For questions about or involving the absolute value function.

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1answer
20 views

Given $f\in L^1(\mathbb{R})$ with $||f||_1 < \infty$, is it true that $\int_{\mathbb{R}} ||f||_1 - f(x) \, dx = 0$?

According to my intuition so far, the answer should be yes, hinging very important on the assumption that $||f||_1 < \infty$. To speak very roughly, if the $L^1$ norm of $f$ is finite, it seems ...
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2answers
20 views

Absolute Value Algebra with inverses

I noticed the following equality in some material regarding limits and infinite series. $$ \left |\frac{x}{x+1} - 1 \right| = \left |\frac{-1}{x+1} \right| $$ And I'm honestly stumped (and slightly ...
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2answers
32 views

Solving Equations And Inequations Based On The Absolute Function [on hold]

Today I came across some equations and inequations based on the absolute function. These were $|x^2+4x+3|+2x+6=0$ $|x^2+6x+7|=|x^2+4x+4|+|2x+3|$ $1\le |x-1|\le 3$ $\frac{2}{|x-4|}\gt 1$ $||x|-1|\le ...
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0answers
20 views

Adjustment of a percentage

Suppose my formula for something used to always be $$\frac{85}{1000} \frac{val}{1-P}$$ It turns out that $1000$ should really be $900$. Rather than changing the formula to $\frac{80}{900}$, I want ...
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0answers
26 views

Consider fourier transformations of $|p(\mathbf{r})|^2$

If we have $\mathbf{k}=(k_x,k_y,k_z)=\frac{2\pi}{L}(j,s,l)$ with $j,s,l \in \mathbb{Z}$ and we have $$p(\mathbf{r})=\sum_{\mathbf{k}}\tilde{p}(\mathbf{k})e^{-i\mathbf{k}\cdot \mathbf{r}} \implies ...
2
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1answer
20 views

Differentiability of an absolute function.

Check the differentiability of $f(x)=x|x|$, $x$ is in $\mathbb{R}$. I know that it is differentiable when $x>0$ and $x<0$. I am not sure about the case when $x=0$. I found that as $$\lim ...
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0answers
33 views

Simplifying a function with absolute value and two variables

I am trying to simplify the following $ f(\alpha, \beta)= \big|1-(\frac{\alpha+\beta}{2}+\frac{|\beta-\alpha|}{2})\big|-2 \big|1-\alpha-(\frac{\alpha+\beta}{2}+\frac{|\beta-\alpha|}{2})\big|+ ...
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0answers
31 views

Solve this inequality for B

I am working on a program that is supposed to qualify a value as "in range" and I have come up with the expression: $$\lvert a-b\rvert \leq c$$ to determine the value. Plugging in test numbers ...
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1answer
33 views

How to graph $|z-1| <2$

Am I correct to rearrange this to $(z-1)^2 < 4$, and hence just graph as a circle or am I completely off?
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1answer
28 views
0
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1answer
27 views

Decomposing absolute value terms

I have something like the following term: 7x1 + 9x2 + | 10 - 7x1 | + | 15 - 11x2 | I want to make it into something like this: Ax1 + Bx2 , where A and B are constants For two values x1 and x2, ...
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3answers
71 views

Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$

Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$.What will be the easiest techique to solve this sum ? Just wanted to share a special type of equation and the fastest way to solve it.I am not asking ...
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2answers
51 views

A difficult trigonometric integral involving absolute value

$$ \int_{0}^{2\pi}\lvert\sin(x)\rvert\cos(nx)\,dx= -\frac{4\cos^2\bigl(\frac{\pi n}{2}\bigr)\cos(\pi n)}{n^2-1} $$ I'm not actually trying to solve this myself. The answer appears in my lecture notes ...
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4answers
56 views

Prove that for $x\in\Bbb{R}$, $|x|\lt 3\implies |x^2-2x-15|\lt 8|x+3|$.

The problem I have is: Prove that for real numbers $x$, $|x|\lt 3\implies |x^2-2x-15|\lt 8|x+3|$. Since there aren't really any similar examples in my book, I've been unsure how to first approach ...
0
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1answer
30 views

how to calculate $\int (|1+x|-|1-x|) dx $ and $\int$ max {${1-x^2,0} $}?

I'm used to integrate normal functions, but here I got quiet confused because these integrals : $\int (|1+x|-|1-x|) dx $ $\int$ max {${1-x^2,0} $} Include absolute value, and and option to choose ...
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4answers
52 views

How do you algebraically derive “x <= 0” from “-x = | x |”

A = "-x = | x |" B = "x <= 0" If A, then B. By plugging in numbers or testing ranges less than zero, greater than zero, and equal to zero, I can verify that A ...
6
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3answers
51 views

Using the Mean Value Theorem, prove that $|\sin{a} - \sin{b}| \leq |a - b|$ $\forall a, b \in \mathbb{R}$

Using the Mean Value Theorem, prove that $|\sin{a} - \sin{b}| \leq |a - b|$ $\forall a, b \in \mathbb{R}$. I'm working towards figuring out an approach for finding that $|\sin{a} - \sin{b}| \leq ...
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3answers
480 views

A unique solution

Find the sum of all values of k so that the system $$y=|x+23|+|x−5|+|x−48|$$$$y=2x+k$$ has exactly one solution in real numbers. If the system has one solution, then one of the three $x's$, should be ...
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5answers
251 views

How to write an expression in an equivalent form without absolute values?

