For questions about or involving the absolute value function.

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0answers
9 views

Ostrowski: an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$

I have included a bolded comment in a step in the part of Gouvea's proof of Ostrowski's theorem where he shows that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$. I will ...
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1answer
18 views

What would the multifunctional inverse of $F(x)=|x|$ be?

What would the multifunctional inverse of $F(x)=|x|$ be, assuming $x$ is on the complex plane. Also, how would this usually be represented? Note that this won't be a 'true' function. (But assume a ...
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1answer
29 views

Doubt with Intervals and Inequalities

This doubt has been bothering me for ages. I would be truly grateful for any help. Problem 1: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals Solution: $x\in(2,4)\cup(4,6)$ ...
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6answers
107 views

Proof of $ |a-b| = |b-a| $ [on hold]

While working out some intriguing qualities of absolute values for my studies of calculus, I frequently used the formula below. I know that the formula below is clearly correct but how would I prove ...
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3answers
46 views

find the derivative of the integral

Prove that the following integral $F(x)$ is differentiable for every $x \in \mathbb{R}$ and calculate its derivative. $$F(x) = \int\limits_0^1 e^{|x-y|} \mathrm{d}y$$ I don't know how to get rid of ...
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4answers
47 views

Solve $|1 + x| < 1$

I'm trying to solve $|1 + x| < 1$. The answer should be $ -2 < x < 0$ which wolframalpha.com agrees with. My approach is to devide the equation to: $1+x < 1$ and $1-x < 1$ and then ...
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1answer
22 views

Has this question on Absolute Value's been asked wrong?

I was going over some basics on Khan's Academy in preparation for a test. To my surprise I got this wrong: Has this been worded wrong? Surely the person farthest from sea level is Howard? This ...
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4answers
29 views

Soft absolute value

I'm looking for a "soft absolute value" function that is numerically stable. What I mean by that is that the function should have $\mp x$ asymptotes at $\mp\infty$ and behave smoothly in $[-1,1]$. ...
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1answer
39 views

Complex integration along a square box — is | * | absolute value or modulus? [closed]

Say I am integrating along the lower edge of the box, from -n to n, crossing from the third quadrant to the fourth. So the parametrization would be something like: z(x) = x-in, for x ranging from -n ...
2
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2answers
46 views

Solve differential equation $y' = |1.1 - y| + 1$

How can the following differential equation be solved analytically? \begin{equation*} y' = |1.1 - y| + 1, \\ y(0) = 1. \end{equation*} I guess one must rewrite the differential equation piecewise ...
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5answers
863 views

If three complex numbers $z_k$ have modulus $1$, then $|z_1+z_2+z_3| = |\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$

Our teacher gave us a hard question (according to her, it is pretty hard for our level): Given that $|z_1| = |z_2|= |z_3|=1,z \in\mathbb{C}$, prove that $|z_1+z_2+z_3| = ...
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4answers
107 views

Can some explain very quickly what $ |5 x + 20| = 5 $ actually means?

I don’t mean to bother the community with something so easy, but for the life of me, I can’t remember how to do these. I even forgot what they were called, and I’m referring to the use of the “$ | $” ...
3
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1answer
131 views

mean value theorem sin(b) - sin(a)

It's too much hassle to post it here as latex, to so here's the screenshot. I don't understand why |cos(c)| = 1 Why 1? Why not $\frac {\sqrt{3}}{2}$? Why absolute value assumes the max value a ...
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3answers
64 views

What is $\lim_{x\to 7^-} \frac{\left|x-7\right|}{x-7}\,$?

$\displaystyle \lim_{x\to 7^-} \frac{\left|x-7\right|}{x-7} = $ Writing absolute value as: $x-7 > 0$ $x > 7$ which means $x - 7$ when $x > 7$ then: $ -(x - 7) < 0$ $-x + 7 < 0$ ...
2
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1answer
19 views

non-archimedean absolute value (Ostrowski's theorem)

I'm reading the proof of Ostrowski's theorem in Gouvea's book on p-adic numbers and there is one step that I don't understand. Let $|\cdot|$ be a non-archimedean absolute value and $n=rp+s$ where ...
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2answers
30 views

Evaluating $\int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y$

I'm always having the wrong result from the following: $$ \int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y $$ I would really appreciate some guidance on how to go ...
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5answers
69 views

Any tips for solving $\frac{|4x-2|}{|2x+1|} \le 1$ as succinctly as possible?

