For questions about or involving the absolute value function.

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0
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0answers
63 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
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1answer
106 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 ...
0
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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-...
0
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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
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1answer
15 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
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3answers
529 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
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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
54 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
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1answer
27 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
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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
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1answer
59 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
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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{...
0
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1answer
37 views

Does equality of the sum of two such series imply equality of each term of that series?

Let a(1)< a(2) < ..< a(m) and b(1)< b(2)<..< b(n) be real numbers such that $$\sum_{i=1}^m |a(i)-x| = \sum_{j=1}^n |b(j)-x|$$ for all x belonging to R. Show that m=n and a(i)=b(i),...
1
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1answer
75 views

$\lim_{x\to a}(f(x)+\frac{1}{|f(x)|})=0$. Find $\lim_{x\to a}f(x)$ [duplicate]

Let a function $f$ be defined in a hollow neighborhood of $a\in \mathbb{R}$, and suppose : $$\lim_{x\to a}\left(f(x)+\frac{1}{|f(x)|}\right)=0$$ Find $\lim\limits_{x\to a}f(x)$ and prove that this ...
0
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1answer
24 views

integral of two functions absolute

I've got the two function: f(x) = -4x + x³ and g(x) = 5x they meet each other at -3, 0 and 3, where the areas between -3 and 0 ...
1
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2answers
38 views

Help to solve absolute value inequality

The inequality I have is $\frac {\mid x-1 \mid} {(x+2)} <1 $ what I'm not sure is how I am supposed to proceed. I cannot multiply by (x+2) because it is unknown whether it is positive or negative. ...
0
votes
1answer
38 views

Maximum value and the absolute value

Let $f\colon X \rightarrow \mathbb{R}$ be a function such that $\max_{x\in X} f(x) + \min_{x\in X} f(x) = 0$. Does it then follow that $\max_{x\in X} f(x) = \max_{x\in X} |f(x)|$? I'm quite sure it ...
25
votes
1answer
289 views

Is there an easy way to see that this simple recurrence is 9-periodic?

In a colloquium talk yesterday, Robert Bryant pointed out that for all initial values $a_0, a_1 \in \mathbb{R}$, the sequence generated by the recurrence relation $$ a_{n+1} = |a_n| - a_{n-1} $$ turns ...
0
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4answers
41 views

What is the solution to this inequality?

This is the given inequality I've been trying to solve $$1/6\leq \frac{1}{\mid x \mid} \leq 1/2$$ However the answer I get is $(0,6] \cup [2,6]$ which is not the answer given in my book. Could you ...
0
votes
1answer
28 views

Absolute value inequality explanation

I was solving the inequality $-4 \le \left|\frac {x+4} {2-x} \right| \le 4$ and I first wrote the domain, which is $(-\infty,-4] \cup (-4,2) \cup (2, \infty)$ and I got the solution that $x \le \frac ...
3
votes
3answers
66 views

What are the solutions of $|x+y|=|x|+|y|$?

So I am having a problem in solving this type of equation. The problem I am dealing with is... $$\left|(2x-1) + \frac{3x-1}x\right| = \left|2x-1\right| + \left|\frac{3x-1}x\right|$$ Please help me ...
2
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1answer
34 views

Does every non-Archimedean absolute value satisfy the ultrametric inequality?

The Archimedean property occurs in various areas of mathematics; for instance it is defined for ordered groups, ordered fields, partially ordered vector spaces and normed fields. In each of these ...
-1
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1answer
34 views

How to calculate the modulus of a complex number? [closed]

I know that for an equation of real numbers you could calculate the modulus as follows (if I am not mistaking): $$ x = a + b$$ $$|x| = \sqrt{a^2+b^2}$$ But now I found this equation with this result:...
0
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4answers
70 views

What region in $\mathbb{C}$ does $\left|{z-1}\right|+\left|{z+1}\right|$ = 2 describe?

I have played around with this a bit and keep getting something that doesn't seem right. Perhaps I'm overlooking something. Using the definition of distance in the complex plane I transform my ...
12
votes
3answers
140 views

Why $|x|$ is not rational expression?

I'm 9th grade student, and my teacher said that $|x|$ is not rational expression ( expression like $\frac{p(x)}{q(x)}$ s.t $p(x)$ and $q(x)\neq 0$ are polynomial) but he didn't have convincing reason. ...
6
votes
3answers
875 views

A basic inequality: $a-b\leq |a|+|b|$

Do we have the following inequality: $$a-b\leq |a|+|b|$$ I have considered $4$ cases: $a\leq0,b\leq0$ $a\leq0,b>0$ $a>0,b\leq0$ $a>0,b>0$ and see this inequality is true. However I ...
2
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2answers
56 views

How to solve the differential equation $y'=y(1-y)$.

Up until now, we simply rearranged and integrated both sides, so $$y'=y(1-y)$$ $$\frac{dy}{dx}=y(1-y)$$ $$\frac{dy}{y(1-y)}=dx$$ $$\int\frac{dy}{y(1-y)}=\int dx$$ With partial fraction decomposition ...
0
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0answers
11 views

generalised eigenvalue problem with absolute value

Problem: $\max_w |w^t A w|-|w^t B w|$ s.t $w^t C w=1$ If there was no absolute values, i.e. if the problem was $\max_w w^t A w-w^t B w$ s.t $w^t C w=1$ this would, by using the appropriate Lagrange ...
2
votes
4answers
51 views

How can an absolute value equation with a variable have both a positive, and negative answer?

