For questions about or involving the absolute value function.

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0
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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 a(i)=b(i),...
1
vote
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
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
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
votes
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
votes
1answer
32 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
33 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
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
139 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
874 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
55 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
votes
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
44 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
42 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
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
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
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
60 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
votes
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
votes
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
54 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
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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
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
58 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
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 ...
2
votes
1answer
53 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
32 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
255 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? ...
1
vote
1answer
36 views

What are the solutions to the inequality $|x|(ax+1)<2$?

Solve absolute value inequalities depending on the parameter $a$. $$|x|(ax+1)<2$$ In the first case where we have $x>0,\;a>0$ we get: $$ax^2+x-2<0$$ I get that $x$ is in the interval $\...
7
votes
3answers
90 views

Solving modulus inequality $|x - 1| + |x - 6|\le11$ geometrically

Find all possible values of $x$ for which $x$ for which the inequality $$|x - 1| + |x - 6|\le11$$ is true. I know this can be easily solved by taking $3$ cases for$x$ and then taking the intersection ...
1
vote
4answers
61 views

Absolute value, and multiplying by x

How would you solve $$\left|\frac{-2}{x}\right|<1?$$ I did $-1<-2/x<1$ but my teacher always tells me not to multiply by variables, so what do I do?
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votes
2answers
38 views

Is there any complex value for $x$ where $|x| < 0$?

What I'm really asking is if I get to a point in a calculation where I have $|x| = -4$, do I say There is no solution for $x$ or do I say There is no solution for $x ∈ ℝ$
0
votes
1answer
39 views

Magnitude of complex function

I was going through an example in a book and and it says to take the magnitude of the function. What it shows is $$X(\omega)=\frac 1{\alpha+j\omega} \implies |X(\omega)|=\frac 1{\alpha^2+\omega^2}$$ ...
1
vote
1answer
293 views

A proof that $|x+yi|=\sqrt{x^2+y^2}$, based on the given the conditions

If we attempt to define $|x+yi|$ by following conditions: $|x|=|xi|=x\operatorname{sgn}(x)$ (implicitly meaning the result will always be $\ge 0$) $|xz|=|x||z|$ $|z^x|=|z|^x$ for $x \in \mathbb{R}...
-1
votes
2answers
43 views

Determine if the function is injective. [closed]

prove that $f(x)=\frac{1+|4x+1|}{2}$ is injective or not, thanks. I can think of counterexamples of it being not injective, but only with non-integers, but $x,y$ must be integers. Suppose that f(x)=f(...
1
vote
1answer
37 views

$\lim_{x \to -1} \frac{|x + 1|} { x^2 + 2x + 1}$?

Is the solution: "no limit" Because: $$\frac{|x + 1| }{ (x^2 + 2x + 1)} = \frac{1}{(x + 1)}$$ Or is factoring different for $|x + 1|$?
2
votes
1answer
35 views

Range of values which satisfy this inequality

Consider the following inequality: $$|f(a)| = \left|\frac{1}{2}(a \pm \sqrt{a^{2}-2})\right| \leq 1$$ I got this inequality while doing stability analysis of a fixed point of a certain discrete ...
-2
votes
1answer
53 views

Absolute value proof for $|x−y| = ||x|−|y||$ [closed]

How would you prove $|x−y| = ||x|−|y||$ please?
0
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0answers
15 views

determine the absolute difference in bits between using an indexed colour

Given the following 4 x 4 image, determine the absolute difference in bits between using an indexed colour (where the index uses the lowest number of bits possible, but the colour is represented as ...
0
votes
1answer
23 views

Proving the existence of a inequality (concerning with distance of numbers) from other three inequalities

This is an exercise from Terence Tao's analysis 1 book, chapter of integers and rational numbers. Let $x$, $y$ and $w$ be rational numbers, and let $d(a,b):=|a-b|$ for any rationals $a$ and $b$. If ...
0
votes
2answers
44 views

finding median with cumulative distribution function (absolute value)

I am currently working on distribution with density function $$f(x)=\begin{cases} \frac{2}{5}|x-2|,& \text{0 ≤ x ≤ 3} \\\\0 & \text{otherwise}\end{cases}$$ I have found that cumulative ...