# 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.

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.

• +1 for one of the best books for undergraduates there is on basic inequalities. Everyone should work through this book before tackling rigorous analysis for the first time. Apr 7, 2012 at 3:50
• These have been the bane of my experience in my real analysis course. I'm going to go through this book with vengeance and get rid of that weakness. Thank you! Oct 5, 2014 at 22:41
• I never had heard of this book before, it would indeed have been useful for analysis. Maybe it would still pose interesting as a light read :)
– user459879
Aug 16, 2019 at 7:49

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 $| x_1 y_1 + x_2 y_2|^2 \leq (x_1^2 + x_2^2 ) (y_1^2+y_2^2)$
• One I often used was a combination of the triangle inequality and the two relations: $(x-y)^2=x^2 +y^2 -2xy \geq 0 \implies x^2 +y^2 \geq 2xy$ combined with: $(x+y)^2=x^2 +y^2 +2xy \leq 2x^2 +2y^2$ to get rid of linear cross-tems in a question about the Banach fixed point theorem. Here you work with inequalities often derived from the mean value theorem and this trick is used to bring everything back to squares so you can get norms of vectors again.
– user459879
Aug 16, 2019 at 7:44

Try this one, i am using it and find it exceptional for HS student: http://www.amazon.com/Inequalities-Mathematical-Radmila-Bulajich-Manfrino/dp/3034600496/ref=pd_sim_b_3

Once you become more familar with that book, use this(give you more insightful more rigorous proof and more advanced stuff): http://www.amazon.com/Inequalities-Cambridge-Mathematical-Library-Hardy/dp/0521358809/ref=sr_1_8?s=books&ie=UTF8&qid=1331606830&sr=1-8#_

• i do not understand why a person that seems to have problems with basic facts about inequalities should read books that deal with mathematical olympiad problems on this sector. That does not make any sense to me. Apr 7, 2012 at 0:32