For questions about or involving the absolute value function.

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1answer
37 views

A question about the absolute value in integrals

I do really understand why we put the absolute value when integrating functions leading to $\log$ function for example: $$ \int{\dfrac{\mathrm dx}x}=\log\lvert x\rvert + C$$ , it is very common in ...
1
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2answers
51 views

Express the function $ f $ without using absolute value signs $\left|\frac{x-2}{x+3}\right|e^{\left|x-2\right|}$?

Good evening to everyone: This is the equation $$ f(x) = \left|\frac{x-2}{x+3}\right|e^{\left|x-2\right|} $$ What I've tried is: $$ \frac{x-2}{x+3}\ge 0 => x-2 \ge 0 => x \ge 2$$ Then $$ \frac{-...
-5
votes
5answers
59 views

On the equation $|x|^2+|x|-6=0$

Which of the following are true for $$|x|^2+|x|-6=0$$ 1. It has $4$ roots 2. The sum of the roots is $-1$ 3. The product of the roots is $-4$ 4. The product of the roots is $-6$ Only one of the ...
0
votes
1answer
22 views

applying exponents in an absolute value brackets

So part of the problem I'm trying to solve is this: $|2-9|^{3{^3}}$ being the exponent, to the power of 3. Do I have to apply the exponent to everything in the bracket? 2*2*2 and 9*9*9 or does the ...
1
vote
2answers
41 views

Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$

Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$, if $z = a+ib$ then, I tried to do that with $|z| = (a+ib)(a-ib)$ then i multipled it $z$ with $z^-$ and then I got stuck. answer is $|z| = 5$
1
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4answers
61 views

How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$?

How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$ ? I've solved equations like $|a|+|b|=|a+b|$ where the condition must be that $a$, $b$ must be of same sign. But in case of three terms ...
1
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1answer
28 views

Is there a derivative for $|x|$ at $0$ specifically “in the direction” of positive $x$?

I know that $|x|$ is not differentiable at $x=0$ because there is potentially an infinite number of tangent lines going through that point. But let's say we were interested in the motion of an object ...
3
votes
3answers
30 views

reciprocal factor of absolute value when evaluating a square root expression

Learning with an old russian math book, i found the following evaluation for the function $f(x)=\sqrt{1+x^2}$: $f(\frac1x)=\vert x \vert^{-1}\sqrt{1+x^2}$ My evaluation gave me $\sqrt{1+\dfrac1{x^...
0
votes
3answers
50 views

Why is $\sqrt{x/x^{-1}}$ OR $\sqrt{x/{1/x}}$ = $\lvert x\rvert$ and not just x

I have this task: Find equal expression to square root of fraction of x and its inverted value (this is translated from my mother tongue so I'm sorry if I've used incorrect terms). Anyway the starting ...
1
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2answers
41 views

Rejecting a Solution to a Modulus Question

Why is the solution of $|1+3x|<6x$ only $x>1/3$? After applying the properties of modulus, I get $-6x<1+3x<6x$. And after solving each inequality, I get $x>-1/9$ and $x>1/3$, but why ...
0
votes
1answer
25 views

Find the solution set of $ \left\lvert 100\left(\frac{x-y}{y}\right) \right\rvert - \left\lvert 100\left(\frac{y-x}{x}\right) \right\rvert < 1 $

I apologize if this is a rather trivial question however I was wondering if I can have a bit of guidance in tackling this problem: $$ \left\lvert 100\left(\frac{x-y}{y}\right) \right\rvert - \left\...
1
vote
1answer
30 views

Integrals of a function and its absolute value

Is the following proposition true? Let $f(x)$ be a real-valued function defined on $[a,b] \subset \mathbb{R}$, and suppose that the integral, $$ I = \int_a^b f(x) dx, $$ exists in the sense of ...
0
votes
1answer
38 views

Joint pdf of X and Y with absolute value

Question. Joint probability function of continuous probability X, Y is here : $f_{X,Y}(x,y) = k(|x|-|y|) \ \ \ \ \ \ \ \ \ \ (-1< y< x< 2)$ Then what is k? I mean how can I differentiate ...
1
vote
1answer
101 views

How to minimize $|Ax+By + C|$ given that $x \geq 0$ and $y\geq 0$ [duplicate]

I am trying to solve problem related to absolute value function, i.e given $Z(x,y) = |Ax + By + C|$ , what is the minimum value of $Z$, if $x \geq 0$ and $y\geq 0$ and x,y belongs to integers
2
votes
4answers
109 views

Prove that $||a|-|b||$ is smaller or equal to $|a-b|$

I am stuck with this question: Show that $\vert \vert a\vert - \vert b\vert\vert \le \vert a-b\vert$ I had tried proving this using the following method below: $\vert a\vert+\vert b\vert \ge \vert ...
2
votes
3answers
51 views

Solve the equation $|2x-1| -|x+5| = 3$

Problem : Solve the equation $|2x-1| - |x+5| = 3$ In my attempt to solve the problem, I only manage to get one of the solutions. Attempted Solution $$\begin{equation} \begin{split} |x|-|y| & \...
0
votes
0answers
24 views

On the Arrangement of Intermediate Subgroups

I am trying to find a journal paper 'On the Arrangement of Intermediate Subgroups' by M.S. Bah and Z.I. Borevich appearing Rings and Linear Groups, Krasnodar (1988), 14-41. This is a Russian text and ...
1
vote
1answer
39 views

what is the result for the following Integral?

