For questions about or involving the absolute value function.

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2answers
21 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
58 views

I need help solving: $-2|x| = 8$ [on hold]

I need to determine whether or not the following equation is linear, then I have to find its solutions. I don't know how to do that. $$ -2|x| = 8 $$
0
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1answer
39 views

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 ...
0
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0answers
29 views

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 ...
2
votes
2answers
82 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) = ...
0
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1answer
15 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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0answers
30 views

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 ...
0
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0answers
30 views

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
49 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
42 views

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 ...
3
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4answers
75 views

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 ...
2
votes
3answers
43 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 ...
3
votes
3answers
113 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 ...
2
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2answers
51 views

Advanced Algebra Manipulation/Inequality Proof

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 ...
1
vote
2answers
339 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:)
2
votes
2answers
65 views

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 ...
0
votes
1answer
21 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 ...
1
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5answers
319 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|) ...
1
vote
3answers
46 views

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
57 views

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 ...
0
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2answers
29 views

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 ...
3
votes
1answer
57 views

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 ...
1
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1answer
38 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 ...
2
votes
6answers
66 views

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 ...
15
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5answers
2k views

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 ...
1
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0answers
39 views

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
54 views

$|f(x)g(x)| = |(f(x)||g(x)|$ [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 ...
4
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1answer
40 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
votes
1answer
51 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) ...
0
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0answers
29 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 = ...
0
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1answer
13 views

Normal Distribution $r-1$ th moment with absolute value

I was stuck for this problem whole night and I tried numerical solution using MATLAB and the following result seems hold for x follow normal N(0,1) and for any positive number (not integer only) ...
0
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1answer
23 views

2nd derivative of a functions absolute value

So on wolfram alpha I am told that if $y=y\left ( x \right )$ then $\frac{d^{2}}{dx^{2}} \left | y \right |= \frac{y}{\left | y \right |}y^{''}+2\delta \left ( y \right )y^{'2} $ See it at this link ...
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0answers
24 views

Least Square Approximation Using Legendre Polynomials

Obtain a fourth degree least squares polynomial for $f(x) = 1/|x|$ over $[-1,1]$ by means of Legendre Polynomials I got stuck when trying the integral over the given interval. Is there another way ...
0
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1answer
58 views

Integral of reciprocal absolute value function

I'm having issues with the integral $$\int_{-1}^1 \frac{1}{|x|}dx$$ Solving it conventionally gives me values such as $\ln 0$ and $\ln(-1)$ which are indeterminate on the real plane. Is there a way to ...
3
votes
3answers
305 views

Number of real roots

Find number of real roots of the equation $$3^{|x|}-|2-|x||=1$$ My try:I have tried to remove the modulas by assuming x in some intervals and moved the linear part to RHS and drawn the rough graph ...
1
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2answers
26 views

Finding first and second derivative of an function with an absolute value

Given the equation $f(x)= |x^2-9|$ where $-4\le x\le 5$, I must find the extremes, as well as the concavities. This I know how to do. The issue is I'm unfamiliar on how to find the first and second ...
0
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2answers
53 views

Solve $1<\left(\dfrac{3x^2-7x+8}{x^2+1}\right)\leq 2,\ \ x\in\mathbb{R}$

Solve $1<\left(\dfrac{3x^2-7x+8}{x^2+1}\right)\leq 2,\ \ x\in\mathbb{R}$ options $a.)\ 1<x<6\\ b.)\ 1 \leq x<6\\ c.)\ 1<x\leq 6\\ \color{green}{d.)\ 1\leq x \leq 6}$ I ...
2
votes
5answers
37 views

solve $\dfrac{x^2-|x|-12}{x-3}\geq 2x,\ \ x\in\mathbb{R}$.

solve $\dfrac{x^2-|x|-12}{x-3}\geq 2x,\ \ x\in\mathbb{R}$. options $a.)\ -101<x<25\\ b.)\ [-\infty,3]\\ c.)\ x\leq 3\\ \color{green}{d.)\ x<3}\\ $ I tried , Case $1$ ,for $ ...
0
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1answer
22 views

Solving a system of equations with an absolute value term

$x$ and $y$ are two integer numbers and $x \geq y$. The values of $x$ and $y$ are positive or negative integers. When the sum of these two numbers are multiplied by $y$ we obtain $P$ and when the ...
2
votes
1answer
92 views

Why is the definition of the absolute value $|x+1|$ the way it is?

In my notebook it is given that for the above function, we would have: $f(x) = {-(x+1), x<-1; (x+1), x\geq-1}$ What I don't get is why did we take $-1$ instead of $0$ as is the case for the ...
2
votes
2answers
34 views

Explaining why the absolute value of an odd function is even.

For the following: If $f(x)$ is an odd function, then $|f(x)|$ is _____. I said even, because I graphed an odd function and then the absolute value of it and ended up with an even function. The ...
1
vote
1answer
102 views

Quick question about absolute value

Hello I am just having a quick question in the textbook intro to real analysis, during one of the limit examples the author notes, if $$|x-c| \lt 1$$ then $$|x| \lt |c| +1$$ What rules are used to ...
0
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5answers
34 views

Roots of Unity: second largest value and absolute value

Consider the $n$th roots of unity $e^{2 \pi i k/n}$ for fixed integer $n \geq 2$ and $0 \leq k < n$. Now I am interested in the second largest value (in absolute value) of the values ...
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votes
2answers
95 views

Under what conditions is $|x+y|=|x|+|y|$ true? [duplicate]

What instance that this equation would be true? $|x+y|=|x|+|y|$ Given that $x$, $y$ are elements of real numbers.
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1answer
21 views

Jargon for maximum/minumum absolute value in a set

Given a group of numbers $-5,-3,1,2$, the maximum is 2, the minimum is -5. What is the mathematical jargon for the maximum and minimum in absolute terms (i.e. -5 and 1 respectively)? Basically, I ...
2
votes
2answers
88 views

Distribution of minimum absolute value

Consider $K$ independent Laplace variables $X_k, k=1,\ldots,K$, with mean 0 and scale $\lambda$ (so that their PDF is $f(x)=\frac{1}{2\lambda}e^{-\frac{|x|}{\lambda}}$. Let $Y$ be the variable taking ...
3
votes
3answers
81 views

Why is $\max(x, x')$ equivalent to $\frac{1}{2}( x + x' + |x - x' |)$?

Why is it that $$\max(x, x') = \frac{1}{2}( x + x' + |x - x'|)$$ is true? Is it supposed to be obvious? Because it seems to come out of thin air for me. Anyway, I've verified this by plotting it in ...
1
vote
2answers
54 views

Quadratic Absolute Value Equation

Problem: Find all $x$ such that $|x^2+6x+6|=|x^2+4x+9|+|2x-3|$ I can't understand how to get started with this. I thought of squaring both sides of the equation to get rid of the modulus sign, ...
3
votes
3answers
39 views

solve $|x-6|>|x^2-5x+9|$

solve $|x-6|>|x^2-5x+9|,\ \ x\in \mathbb{R}$ I have done $4$ cases. $1.)\ x-6>x^2-5x+9\ \ ,\implies x\in \emptyset \\ 2.)\ x-6<x^2-5x+9\ \ ,\implies x\in \mathbb{R} \\ 3.)\ ...
0
votes
4answers
112 views

I have discovered a way to calculate the absolute value (area,volume, etc) of a n-dimentional shape, using it's coordinates only, how do I publish it?

Firstly, I want to preface by saying that I am no experience with the maths community at all, however I did take Maths and Further Maths for my A-Levels. What I have discovered is a way of using ...