For questions about or involving the absolute value function.

learn more… | top users | synonyms

0
votes
2answers
41 views

Proving uniform continuity of absolute value

Prove that the function $f(x) = |x-a| - |x-b|$ is uniformly continuous on $\mathbb{R}$.
0
votes
1answer
19 views

Modulus Inequality

Solve the inequality $$2|x-3| > |3x+1|$$ Is sketching the only way I can solve ALL modulus equations and inequalities? Does an algebraic technique work for all modulus equations and inequalities?
1
vote
1answer
38 views

Solve the inequality…

Can you please show me how can I solve this inequality. I would like to see how it can be done without the graph of the functions. Thank you! $$2\sqrt{(x-1)(x+2)}\ge|x+1|-2$$
0
votes
1answer
48 views

Solve the following inequality…

Can you please verify if I've done this exercise correctly, and if you have a better solution, please, show it to me. Thank you! (The exercise is in the left top corner.)
1
vote
1answer
33 views

Triangle inequality frobenius norm

I'm trying to show that the frobenius norm is a norm. however it appears as if triangle inequality isnt met. $$||A+B||_F = \sqrt{\sum_{i,j=1}^n |a_{ij}+b_{ij}|^2} \leq \sqrt{\sum_{i,j=1}^n ...
0
votes
1answer
28 views

Equivalent form not using absolute values

Looking at the solution of Trench´s Introduction to Real Analysis exercises, I am struggling with this: Write the following expression in equivalent form not involving absolute values: $a + b + 2c + ...
0
votes
0answers
40 views

Simple proof explanation - Possibly triangle inequality involved

I'd like some help with understanding the following statements...I saw it on the internet while searching for a proof, and I'd like to understand why its true: let $A$ be a diagonally dominant matrix ...
-1
votes
1answer
23 views

Help determining if an equation is a function of x

Graph: ${y\over|y|}={x\over|x|}$ ${\lfloor x \rfloor \lfloor y \rfloor = 1}$ Determine if each graph represents a function of x and explain your answer. I've never seen anything like the before ...
0
votes
1answer
39 views

Periodic absolute value function

Define $$h(x)=|x|$$ on the interval $[-1,1]$ and extend the defintion of $h$ to all of $\mathbb{R}$ by requiring that $h(x+2)=h(x)$. Now define the function: $$h_n (x)=\frac{1}{2^n} h(2^n x)$$ ...
2
votes
2answers
63 views

Solving inequalities with absolute values

This is the question: $$ \left| \frac{x+2}{3(x-1)} \right| \leq \frac{2}{3} $$ And this is my working out, first I squared both the numerator and denominator, then solved it as if it was a normal ...
0
votes
0answers
31 views

Looking for a counter example: limit of absolute value of $f(x)$

Consider the following: $$\lim_{x\rightarrow a}f(x)=L\Rightarrow \lim_{x\rightarrow a}|f(x)|=|L|$$ I proved it using the "second triangle inequality", but I tried to think why is the reversed ...
0
votes
5answers
87 views

Why is $f(x)=|x|$ not differentiable?

Consider the function $f(x)=|x|$, I know that $f$ is not differentiable at $x=0$, but still, when you try to differentiate $f(x)=\sqrt{x^2}$ (which is exactly the same), you get: ...
0
votes
1answer
17 views

Steps to Graph Exponential Equations & Absolute Value

how to sketch: $-e^{|-x-1|} + 2$ Can someone clarify: $|f(x)|:$ we draw $f(x)$ and then reflect the ($-y$ parts) in the $x$-axis $f|(x)|:$ we draw $f(x)$ and then reflect the ($-x$ parts) in the ...
1
vote
2answers
30 views

I need help on this differential equaion problem?

Let equation $(1)$ be $\overrightarrow{F}= m \cdot \overrightarrow{a}$ and equation $(2)$ be $\overrightarrow{F}= \frac{-G \cdot M \cdot m}{ | \overrightarrow{r^2} |} \frac{\overrightarrow{r} }{ ...
0
votes
1answer
57 views

Derivatives of Norms and Absolute Values (distributions)

For example we have for $x \in \mathbb{R}$, $$\frac{\partial}{\partial x}\left| x\right| = 2\Theta(x) -1 $$ and thus $$\frac{\partial^2}{\partial x^2}\left| x\right| = 2\delta(x) $$ We also have, ...
0
votes
1answer
30 views