The question I have in front of me is the very first problem in Trench's Introduction to Real Analysis: Write the following expression in equivalent form not involving absolute values: $a+b+|a-b|$ ...
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2answers
29 views

Inequality with absolution value for complex number

How to show that inequality: $|1-\bar{\alpha} z| \ge |z-\alpha|$ $z$ and $\alpha$ are complex number, $\alpha$ is constans and $|z|<1$, $| \alpha| < 1$ I can proof that by using substition ...
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1answer
30 views

Metric and Absolute value function on $\mathbb R$ [closed]

I'm contemplating the notion of the absolute value function on $\mathbb R$ as well as of the usual metric on $\mathbb R$. It seems to me that each one of those can be seen in light of the other. ...
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4answers
68 views

is the following true: $|a-b| = ||a|-|b||$?

is $|a-b| = \bigg||a|-|b|\bigg|$ ? I have tried a few examples and they seems to come out true, but I can't find any rule stating it. Is it true for all $a$ and $b$? Or am I missing something? ...
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2answers
43 views

How to start proof of triangular inequality? [duplicate]

$$\left| {\left| a \right| - \left| b \right|} \right| \le \left| {a \pm b} \right| \le \left| a \right| + \left| b \right| $$
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3answers
82 views

Solve $|3-x|=x-3$.

Solve: $|3 - x| = x - 3$. Answer: $|u| = -u$ when and only when $u \le 0$. So, $|3 - x| = x - 3$ when and only when $3 - x \le 0$; that is, $3 \le x$. Hi! I'm new here. I'm working out of ...
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0answers
53 views

two dimensional Gaussian integral with complex exponent of an absolute value

I am trying to solve the following two dimensional integral: $$\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}{e^{ia(\left|x\right|-\left|y\right|)} \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{x^2+y^2-2\rho ...
2
votes
2answers
49 views

Prove triangle inequality using the properties of absolute value

So I was given the task of proving the following variant of the triangle inequality using only the properties of the absolute value: $\vert\lvert x\rvert -\lvert y \rvert \rvert \leq \lvert ...
2
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0answers
44 views

Simplifying an expression with absolute values

I am trying to simplify the function $D(\alpha,\beta)$ shown below (with $\alpha,\beta>0):$ $$ D(\alpha,\beta)=\frac{1+\alpha+2\beta}{2} + \frac{|\alpha-1|}{2} - 2 ...
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2answers
17 views

Build the graph of a function with absolute value.

The function is: And my idea of graphic (i did it using two graphs and deleting some parts) Is it correct?
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0answers
27 views

How to solve an equation with absolute value and x as exponent.

The inequation is: I thought that the solution would be the same as if there was no x as exponent, but in Microsoft Math it says it has no solution and about the equation it said it has solutions. ...
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1answer
24 views

Exercise on inequalities in bounded derivatives (from Spivak)

Suppose $f$ is two times differentiable in $(0,\infty)$ and that: $|f(x)| \leq M_{0}, \forall x>0$; $|f''(x)| \leq M_{2}, \forall x>0$. a) Show that $$|f'(x)| \leq ...
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1answer
20 views

A general method for solving inequations with absolute values

I've been asked to find which $b$ satisfy $|a + b| = |a| + |b|$ for $a \geq 0$. I'm familiar with the method described here and I tried to apply it but I'm confused about what I should do with the ...
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0answers
26 views

equivalence of norms vs equivalence of absolute values

Consider a field, $k$. A norm on $k$ is a map, $|-| : k \to \mathbb{R}$ such that $(1)$ $|v|= \iff v=0$ $(2)$ $|vw| = |v||w|$ $(3)$ $|v+w| \leq |v|+|w|$ and we say two norms $|-|_1, |-|_2$ are ...
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5answers
126 views

Proving $|a-1|+|a-2|+|a-3| \ge 2$

I need to prove the following sentence for $a\in\mathbb{R}$: $$ |a-1|+|a-2|+|a-3| \ge 2$$ Breaking the equation into cases it does work, i.e. for $a\le 1$: $$-a+1-a+2-a+3\ge 2$$ $$-3a \ge -4$$ $$a ...
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0answers
18 views

When does $\overline{U(0,1)}=B(0,1)$ hold?