I would like to find the result for the following integral $$ \int_{-\infty}^\infty x e^{-|x|/a}\cdot e^{-|x-y|/b} \, dx $$ where $a$ and $b$ are constants. $x$ and $y$ are variables
0
votes
1answer
18 views

Finding the set of values for k of a modulus function.

"Find the set of values of k for which |(x-4)(x+2)| = k has four solutions." EDIT: Ok so I thought I'd start with setting the modulus function equal to k and -k to get the two set of results. Doing ...
0
votes
1answer
28 views

I need to find the following Integral?

I would like to ask if any one can find the following integral $$ \int_c^d x . e^{-{\frac{\vert x \vert}a}}.e^{-{\frac{\vert x-y \vert}b}} dx $$
4
votes
4answers
73 views

Show that $f(x)=f(y)$ then $|x|=|y|$, where $f(x )=\frac{1+|x|}{x}$

Let $f: \mathbb{R}^{*}\to \mathbb{R}$ function definied by $f(x )=\dfrac{1+|x|}{x}$ Show that $f(x)=f(y)$ then $|x|=|y|$ Indeed, $$f(x)=f(y)\\ \iff \\\dfrac{1+|x|}{x}=\dfrac{1+|y|}{y} \\ \iff \\ \...
1
vote
1answer
29 views

How do I represent $f(x) = \int_{-1}^{0} |x + t| dt, 0 \leq x \leq 2$ in an integrated form?

Given the following function, how do I define it without the integral symbol? $$f(x) = \int_{-1}^{0} |x + t| dt, 0 \leq x \leq 2$$ I don't understand how I determine when $x + t$ is positive and ...
1
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2answers
52 views

build absolute value equations know solution

We have absolute value equations with unknown coefficients: $$|x + a| = b$$ and we know the solutions: $$x = 11 \text{ and } x = 5$$ We need to find $a$ and $b$. From $$11 + a = b \\ 5 + a = -b$$ we ...
-1
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2answers
36 views

Normal distribution, probability and modulus question [closed]

Say $X$ is a random variable which is normally distributed with mean $0$ and variance $1$. How do I find $k$ such that $$\mathbb{P}(|X-k| < |X+k|) = 0.7$$
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0answers
35 views

Find the product all real numbers in an equation

What is an easy and fast way to solve the problem without going through all these possibilities: a) $n^2-9n+20>0, 16-n^2>0$, b) $n^2-9n+20>0, 16-n^2<0$, c) $n^2-9n+20<0, 16-n^2>0$, ...
1
vote
1answer
37 views

Is there an easy way to solve this absolute values problem?

This is a simple problem. What I want to know is whether there is an easy and fast way to solve the problem. I solved this problem by considering four situations: a) $x>1$, b) $0<x<1$, c) $x&...
0
votes
3answers
36 views

How to solve this problem on absolute value function?

If $a,b\in \mathbb R$ and be distinct numbers satisfying $$|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$$ then the minimum value of $|a-b|$ is ? ($|...|$ represents absolute value) I tried solving the equalities ...
1
vote
2answers
40 views

If $a\le b\le c$, then $|b-d|\le\max\{|a-d|,|c-d|\}$

I am trying to find a simple way to show the fact that if $a\le b\le c$, then $|b-d|\le\max\{|a-d|,|c-d|\}$ for any number $d$. Is there a way to do this besides breaking it up into the cases 1) $...
1
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0answers
40 views

Uniform convergence fourier series |sin(x)|

As an exercise we have to calculate the fourier series of |sin(x)| (was no problem) and after that we are meant to show that this series converges uniformly towards |sin(x)|. After thinking about it ...
0
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0answers
25 views

Is complex multiplication the only multiplication operation on $\mathbb{R}^2$ that works with the Euclidean norm?

What I'm asking is: viewing complex multiplication as binary operation on $\mathbb{R}^2$, is usual multiplication of complex numbers the only operation $\otimes$ on two vectors $\vec{u}$ and $\vec{v} \...
2
votes
1answer
26 views

why is $\left|xy\ log(\left|x\right|+\left|y\right|)\right|\leq\left|(\left|x\right|+\left|y\right|)log(\left|x\right|+\left|y\right|)\right|$?

