towards some more practice problems:
Use the triangle inequality and other results to find an upper and lower bound for the following functions on the given interval:$$y=\frac{2x^2 + 1}{x+ 3}, |x|<1$$ $$y=\frac{x^3+3x+1}{10-x^2}, |x+1|<2$$
You could use the triangle inequality and the "reverse triangle inequality", $$|a\pm b|\le|a|+|b|\qquad\hbox{and}\qquad |a\pm b|\ge|a|-|b|\ .$$ For example, if $|x|<1$ then $$|2x^2+1|\le 2|x|^2+|1|<3\ ;$$ also $$|x+3|\ge|3|-|x|>2\ ,$$ so $$\Bigl|\frac{1}{x+3}\Bigr|<\frac{1}{2}$$ and therefore $$\Bigl|\frac{2x^2+1}{x+3}\Bigr|<\frac{3}{2}\ .$$ So $\frac{3}{2}$ is an upper bound for this expression when $|x|<1$. Note that any larger number will also be an upper bound, and maybe if we did some more careful work we could find a smaller number which is still an upper bound. This kind of question always has more than one answer.
Another approach would be to start by rewriting the expression algebraically: $$\frac{2x^2+1}{x+3}=\frac{2(x+3)(x-3)+19}{x+3}=2(x-3)+\frac{19}{x+3}\ .$$ If $|x|<1$ then$$2(x-3)<-4\quad\hbox{and}\quad \frac{19}{x+3}<\frac{19}{2}$$ so $$\frac{2x^2+1}{x+3}<\frac{11}{2}\ .$$ In this case, this result is not as precise as the one we obtained by the other method.