Questions tagged [absolute-value]

For questions about or involving the absolute value function.

3
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0answers
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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: ...
1
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1answer
476 views

How to solve equations with absolute value and using the Archimedean property

I'm trying to learn Real Analysis on my own, but I found that i'm a bit rusty with the elementary stuff. How do I solve equations like $|x| + |x+1| = 1$ and $|x-1| + |x+1| = 2$? I don't want the ...
3
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3answers
155 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 ...
0
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5answers
336 views

Solving an equation with absolute values

The equation I am trying to solve is this : $\newcommand\abs[1]{|#1|}\abs{3y+7}=\abs{2y-1}$. My conventional approach is to split this into three intervals with $1/2$ and $-7/3$ being the two "split" ...
4
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1answer
1k 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 ...
0
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1answer
2k views

derivative of absolute value of a complex function

If $f:U\subset\mathbb{C}\mapsto\mathbb{C}$, where $f(x+iy)=u(x,y)+iv(x,y)$ is a meromorphic function and if $f$, $f'$, and $f''$ are not zero in the strip $a<x<b$, can we get $\frac{\...
5
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1answer
4k views

Prove variant of triangle inequality containing p-th power for 0 < p < 1

Sorry if this is a trivial question, but I am kind of stuck with proving the following inequality and have been searching for a while: $\rho \left( \sum\limits_i^n d_i \right) \leq \sum\limits_i^n \;\...
2
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1answer
640 views

When should I put an absolute value sign around a function?

In Griffiths' introduction to QM book, I see $\int f^* f\ dx$ often written as $\int | f |^2 dx $, whereas $\int f^* g\ dx$ is understandably just that. Should I always write the absolute value sign ...
1
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2answers
2k views

Joint pdf of X and Y with absolute value range

I have the following joint pdf: $f(x,y)=0.5$ where $0 \leq|x|\leq|y|$, $0 \leq|y|\leq1$, and $0$ otherwise The question is: are $X$ and $Y$ independent and uncorrelated? I know that if $f(x)$*$f(y)...
10
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5answers
563 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 ...
2
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2answers
89 views

How do I find this limit?

How would i find the limit as $\lim\limits_{x\to3}\frac{4x(x-3)}{|x-3|}$? that is the absolute value of x-3 in the denominator. I thought my professor told my class that we were able to omit the ...
2
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1answer
1k views

$|x-y| + |y-z| = |x-z|$ then $x \le y \le z$

I'm doing this exercise from Robert Bartle's Introduction to Analysis, it's a if only if excersise and I've done the half part, but I can't figure this part of the ...
3
votes
2answers
440 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, $\left| ...
4
votes
1answer
228 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 $|x-...
4
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1answer
3k 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 ...
2
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1answer
1k 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
1k 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 ...
3
votes
1answer
370 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 $xy\geq 0$ if ...
5
votes
2answers
23k 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 ...
3
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1answer
247 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 $\delta = \frac{-x^{2}M}{2}$ ...