What is a good book to learn how to manipulate inequalities with absolute values?

I've noticed that I'm really having trouble with limits because I've had very little experience manipulating inequalities and I really have little to no idea how to manipulate inequalities involving absolute values. I really didn't learn it in high school (along with analytic geometry). I don't really know where people have learned when it is okay to for example pull the exponent out of the absolute value bars and such. I'm trying to pick some things up from my Spivak calculus book, but I want to know that I know ALL the rules of manipulation. My question is what is a good book that explains how to manipulate inequalities involving absolute values? I would prefer a book that deals with this extensively.

Solution 1:

The book The Cauchy-Schwarz Master Class by Steele is focused specifically on manipulating inequalities and contains detailed solutions to all of the exercises, making it a good choice for self-study. I, too, am lousy with inequalities and the (regrettably small) amount of time I have spent with this text has been profitable.

Solution 2:

Here are some of the basics.

1. If $|a|\leq b$ (where $b\geq 0$) then $-b \leq a \leq b$
2. If $|a|\geq b$ (where $b\geq 0$) then either $a \geq b$ or $a\leq -b$
3. $|ab|=|a||b|$ and, as a consequence, $|a^n|=|a|^n$
4. the triangle inequality $|a+b| \leq |a| + |b|$
5. If $f(x)$ is monotone increasing, then if $x\leq y$ then $f(x)\leq f(y)$. See the wikipedia article on inequalities
6. Somewhat less used in undergrad calculus is the Cauchy Schwarz inequality $| x_1 y_1 + x_2 y_2|^2 \leq (x_1^2 + x_2^2 ) (y_1^2+y_2^2)$