I thought absolute values are supposed to be a number's distance from 0, which is always positive. So I had this equation: | 7 – y | = 12 According to practice tests they say this, This ...
0
votes
1answer
14 views

How to prove $|x-y|\le \delta\lor |x^2-y^2|\gt \epsilon$ with the following condition?

How to prove $\forall \epsilon\in \Bbb{R^+},\exists \delta\in\Bbb{R^+},\forall x\in\Bbb{R^+}, \forall y\in\Bbb{R^+}, |x-y|\le \delta\lor |x^2-y^2|\gt \epsilon$. My try: Pick $\delta=\frac{\epsilon}{...
3
votes
1answer
43 views

Confirmation on a monotonicity formula?

After a long series of difficult problems (which are completely irrelevant) I found myself experimenting with a way to convert a graphed function to a purely monotonic form (I hope I didn't butcher ...
0
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1answer
25 views

Why can't critical value/transitional points approach be used to solve this question?

Consider the following question: What is the sum of all possible solutions of the equation $|x + 4|^2 - 10|x + 4| = 24$? The answer is $-8$. I was able get $-8$ by doing it the regular way - ...
0
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1answer
30 views

Differential Equations: When do constants combine to be another constant?

I'm trying to isolate $y$. I have a constant times a negative one? Do I ignore the negative and leave is as a constant? Here's what I'm working with... $$|(6-2(y^3))| = ke^{-3\times2}$$ For a ...
1
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2answers
52 views

evaluate $\lim_{x\rightarrow 0}\frac{x}{\left | x-1 \right |-\left| x+1\right|}$

I just have a quick question about limits like this one: $$\lim_{x\rightarrow 0}\frac{x}{\left | x-1 \right |-\left| x+1\right|}$$ leaving it as is i get $$\lim_{x\rightarrow 0}\frac{x}{\left ( x-1 \...
3
votes
2answers
62 views

1987 AIME Problem #4

The Question: Find the area of the region enclosed by the graph of $|x-60|+|y|=|\frac x4|$. Answer: What I know: Because of all the absolute values I only need to find one side of the graph of ...
0
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0answers
31 views

Sketch subset of $\mathbb{C}$ which satisfies $|z-3-4i|=5$

I proceeded by plotting $z$ on the complex plane, and the modulus of $z-3-4i$: From this I deduced that: $$\begin{align}&\mathrm{Re}(z)=5\cos\theta+3\\&\mathrm{Im}(z)=(5\sin\theta+4)i\end{...
0
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1answer
22 views

Integral of reciprocal of absolute value

I am having trouble with the following question. For which values of $n, p$ is the integral $\int_{\mathbb{R}^n}\frac{1}{|x|^p}$ convergent? I need to derive a relationship between $n$ and $p$, i.e. ...
2
votes
3answers
55 views

Inequality involving the min function

I'm trying to prove the following inequality: $$ \left|y_{1}\land x_{1}-y_{2}\land x_{2}\right|\leq\left|y_{1}-y_{2}\right|+\left|x_{1}-x_{2}\right|, $$ where $x\land y=\mbox{min}(x,y)$. By ...
0
votes
1answer
54 views

Function inequalities

I want to resolve this inequality, any help? $$\left\lvert\frac {2+\sin (x)}{x+4} \right\rvert<k$$ for $k>0$. (I am sorry for my English. It's not my first language)
1
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1answer
42 views

What is the relation between the square root of the sum of squares and the sum of the absolute values?

I want to prove that $\sqrt{\sum a_{i}^{2}} \geq \sum \left | a_{i} \right |$, is it possible ?
1
vote
6answers
59 views

Solve for the values of $x$ in $|x+k|=|x|+k$ where $k$ is a positive real number

The question asks me for which values of the real number $x$ is $|x+k|=|x|+k$ where $k$ is a positive real number. How do I go about this? Can I square both sides to get rid of the absolute value ...
0
votes
1answer
26 views

Does there exist a perfect square in the form: $|x^2+52x|$, where $x\in \Bbb Z$? [closed]

Does there exist a perfect square in the form: $|x^2+52x|$, where $x\in \Bbb Z$? $x<0,x\neq-52$
1
vote
3answers
81 views

Does |x| = |y| requires checking conditions while solving?

I am trying to solve this equation $\lvert 2x \rvert = \lvert x - 2 + y \rvert$ (specifically, find set of all points $(x, y)$ satisfying equation). $\lvert 2x \rvert = \lvert x - 2 + y \rvert$ is ...
1
vote
2answers
35 views

Velleman exercise 6 in section 4.3

I am stuck on exercise 6 section 4.3, in Daniel J. Velleman's book "How To Prove It". I just need to prove the following, but cannot do it. The free variables $r$ and $s$ are arbitrary positive real ...