How to take derivative of sums of absolute values

Take the derivative of $f(m) = \sum_i | x_i - m |$. I've been told that derivative of each term is +1 or -1. How do you show that?
1
vote
4answers
104 views

Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$

Let $a,b,c$ real numbers. Prove the inequality $|a-b| \leq |a-c| + |c-b|$. Prove that equality holds if and only if $a \leq c \leq b$ or $b \leq c \leq a$. I've tried starting with just $a \leq ...
2
votes
1answer
23 views

Real parameter equation

I'm having a problem with this question: For which values of the real parameter a the equation: $$||x|-1|=a$$ has exactly 4 solutions? The solution is this: $$0 < a < 1$$ What I tried was ...
-1
votes
4answers
51 views

A function where absolute maximum is also absolute minimum?

What is an example of a real-valued function where an absolute maximum is also an absolute minimum?
0
votes
0answers
24 views

Formula to convert value to absolute value

This is probably a 'dumb' question (it's a while since I studied maths) but is there a way to convert a value to an absolute value using only the +,-,x,/ symbols? I'm pretty certain that the only way ...
2
votes
1answer
52 views

$|x|^{|x|}$ is continuous in $\mathbb{R}$

I'm trying to show this now my self, but still no go. There isn't really a concrete attempt that I can show.. Help?
2
votes
7answers
116 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
0
votes
1answer
47 views

How do I prove that $|x+y| \ge \big||x|-|y|\big|$?

I don't know where to start with proving this. Any help will be greatly appreciated.
2
votes
0answers
37 views

Fourier Transform of inverse powers of the absolute value

I don't think this question has been asked previously, so here goes. I need to evaluate the following integrals - $$ ...
0
votes
1answer
20 views

finding an absolute value inequality

The question asks, "find an absolute value inequality whose solution's are x>2 and x<-12". I have no idea where to start and was wondering if anyone could help
1
vote
1answer
24 views

Rearranging absolute values (limit proof)

My textbook ends a proof with the following: $|x-9| \over \sqrt(x) + 3$ < $\epsilon$ can be rearranged to conclude: |$x-9 \over \sqrt(x) -3$ - 6| < $\epsilon$ However, I don't understand ...
0
votes
2answers
21 views

Determine a value $t_0$ such that $|u(x,t)| = |-4e^{-2t/5}\cos2x\;| < 0.0001$

My problem is the following: Determine a value $t_0$ such that $|u(x,t)| = |-4e^{-2t/5}\cos2x\;| < 0.0001,$ for $t > t_0$, with $0<x<\pi$. How to approach this problem? According to my ...
0
votes
1answer
30 views

How to take the laplace of $e^{-|t|}$

I seem to be having some trouble trying to compute the laplace transform of this function. I looked on Wolfram and it said the answer was simply $$\dfrac{1}{s+1}$$ but I highly doubt that is the ...
0
votes
1answer
31 views

Finding the solutions of nonlinear system with absolute value

I need to show that the initial value problem: $\dot{x}=|x|^{1/2}$ $x(0)=0$ has 4 different solutions through the point (0,0). The problem also says that I have to sketch the solutions in the ...
0
votes
2answers
47 views

Proving absolute value inequality by contradiction

Prove that for $|x|, |y|, |z| \geq 2$ the following holds: $|x^2 + y| + |y^2 + z| + |z^2 + x| \geq |x| + |y| + |z|$ So I thought about a simple proof by contradiction but am not sure whether it's a ...
1
vote
1answer
43 views

Explanation for |x|=-x if x<0

Can someone explain $|x|=-x$ if $x<0$ . I've proven various theorems in my real analysis text for homework, but I cannot see how $|x|=-x$ if $x<0$ makes sense.
0
votes
3answers
76 views

Proof for absolute value inequality of three variables: $|x-z| \leq |x-y|+|y-z|$ [duplicate]

$|x-z| \leq |x-y|+|y-z|$ We know that both LHS and RHS are non negatives. So, I thought of proving this by comparing the squares of both sides but can't advance beyond that step. Any help would be ...
0
votes
1answer
40 views

How do I expand absolute values?

