For questions about or involving the absolute value function.

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0
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1answer
19 views

Real Analysis Absolute values [on hold]

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
479 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
19 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
33 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
19 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
18 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
40 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
27 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
39 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
19 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
27 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)\}. ...
0
votes
1answer
35 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 ...
1
vote
1answer
70 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
votes
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
vote
2answers
30 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 ...
24
votes
1answer
272 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
votes
4answers
39 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
24 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
62 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
votes
1answer
24 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
votes
1answer
31 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 ...
0
votes
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
136 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
859 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
votes
2answers
53 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
42 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 ...
-5
votes
1answer
29 views

Solve for n when y=4 [duplicate]

Determine value of n that satisfy the equation when $y=4$ Question 1) $$Y=n^2-8n+16$$ When $y =4$ I had help with another equation but I still don't understand how to completely solve for $n$
2
votes
0answers
25 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
votes
1answer
24 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
votes
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
vote
2answers
51 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
54 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
29 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: ...
0
votes
1answer
20 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. ...
0
votes
0answers
43 views

Solve for $\phi$

I want to find value of $\phi$ in this situation $$2 a^2 \cos \phi \sin \phi=\gamma\frac{\partial}{\partial \phi} (|a \cos \phi- \gamma|)$$ What I did is like this $$2 a^2 \cos \phi \sin \phi ...
2
votes
3answers
51 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
vote
1answer
41 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
55 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
25 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
79 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 ...
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vote
2answers
34 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 ...
2
votes
1answer
30 views

The time derivative of the absolute value of a gradient.

I am interested in finding out the time rate of change of the absolute value of the density gradient, such that the directional change of the density gradient does not affect the final sign of the ...
3
votes
1answer
26 views

Evaluate x in this absolute value form equation [closed]

|x-1|+|x-2|=|x-3| Can you show me the solution to this equation?
1
vote
1answer
37 views

Graph the solutions of $ | z-2| + |z+2| < 5 \quad z\in \mathbb{C} $ [closed]

I really don't get how to solve this kind of equations and inequalities on complex numbers. Can someone solve this as an example, or others similars to teach me how to do it please? Thanks a lot.
1
vote
0answers
33 views

Usefulness of absolute value in optimization algorithms

In a course of Optimization Algorithms at university, professor said that in every algorithm the objective/object function/function cost is defined as: $$f(\bar x)=\lvert x_0 - g(\bar x)\rvert^{2}$$ ...
3
votes
1answer
253 views

The case $x < - 3$ in the absolute value equation $|x + 3| + |x - 2| = 5$

In the absolute value equation $|x + 3| + |x - 2| = 5$, why do we replace $|x + 3|$ by $-x - 3$ rather than $3 - x$ when $-\infty < x < -3$? $$|x+3|+|x-2|=5$$ What is the result set? ...