For questions about or involving the absolute value function.

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Functional Derivative of Complex Absolutes

It is said, that the lowest order complex amplitude equation shows potential dynamics with the functional \begin{align} V[A] = \int\limits_{a}^{b} \text{d}X\left[- \vert A \vert^2 + \frac{1}{2} \vert ...
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absolute values and integals

I have the following integral $$\int_{- \infty}^\infty e^{-|x|} dx$$ and the following two questions (1) Since the preimages $x$ determine the the images $e^{-|x|}$ for nonnegative and negative ...
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2answers
54 views

Absolute value inequality $3 > |x + 4| \geq 1$

I've just started with absolute value equations and I have a real hard time understanding how to solve this. I got the following question, and I can't make heads or tails out of it. Assume that $x, ...
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2answers
44 views

How to solve complex equation with same variable on two sides

What is the analytic solution of X for the equation below? $$conjugate(X)= \frac{-2\times A}{B\times X} $$ A, B and X are complex numbers. Would the magnitude of X be given by this? $$ |X| =\sqrt ...
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3answers
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If {$b_n$} converges to b, then prove that {|$b_n$|} converges to |b|.

Is my proof good or does it need more work? Let $\epsilon$ > 0, we want N s.t. $\forall$ n $\geq$ N $\subset$ ||$b_n$| - |b|| < $\epsilon$. If |$b_n$| $\longrightarrow$ |b|, then $\exists$ N $\in$ ...
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2answers
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Prove that |a -b| $\geq$ ||a| - |b|| $\forall$ a,b $\in$ $\mathbb{R}$

Here is my proof can anyone check if it needs more detail or if it's good? Since a = a - b + b, then by the triangle inequality, |a| $\geq$ ||a| - |b|| + |b| so that |a| - |b| $\geq$ ||a| - |b|| ...
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1answer
42 views

Show that $|a+b|=|a|+|b|\Leftrightarrow ab\geq0$

Assume $a, b \in \mathbb{R}$. Show that $$|a+b|=|a|+|b|\Leftrightarrow ab\geq0$$ The triangle inequality says that for $a,b\in\mathbb{R}$, $|a+b|\leq |a|+|b|$. So I belive I can say ...
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Around an inequality

I have a very general question, hopefully not too general. Assume that we have real numbers $a_{ij}, b_{ij}$ $(1 \leq i, \: j \leq n)$ such that $-1 \leq a_{ij}, b_{ij} \leq 1$ for all $i,j,$ for ...
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1answer
65 views

Need help using the triangle inequality

Use the triangle inequality and the reverse triangle inequality to find an upper bound for the set of all numbers of the form $\left\lvert\frac{x^2-3}{x-2}\right\rvert$ as $x$ ranges over the interval ...
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2answers
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Solving the absolute value equation $2-3|x-1| = -4|x-1|+7$

$$2-3|x-1| = -4|x-1|+7$$ This is an example from my text book, and I do not understand how they got the answers. Solution: (this is the solution in my textbook) Isolate the absolute value of ...
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1answer
33 views

If we bound x on an interval, how can we bound |x|?

We are given $a<x<b$, where $a$ and $b$ are constants. This means that $x$ belongs to the interval from $x=a$ to $x= b$ excluding $a$ and $b$. What is the interval that $|x|$ (the absolute ...
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Minimizing the absolute value of the sum, |sum(cx)| ,by linear program

Now I need to use linear program to minimize the absolute value of the sum.The mathematical model of linear program can be expressed as Min |sum(cx)| Subject to |Ax|<=b ...
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1answer
27 views

Absolute value in Hasse's theorem

The Hasse's theorem says that for an elliptic curve $E$ defined on $\mathbb{F}_p$ where $p$ is a prime number, we have: $|n-(p+1)| < 2\sqrt{p}$ with $n$ the order of $E$. I am wondering why the ...
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4answers
60 views

Solve $\vert x-2\vert+2\vert x-4\vert\leq \vert x+1\vert$

I was helping someone with abolute values and inequalities and found this question. What is the easiest way to solve this? The only thing I thought of is to add the L.H.S and graph it with the R.H.S ...
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1answer
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Solving an absolute value equality without using a graphical method [closed]

How to solve this equality without using a graphical method ? $|1-x|-|x+1|=2$
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1answer
58 views

Lower bound for norm of sum of vectors

It is common to use: $$\| \sum_{i=1}^n \bar X_i \| \le \sum_{i=1}^n \|\bar X_i \|. $$ Now for 2 dimensions, I know that: $$ \|\bar A + \bar B \|^2 = (\bar A + \bar B)\cdot(\bar A + \bar B)$$ $$ = ...
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1answer
26 views

Absolute change and percentage change.

