For questions about or involving the absolute value function.

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3
votes
2answers
308 views

Why exactly can you take the absolute value of one side of this inequality and assume it is still true?

Exercise: Show that if $(b_n) \to b$, then the sequence of absolute values $\left| b_n \right|$ converges to $\left| b \right|$. Solution (partial): By the triangle inequality, ...
3
votes
1answer
165 views

Inequality with absolute value

I am unsure if have solved the following inequality correctly: $ \dfrac{2x+3}{x+5} \leq \dfrac{x+1}{|x-1|} \tag{1}$ I've proceeded as follows. If $x>1$ then $|x-1|=(x-1)$ If $x<1$ then ...
3
votes
1answer
657 views

Proving Absolute Value Inequality

I had posted a portion of this earlier asking about how to interpret min(). I received some excellent answers, however, I have run into problems and feel stuck. I am posting the question in its ...
4
votes
1answer
672 views

integral from 0 to $2\pi$ of $|\cos x|\operatorname{d}x$ not integrating as I'd expect

I drew a rough sketch of $|\cos x|$ and would guess the correct answer to this integral is $4$ because I know the area under the curve of $\cos x$ from $0$ to $\pi/2$ is $1$, and there are $4$ such ...
3
votes
3answers
116 views

Solving $ \left| \frac{-2x-6}{4} \right| \le 5$ for $x$

Say I have a statement like: $$ \left| \frac{-2x-6}{4} \right| \le 5. $$ And I want to find the closed interval form of $x$. i.e. I want to know what the maximum and minimum $x$ can be. How do I ...
2
votes
2answers
2k views

Square root of simple binomial function

Let's say I have the following formula: $$\sqrt{a^2-2ab+b^2}=\sqrt{(a-b)^2}=\sqrt{(b-a)^2}$$ When do I know which one of the following I should use?: $$\sqrt{(a-b)^2}=a-b\qquad\text{ or }\qquad ...
6
votes
2answers
2k views

How to use triangle inequality to establish Reverse triangle inequality

I need to use $|a+b| \leq |a|+|b|$ to show that $||a|-|b|| \leq |a-b|$. I have tried to represent $||a|-|b||$ as $||a|+(-|b|)|$, and then get $||a|+(-|b|)| \leq |a|+|-|b||$, but that isn't leading ...
2
votes
1answer
432 views

Rules applying to nested absolute values

I'm trying to use some algebra get $||x-5|-10|<\epsilon$ into a more manageable form (I'd like it in terms of $0<|x+5|<\delta$) but I'm not sure where to begin. I don't really know the rules ...
2
votes
3answers
664 views

Trouble with absolute value in limit proof

As usual, I'm having trouble, not with the calculus, but the algebra. I'm using Calculus, 9th ed. by Larson and Edwards, which is somewhat known for racing through examples with little explanation of ...
10
votes
5answers
460 views

Is -5 bigger than -1?

In everyday language people often mix up "less than" and "smaller than" and in most situations it doesn't matter but when dealing with negative numbers this can lead to confusion. I am a mathematics ...
3
votes
1answer
242 views

Problem about absolute value

$$\begin{align*} |x|=x &\text{if }x\geq 0\\ |x|=-x &\text{if }x\lt 0. \end{align*}$$ Show that $|xy|=|x||y|$. I try to prove it as follows: $|xy|=xy$ if $xy\geq 0$, but $xy\geq 0$ ...
1
vote
4answers
3k views

Absolute values of 1-10 in a pyramid form

_ _ _ _ \/ \/ \/ _ _ _ \/ \/ _ _ \/ _ you have numbers 1-10. you can only use each number once and the number below is equal to the absolute ...
1
vote
1answer
4k views

Solving inequality with two absolute values

Hey, ! In my pre-calculus class the teacher showed the solution of the following example: \begin{align} \vert x-3 \vert \lt \vert x - 4 \vert + x \end{align} He started by stated the domains ...
0
votes
2answers
7k views

Absolute value of all values in a matrix

How do I express the matlab function abs(M), on a matrix M, in mathematical terms? I thought about norms or just |M|, but these return scalars, not another matrix of the same size as M. Sorry for ...
2
votes
0answers
159 views

Proof that there's a unique division quaternion algebra over a locally compact field?

There are many proofs that there is a unique division quaternion algebra over a locally compact field that is not $\mathbb{C}$. For instance this set of notes/book by John Voight contains two proofs: ...
10
votes
1answer
301 views

How to determine all valuations of the field $\mathbb{Q(\sqrt[n]{2})}$?

This is an exercise from the book Algebraic Number Theory by Jurgen Neukirch, on page 166. And, after solving several previous exercises, I found this to be particularly difficult to solve. I am ...
1
vote
4answers
211 views

Understanding $y=|mx+n|$

The diagram shows the graph of $y=|mx+n|$ (i tried my best to do the same thing as my exercise book, actually 1 is propotional to 1 and 3 is propotional to 3, but 2 is not propotional to 2) Find ...
3
votes
1answer
198 views

Is the book wrong about this left-hand limit with absolute value? (But, my delta depends on x.)

The book says that $$\lim_{x \rightarrow 0^{-}} \left( \frac{1}{x} - \frac{1}{|x|} \right) \mbox{does not exist}$$ But, given any $M \lt 0$ of large magnitude, if I choose $\delta = \frac{-x^{2}M}{2}$ ...