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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.

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up vote -1 down vote accepted

Try this one, i am using it and find it exceptional for HS student:

Once you become more familar with that book, use this(give you more insightful more rigorous proof and more advanced stuff):

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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. – miracle173 Apr 7 '12 at 0:32

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.

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+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. – Mathemagician1234 Apr 7 '12 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! – Mathematacticool Oct 5 '14 at 22:41

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 $| x_1 y_1 + x_2 y_2|^2 \leq |x_1 + x_2 |^2 |y_1+y_2|^2$
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