This page says- absolute change = new quantity – old quantity. For ex- Example 1: On September 20, a gallon of gas at my usual gas station cost $1.83. On September 30, I noticed that the price had ...
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1answer
36 views

Absolute Value $|1-(1/x)| = |(1/x)-1|$

Can someone please explain to me how: $$|1-(1/x)| = |(1/x)-1|$$ Im working on a limit problem in my calculus book and I cant seem to understand how they reversed this and it equals the same thing. ...
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1answer
68 views

Need help with proof with absolute value and complex numbers. [duplicate]

Had some trouble trying to prove the following problem. Prove that if $|z| < 1$ and $|w| < 1$, then $$ \frac{|z-w|}{|1-\overline{z}w|} < 1 $$ Would appreciate some help.
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2answers
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Polynomial equations with absolute values

How can I solve: $x^2 + 2|x| - 3 = 0$ ? My attempt: $|x| = \frac{3 - x^2}{2}$ $x = \pm \frac{3 - x^2}{2}$ $x^2 \pm 2x - 3 = 0$ The solutions to this 2nd degree polynomial is $x_1 = -3$ $x_2 ...
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2answers
36 views

absolute value binomial split into two absolute values

$$ |a-b| = |a|-|b| $$ I think I might missing something with absolute values. Can I split a binomial into two separate absolute values like above?
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1answer
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How to prove triangle inequality in How to Prove It Sec. 3.5 Question 12c?

(a) Prove that for all real numbers $a$ and $b$, $$|a| \le b \text{ iff } -b \le a \le b.$$ (b) Prove that for any real number $x$, $$-|x| \le x \le |x|.$$ (Hint: Use part (a).) (c) Prove that ...
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Prove that $\left|(|x|-|y|)\right|\leq|x-y|$

Prove that $\left|(|x|-|y|)\right|\leq|x-y|$ Proof: $$\begin{align} \left|(|x|-|y|)\right| &\leq|x-y| \\ {\left|\sqrt{x^2}-\sqrt{y^2}\right|}&\leq \sqrt{(x-y)^2} ...
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Decomposition and valuation rings

I am reading Algebraic Number Theory by A. Fröhlich, M. J. Taylor, it first introduced the theory: $(K,u)$ be a field and its absolute value, $(K_u,\bar u)$ be its completion and absolute value ...
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2answers
98 views

Maxima/Minima of absolute function

Given $a_i=\{a_1,\dots,a_n\}$ and function $$f(x)=\sum_{i=1}^n{|x-a_i|}^3$$ I need to find minimum value of $f(x)$. As far my understanding goes the derivative is given by: $$f'(x) = ...
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1answer
22 views

Prove that $\left|\frac{x^p-1}{p}\right|\leq x+|\ln(x)|$ for all $x\in(0,\infty)$ and for all $p\in(0,1)$

So far I have shown that $$\displaystyle \lim\limits_{p\to 0^+}\frac{x^p-1}{p}=\lim\limits_{p\to 0^+}\frac{e^{p\ln(x)}-1}{p}=\ln(x)\lim\limits_{p\to 0^+}\frac{e^{p\ln(x)}-1}{p\ln(x)} =$$ (L'Hopital) ...
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Complete field and field extension.

$(K,u)$ be a pair of the field $K$ and its absolute value $u$, $(K_u, \bar u)$ denotes its completion and the corresponding absolute value. Let $L$ be a field containing $K$, $\pi:K_u\rightarrow ...
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Unique extension of the absolute value

Let $(K,u)$ be a complete valued field, $u$ be its discrete absolute value (corresponds to a discrete valuation on $K$), then: ($\ast)$ Let $E/K$ is a finite separable field extension, then the ...
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3answers
50 views

How can I prove that if $\lim_{n \to \infty}s_n=s$ then $|s_n-s|< \epsilon$ is equivalent to $s-\epsilon <s_n <s+ \epsilon$

My professor casually mentioned this in class and told us to prove it if we weren't convinced, however, I cannot find how to prove it.
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2answers
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Absolute value of numbers

The absolute value of the sum of -5 and twice a number is 19. Find the number. I have a problem with this question because i do not fully understand absolute value and this question is a little trucky ...
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4answers
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Help with Absolute Value Mathematics

Currently, I am having trouble with the following questions listed below: Solve the equation $$\left\lvert x-2\right\rvert -\left\lvert x+ 3\right\rvert =x^2 - 1$$ For this question, I have drawn ...
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3answers
58 views

How to solve this absolute value equation and summation question??

$$|2x − 3| − |x + 2| = 5$$ I have no idea. I didn't see anything like this in class. It is a practice question and something like it will come up on the exam can someone please show me the full ...
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3answers
134 views

What is the derivative of $|x^3|$?