$\frac{|4x-2|}{|2x+1|} \le 1$ So as I currently see it, I have two choices: 1) Attempt to solve algebraically but that has led me down some long paths when I believe the question should be solvable ...
1
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1answer
34 views

A quadratic polynomial bounded by another

Suppose $p(x)$ and $q(x)$ are two quadratic polynomials in real coefficients such that: $$\lvert p(x) \rvert \leq \lvert q(x) \rvert ~ ~ ~ \text{for all} ~ x \in \mathbb{R} \tag{1}$$ Is the above a ...
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1answer
25 views

“p-adic absolute value” in polynomial ring

I'm working with Gouvea's book on p-adic numbers. In problem 34 I'm asked to give a "p-adic" (I put it in quotes as in my understanding its just a p-adic-like) valuation and absolute value for an ...
3
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2answers
38 views

Is every Pisot-like integer the product of a Pisot integer and a root of unity?

For lack of better terminology, let's call an algebraic integer $\beta$ Pisot-like if $|\beta|_{\mathbf{v}} > 1$ for the place $\mathbf{v}$ of $\Bbb{Q}(\beta)$ corresponding to the embedding $\beta ...
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1answer
21 views

Online calculator for $ p $-adic valuations and absolute values.

Does anyone know a website where I can enter a prime base and a rational and then get the $ p $-adic valuation and the $ p $-adic absolute value? For sure I know how to do it by hand, but I want to ...
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2answers
28 views

How do I upperbound this expression?

With a given condition such as $$|x|^2 > |y|^2$$ Is there any way I can upper bound the following expression $$\log\left(1+\big||y|-x\big|^2\right) \leq \,\,\, ? $$ Thank you
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2answers
76 views

For X,Y random variables, with pdfs that are symmetric around 0, does $V(X)\geq V(Y) \Rightarrow E(|X|)\geq E(|Y|)$?

I need to show the following thing. Consider two continuous random variables $X,Y$ which take values in $[-1,1]$ and are have pdf's that are symmetric around zero. How can I show that $V(X)\geq V(Y) ...
3
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3answers
81 views

Prove that $g(x):=|f(x)|$ is differentiable iff $f'(a)=0$

Suppose that $f(a)=0$. Prove that $g(x):=|f(x)|$ is differentiable iff $f'(a)=0$ Not sure how to go about this at all. The limit definition that I am working with is $$ g'(a)=\lim_{x \rightarrow ...
2
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1answer
42 views

An expression with gcd and abs is transformed magically!

There's a problem to calculate $\sum^{n}_{i=1}\sum^{m}_{j=1}\frac{|i-j|}{\gcd(i,j)}$, whose tutorial gives the following transformation I really don't understand. ...
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1answer
27 views

Given $f\in L^1(\mathbb{R})$ with $||f||_1 < \infty$, is it true that $\int_{\mathbb{R}} ||f||_1 - f(x) \, dx = 0$?

According to my intuition so far, the answer should be yes, hinging very important on the assumption that $||f||_1 < \infty$. To speak very roughly, if the $L^1$ norm of $f$ is finite, it seems ...
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2answers
24 views

Absolute Value Algebra with inverses

I noticed the following equality in some material regarding limits and infinite series. $$ \left |\frac{x}{x+1} - 1 \right| = \left |\frac{-1}{x+1} \right| $$ And I'm honestly stumped (and slightly ...
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2answers
37 views

Solving Equations And Inequations Based On The Absolute Function [closed]

Today I came across some equations and inequations based on the absolute function. These were $|x^2+4x+3|+2x+6=0$ $|x^2+6x+7|=|x^2+4x+4|+|2x+3|$ $1\le |x-1|\le 3$ $\frac{2}{|x-4|}\gt 1$ $||x|-1|\le ...
2
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1answer
20 views

Differentiability of an absolute function.

Check the differentiability of $f(x)=x|x|$, $x$ is in $\mathbb{R}$. I know that it is differentiable when $x>0$ and $x<0$. I am not sure about the case when $x=0$. I found that as $$\lim ...
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0answers
33 views

Simplifying a function with absolute value and two variables

I am trying to simplify the following $ f(\alpha, \beta)= \big|1-(\frac{\alpha+\beta}{2}+\frac{|\beta-\alpha|}{2})\big|-2 \big|1-\alpha-(\frac{\alpha+\beta}{2}+\frac{|\beta-\alpha|}{2})\big|+ ...
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0answers
32 views

Solve this inequality for B

I am working on a program that is supposed to qualify a value as "in range" and I have come up with the expression: $$\lvert a-b\rvert \leq c$$ to determine the value. Plugging in test numbers ...
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1answer
34 views

How to graph $|z-1| <2$

Am I correct to rearrange this to $(z-1)^2 < 4$, and hence just graph as a circle or am I completely off?
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1answer
31 views
0
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1answer
30 views