If we have this expression: $$f = uu-\left( u + \frac{\partial u}{\partial x} \delta x \right) \left( u + \frac{\partial u}{\partial x} \delta x \right)$$ we can expand it to this: $$f = u^2-\left( ...
0
votes
1answer
115 views

Logarithmic inequalities

Full disclosure: This is a homework problem, but my question is regarding a concept that came about during solving the problem, not the actual solution to the problem. Problem: Rewrite as geometric ...
0
votes
2answers
38 views

How to integrate absolute function

I have this absolute e-function, but I don't know how to calculate the integration $$ \int_{-2}^{2} e^{\frac{1}{2}j\omega |x|}dx $$ Any idea?
0
votes
0answers
33 views

Inequality with false solutions. Why? [duplicate]

When you have a question like $|x| = 3x – 2$, why do false solutions occur? if $x>0$, $x = 3x- 2$ $-2x = -2$ $x = 1$ If $x<0$, $x = -3x + 2$ $4x = 2$ $x = 1/2$ The $1/2$ solution is ...
0
votes
2answers
40 views

Absolute value question false solution

|x| = 3x – 2 Why does this statement eventually give you a solution that isn't valid. So this equation comes out: x = 3x - 2 2 = 2x x = 1 OR x = -3x + 2 4x = 2 x = 1/2 However 1/2 doesn't ...
1
vote
1answer
32 views

Continuity of absolute value

Let $f(x)$ be a continuous function. Prove that $\left|f(x)\right|$ is also continuous. Is it correct to say that, by the reverse triangle inequality, $\left|f(x)-f(c)\right| \geq ...
0
votes
3answers
45 views

Absolute value question

Is it true that:$$\left|\,a-b+c-c\,\right|=\left|\,(c-a)+(c-b)\,\right|,$$ or, alternatively, $$\left|\,a-b+c-c\,\right| = \left|\,(a-c)+(c-b)\, \right|?$$ Why is this the case?
0
votes
1answer
56 views

Absolute value and credit card balance

I'm embarrassed to ask this question, but my child has the following homework question: "Use absolute value to describe the relationship between a negative credit card balance and the amount owed." ...
0
votes
2answers
38 views

$|a-b|+|b-c|+|c-a|=2(\max\{a,b,c\}-\min\{a,b,c\})$

Let $a,b,c ∈ \Bbb R$ Show that $|a-b|+|b-c|+|c-a|=2(\max\{a,b,c\}-\min\{a,b,c\})$ Not sure where to start
0
votes
0answers
13 views

Prove that the following function of binary random variables is monotonic

Consider a binary random variable $y$ over the space $\mathcal{Y} = \{+1, -1\}$ such that $\Pr(y = 1) = q$. Consider also $r$ binary random variables $y^1, \ldots, y^1$ over the space $\mathcal{Y}$ ...
2
votes
2answers
50 views

Strategy to solve absolute value inequality

I was wondering if there is any strategy to solve absolute value On both sides inequalities, for example, $$| x^2 -3x + 2 | < | x + 2|$$ Thanks, Eli
0
votes
1answer
40 views

Complex number in polar coordinates

I have to get $\Im$, $\Re$, the absolut value as well as the argument $\phi$ of the complex number $$z = \left(-\frac{1}{\sqrt2}+\sqrt\frac{3}{2}i\right)^8$$ I do this by transforming $z' = ...
0
votes
2answers
43 views

Why is the following simplification possible?

I have seen the following simplification: $$\left|\frac{1}{(-1-\frac{1}{n})^4 - 1}\right| = \frac{1}{\left|-1-\frac{1}{n}\right|^4 - 1}$$ I really don't have a clue why this is possible... I am ...
1
vote
0answers
31 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
0
votes
1answer
21 views

Calculate the area that the following graphs form

I have been trying and trying to solve the following problem (I even used wolframalpha as an extra help, but no success, and I have like 100 calculations in my notebook): The Task: Calculate the ...
10
votes
6answers
1k views

How to calculate with absolute value.

Calculate:$$\frac{ \left| x \right| }{2}= \frac{1}{x^2+1}$$ How do I write the whole process so it will be correct? I need some suggestions. Thank you!
0
votes
0answers
52 views

about vectors norm

in the following article http://blanche.polytechnique.fr/~mallat/papiers/MallatPursuit93.pdf page 3 he say: $$y= \langle y , a_{k_0} \rangle a_{k_0} + R $$ with $a_{k_0}\in D$ with $\forall ...
0
votes
2answers
38 views

What is the Laurent series of the complex absolute value?

What is the Laurent series of the function $f(z) = |z|$? It seems to be ill defined at $z=0$. Are there any other expansion techniques applicable for this function at $z=0$?