I should note that this was used by my book in order to show that the limit of $xy\ log(\left|x\right|+\left|y\right|)$ at $(0,0)$ is $0$. After several attempts in vain, I plotted the function $\...
3
votes
4answers
129 views

The Definition of the Absolute Value

The Absolute Value can be defined in many ways, but these are the two most common : 1. As a Piecewise Function $$ |x|= \begin{cases} -x&\text{if } x < 0\\ x&\text{if } x\geq 0 \end{cases} ...
0
votes
0answers
62 views

Compare difference between mean and actual

My problem is: I have two sets of numbers as follows: $X = {x_1, x_2, ..., x_n}; Y = {y_1, y_2, ..., y_m}$. Where $r$ is the actual value. $x^*$ is the mean of set X, $y^*$ is the mean of set Y, (n!=m)...
3
votes
1answer
103 views

Close approximation for absolute value function

I made a very acurate approximation function for $\sqrt{n^{2}+1}$ It is $\sqrt{n^{2}+1}\approx\frac{2n(n^{2}+1)}{2n^{2}+1}+\frac{2n^{2}+1}{n(4(2n^{2}+1)^{2}+1)}$ From this I can make a very close ...
0
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0answers
14 views

Automorphisms of local field

(1)Suppose that $K$ is a local field but not $\mathbb C$. Then show that any automorphism of $K$ is continuous. (We can assume that $K$ is $\mathbb R$ with classical absolute value or $K$ is a finite ...
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2answers
38 views

An equation with a parameter

Given the equation $(|x+1|+|x-a|)^2-2(|x+1|+|x-a|)+4a-4a^2=0$ find all possible $a$ such that this equation has only one solution. I wanted to solve it like this: $(|x+1|+|x-a|)^2-2(|x+1|+|x-a|)+4a-...
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0answers
19 views

$\sqrt[n]{x}$ as a power series in a complete field with absolute value.

Let K be a field complete with respect to a non-archimedean absolute value $| \cdot|:K^*\rightarrow \mathbb{R}$ and $char(K)=0$ or $char(K)\nmid n$. I want to prove that if $|x|\leq 1$ (i,e $x$ is in ...
1
vote
2answers
56 views

Is my hypothesis correct? [closed]

$$\left| \left|(a^2) - 25\right|-b\right| + b = 0$$ You have to prove that $b<0$ and $b=0$ at the same time I have no problem to prove that $b$ can be $0$ the thing that I need help with is $b<...
1
vote
1answer
13 views

Mixed Integer linear programming - absolute value of a variable not involved n the objective function

I'm looking to find the absolute value of the expression s-t. I have begun by introducing the following constraints: Where A is the absolute value. Unfortunately, A is not involved in the objective ...
0
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1answer
19 views

Real Analysis Absolute values [closed]

Someone please help me with detailed explanation on how to solve this problem. For all $a, b \in \Bbb R$, show that; $$ | a - b | \geq | a | - | b | $$
6
votes
3answers
518 views

Absolute value graph sketching

Where would you start if you were told to plot: $$||x-1|-1|$$ I looked at just $f(x) = |x-1|$ and noticed that the two equations are: $\pm (x-1)$ for $x \geq 1$ and $x < 1$. Extrapolating then: $\...
1
vote
1answer
28 views

Expressing the minimum function in terms of the absolute value in a symmetric manner (generalized to more variables)

It is well known that: $$\max(a,b) = \frac12(a+b)+\frac12|a-b|$$ and similarly: $$\min(a,b) = \frac12(a+b)-\frac12|a-b|.$$ In fact, they are equivalent since $\max(a,b) = -\min(-a,-b)$. We can try ...
1
vote
1answer
46 views

Spivak Calculus 3rd. Edition Chapter 1 Problem 12 (v) and (vi) Proofs Critique

Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next ...
1
vote
1answer
24 views

Spivak Calculus 3rd. Edition Chapter 1 Problem 12 (iii) and (iv) Proofs Critique

Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next ...
2
votes
1answer
22 views

non-Archimedean Valued field extension of $\mathbb{R}$

Let $K$ be a field with non-Archimedean valuation $|\cdot|$. Suppose that $\mathbb{R}\subset K$. Question 1: Is the restriction of $|\cdot|$ to $\mathbb{R}$ the trivial valuation? I guess that the ...
2
votes
1answer
50 views

Spivak Calculus 3rd. Edition Chapter 1 Problem 12 (i) and (ii) Proofs Critique

Here are my "proofs" for Spivak's Calculus Chapter 1 Problem 12. I am new to this level of rigour and I am attempting to intimate myself with more advanced topics of mathematics to prepare for next ...
2
votes
0answers
33 views

Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
2
votes
1answer
44 views

Is there a solution to the absolute value of an expression which results in a negative value?

The equation given: $ \mid x - 4 \mid = -3$. My instinct (and example 2 in this article) tells me that there shouldn't be any solution as there would be no value of x which would result in a negative ...
0
votes
1answer
22 views

absolute value (positive and negative part)

I found a notation that $$|y_i| = y_i^{+}+y_i^{-} $$ where y is y n dimensional vector. what does +/- imply? I understand that we can have both negative and positive value into absolute function, ...
0
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0answers
28 views

The maximum of the absolute value of a real-valued function

For a continuous real-valued function $f$ on a compact set $X\neq \emptyset$, we have \begin{equation*} \max_{x\in X}|f(x)| = \max \{\max_{x\in X} f(x), \max_{x\in X} -f(x)\}. \end{...