Decomposing absolute value terms

I have something like the following term: 7x1 + 9x2 + | 10 - 7x1 | + | 15 - 11x2 | I want to make it into something like this: Ax1 + Bx2 , where A and B are constants For two values x1 and x2, ...
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3answers
80 views

Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$

Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$.What will be the easiest techique to solve this sum ? Just wanted to share a special type of equation and the fastest way to solve it.I am not asking ...
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2answers
62 views

A difficult trigonometric integral involving absolute value

$$ \int_{0}^{2\pi}\lvert\sin(x)\rvert\cos(nx)\,dx= -\frac{4\cos^2\bigl(\frac{\pi n}{2}\bigr)\cos(\pi n)}{n^2-1} $$ I'm not actually trying to solve this myself. The answer appears in my lecture notes ...
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4answers
59 views

Prove that for $x\in\Bbb{R}$, $|x|\lt 3\implies |x^2-2x-15|\lt 8|x+3|$.

The problem I have is: Prove that for real numbers $x$, $|x|\lt 3\implies |x^2-2x-15|\lt 8|x+3|$. Since there aren't really any similar examples in my book, I've been unsure how to first approach ...
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1answer
30 views

how to calculate $\int (|1+x|-|1-x|) dx $ and $\int$ max {${1-x^2,0} $}?

I'm used to integrate normal functions, but here I got quiet confused because these integrals : $\int (|1+x|-|1-x|) dx $ $\int$ max {${1-x^2,0} $} Include absolute value, and and option to choose ...
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4answers
54 views

How do you algebraically derive “x <= 0” from “-x = | x |”

A = "-x = | x |" B = "x <= 0" If A, then B. By plugging in numbers or testing ranges less than zero, greater than zero, and equal to zero, I can verify that A ...
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3answers
59 views

Using the Mean Value Theorem, prove that $|\sin{a} - \sin{b}| \leq |a - b|$ $\forall a, b \in \mathbb{R}$

Using the Mean Value Theorem, prove that $|\sin{a} - \sin{b}| \leq |a - b|$ $\forall a, b \in \mathbb{R}$. I'm working towards figuring out an approach for finding that $|\sin{a} - \sin{b}| \leq ...
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3answers
484 views

A unique solution

Find the sum of all values of k so that the system $$y=|x+23|+|x−5|+|x−48|$$$$y=2x+k$$ has exactly one solution in real numbers. If the system has one solution, then one of the three $x's$, should be ...
5
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5answers
259 views

How to write an expression in an equivalent form without absolute values?

The question I have in front of me is the very first problem in Trench's Introduction to Real Analysis: Write the following expression in equivalent form not involving absolute values: $a+b+|a-b|$ ...
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2answers
30 views

Inequality with absolution value for complex number

How to show that inequality: $|1-\bar{\alpha} z| \ge |z-\alpha|$ $z$ and $\alpha$ are complex number, $\alpha$ is constans and $|z|<1$, $| \alpha| < 1$ I can proof that by using substition ...
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4answers
70 views

is the following true: $|a-b| = ||a|-|b||$?

is $|a-b| = \bigg||a|-|b|\bigg|$ ? I have tried a few examples and they seems to come out true, but I can't find any rule stating it. Is it true for all $a$ and $b$? Or am I missing something? ...
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2answers
46 views

How to start proof of triangular inequality? [duplicate]

$$\left| {\left| a \right| - \left| b \right|} \right| \le \left| {a \pm b} \right| \le \left| a \right| + \left| b \right| $$
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3answers
82 views

Solve $|3-x|=x-3$.

Solve: $|3 - x| = x - 3$. Answer: $|u| = -u$ when and only when $u \le 0$. So, $|3 - x| = x - 3$ when and only when $3 - x \le 0$; that is, $3 \le x$. Hi! I'm new here. I'm working out of ...
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0answers
63 views

two dimensional Gaussian integral with complex exponent of an absolute value

I am trying to solve the following two dimensional integral: $$\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}{e^{ia(\left|x\right|-\left|y\right|)} \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{x^2+y^2-2\rho ...
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2answers
65 views

Prove triangle inequality using the properties of absolute value

So I was given the task of proving the following variant of the triangle inequality using only the properties of the absolute value: $\vert\lvert x\rvert -\lvert y \rvert \rvert \leq \lvert ...
2
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0answers
44 views

Simplifying an expression with absolute values

I am trying to simplify the function $D(\alpha,\beta)$ shown below (with $\alpha,\beta>0):$ $$ D(\alpha,\beta)=\frac{1+\alpha+2\beta}{2} + \frac{|\alpha-1|}{2} - 2 ...
2
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2answers
18 views

Build the graph of a function with absolute value.

The function is: And my idea of graphic (i did it using two graphs and deleting some parts) Is it correct?