Let $f(x)=|x^3|$. I found two ways to differentiate this function. Apparently method 2 is wrong, but I cannot figure out why. So the question is, is method two wrong and why? Method 1 (according to ...
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2answers
84 views

Advanced Algebra Manipulation/Inequality Proof: $\frac{4x^3(x^2+y^2)-2x(x^4+y^4)}{(x^2+y^2)^2} \leq 6|x|$

I need to show that $$\frac{4x^3(x^2+y^2)-2x(x^4+y^4)}{(x^2+y^2)^2} \leq 6|x|$$ by starting with the left side of the inequality and working from there. Hints from the textbook said to work from ...
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390 views

algebra problem, Solve the equation [closed]

a nice problem: Solve the equation $$\left|2x-57-2\sqrt{x-55}+\frac{1}{x-54-2\sqrt{x-55}}\right|=|x-1|.$$ It's just for sharing a new ideas, thanks:)
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2answers
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Prove that if $|x-2|<0.001$, then $|\frac{1}{x}-\frac{1}{2}|<3\times 10^{-3}$

I still have difficulties with absolute value, and even if I manage to solve questions and problems, I do that awkwardly. So, please show me if this is the way to answer this question. Thank you in ...
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1answer
23 views

Will the absolute logarithm always produce the correct real result if one exists?

I'm a computer scientist, so my math skills are a bit rudimentary. The application I'm writing is more or less about solving equations. I'm only interested in real number solutions, so imaginary ...
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5answers
407 views

Computing the absolute value integral $\int_{-1}^{2} (|x|+|1-x|) dx$

I'm having trouble with one of the exercises, I have to split the integral for the absolute value but I can't manage to algebraic find the boundaries for the integral. $$\int_{-1}^{2} (|x|+|1-x|) ...
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Equation with absolute value and parameter

How to solve this equation over real numbers with parameter $p \in \Bbb R$? $$(1 - p)(\left\lvert x + 2 \right\rvert + \left\lvert x \right\rvert) = 4 - 3p$$ I know how to solve absolute value ...
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2answers
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Distinct Roots of $x^2+(a-5)x+1=3|x|$

Problem: $$x^2+(a-5)x+1=3|x|$$ Find 3 distinct solutions to the above problem. A friend of mine at my coaching center came up with this problem which nobody was able to solve. Unfortunately, I ...
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Metric spaces, manipulating the absolute value function.

I have the following problem involving the set $Y$ of infinite sequences that absolutely converge such that, $$\sum_{i=0}^\infty x_i^2 \lt\infty$$ where $x_i$ is the $i$-th term in the infinite ...
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1answer
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Rewriting $|x-10|+|y-5|\leq 7$ so that absolute values disappear - Algebra

Equation 1: $|x-10|+|y-5|\leq 7$ I want to rewrite this equation into equations that do not have the absolute value. $|A|\leq b$ can be written as $A \leq b$ $A \geq -b$ I want to apply the ...
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1answer
50 views

Complex Conjugation problem using the identity $|x|^2=xx^*$

Show that $$|c|^2= \frac{4k^2}{k^2 +\gamma^2}$$ given (1)$$a+b=c$$ and (2)$$ik(a-b)=-\gamma c$$ This was given in a lecture without proof, so there's probably a very simple way of proving the ...
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6answers
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Why does $|x_1| = |x_2| \implies x_1 = \pm x_2$

I was doing a 'prove this is not surjective' practice problem and the step leading from my hypothesis, as listed, to the conclusion was not defined. I don't recall being exposed to a situation where ...
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When I was teaching absolute function properties, I suddenly made this question …

I was teaching absolute function properties in a K-12 class. I made this question in my mind. Suppose $f(x)$ is a one-to-one function, and its definition is $f(x)=max\left \{ x,3x\right ...
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0answers
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Epsilon delta limit to show that [closed]

show that $$\left|\frac{28}{3x+1}-4\right| = \left|\frac{12}{3x+1}\right| \cdot |x-2| $$ using $\epsilon$-$\delta$ definition of a limit. I have no idea where to start since the question is not ...
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2answers
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$|f(x)g(x)| = |(f(x)||g(x)|$ [duplicate]

I was wondering if $|f(x)g(x)| = |f(x)| |(g(x)|$ is true all the time as in the case of real numbers. I was not convinced enough that that was true. But I can't think of any counterexample. Thank ...
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1answer
48 views

Taking out absolute value on the solution to integral equation

I have this equation:$$y=2+\int_2^x (t-ty(t))dt$$ After solving it I got the answer $-\ln|1-y|=\frac {x^2} 2-2$ although the book has the same answer without the absolute value in the logarithm, why ...
3
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1answer
55 views

Difficulty in finding the Range of x

$x^2 - | x-2 | + 6 > 0 $ , where x belongs to $R$ I am not sure about my own approach to this ques. I solved it as: $x^2 + 6 > | x-2 |$ , thereafter i got 2 cases Case 1: $-(x^2 + 6) ...
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42 views

What is the expected value of the absolute value of a Wiener Process?

I am trying to show that the with a Wiener Process $w(t)$, then $\mathbb{E}[|w(t_1)w(t_2)|] = (\frac{2a}{\pi}) \sqrt{t_1 \cdot t_2} (\cos \theta + \theta \sin \theta)$, given $\sin \theta = ...