Given $R$ an absolute valued ring (with unit), sometimes $\overline{U(0,1)}=B(0,1)$ (for example, $\mathbb{Q},\mathbb{R},\mathbb{C},\mathbb{H}$) and sometimes $\overline{U(0,1)}\neq B(0,1)$ (for ...
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4answers
24 views

Absolute value function inequality

I need to find the values of x that satisfy the inequality x|x| > x I know the possible outcomes are ...
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1answer
29 views

Modulus Inequalities Proof

Need to prove that: $$|x-1|+|x-5| \geq 6$$ I've tried squaring but I'm not sure if I'm doing it correctly? Thank you in advance Note: x is real and does not equal 1 or 5
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1answer
69 views

Evaluate $\int_{-1}^1 \frac {1}{\sqrt{|x|}} \text{d}x$

I need some help to solve this integral with absolute value. I'm not sure how to do these types of integrals. $$\int_{-1}^1 \frac {1}{\sqrt{|x|}} \text{d}x$$ Thank you
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3answers
73 views

Absolute value:$ |x-a| \leq \frac{|a|}{2} \Rightarrow |x| \geq \frac{|a|}{2}$

Prove that $$|x-a| \leq \frac{|a|}{2} \Rightarrow |x| \geq \frac{|a|}{2}$$ I've tried by direct proof and contradicion but nothing worked. I would like a hint or a tip of what should I do. Thanks ...
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2answers
64 views

For real numbers a and b, when is the equation |a + b| = |a – b| true?

I put that the statement was true only when a = 0 and b = 0 but the correct answer was that it only held true for a = 0 OR b = 0. With 'and' I figured |0 + 0| = 0 and |0 - 0| = 0. Could someone ...
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2answers
31 views

Finding functions max and min (abs value)

I have the function $$g(x)=|x^2-x-2|$$ which is defined on $$-\frac{3}{2}\leq x\leq \frac{3}{2}$$ I am struggeling with that g(x) has absolute values wrapped around. I taught that I just draw the ...
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1answer
21 views

What is this equal to? : $|A+B|^2$ where $A = P e^{ia}$ and $B = Q e^{ib}$

$A$ and $B$ are two complex numbers: $A = P e^{ia}$ $B = Q e^{ib}$ I would like to know what is this equal to? : $|A+B|^2$ Please also give a small proof if possible.
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1answer
16 views

Deploy the absolute inequality formua

Please help me derive that equation: $$f(x) = \begin{cases} 1, & \text{if $(x-a)^2-(x-b)^2>0$} \\ -1, & \text{otherwise} \end{cases}$$ where $x,a,b \ge 0$ Thank you in advance This is ...
2
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3answers
407 views

A simple inequality looking for a more elegant proof…

I have been given the following statement by a professor: Let $m\leq x\leq M$ and define $K=\max(|m|,|M|)$. Then $|x|\leq K$. Now I can clearly see that this is true, and working through ...
2
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3answers
76 views

Question regarding the square root of a squared number. [duplicate]

I've learnt that the square root of a number squared is equal to the absolute value of that number, but I haven't really understood why. I have looked through other questions on MSE but didn't really ...
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0answers
26 views

why and when t0 use norm instead of abs and vice versa

What is the difference between the norm and abs of an expression.. as far i understand does ||a - z|| mean norm and |a-z| abs , but what is the difference?
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1answer
24 views

Property of absolute value in the real numbers

To prove that $ \lvert a-b \rvert \le c-d $ for $ a,b,c,d $ in the real numbers, what needs to be shown? Is the fact that $a-b\le c-d$ enough? Or is there something more that needs to be shown?
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1answer
21 views

Definite integral of absolute value complex function

Seems pretty straight forward but absolute values have always given me headaches $$\int_0^1 |1 -t + it|^2$$ Now usually I get roots and split up the intervals for when the function is greater or ...
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2answers
42 views

proof : f continuous at a then |f| is continuous at a

Here's my proof, which I am not sure is correct : Assume f is continuous at a $=> \lim \limits_{x \to a} f(x) = f(a)$ $=> \lim \limits_{x \to a} f(x)$ exists $=> \lim \limits_{x \to a} ...
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1answer
24 views

Solve $|x - z_1| = d_1 + y$ and $|x - z_2| = d_2 + y$ simultaneously for $x$ and $y$

Given the two equations $|x - z_1| = d_1 + y$ and $|x - z_2| = d_2 + y$ , and suppose that $z_1, z_2 \in \mathbb{R}$, $z_1 \neq z_2$ and $d_1, d_2, \in \mathbb{R}_{> 0}$ are all known reals, solve ...
1
vote
1answer
26 views

Inequality involving absolute moment and variance

Suppose $X\in\Omega$ is a random variable and $f:\Omega\rightarrow [0,1]$. Is the following true: $$E[|f(X)-E[f(X)]|]^2\leq \operatorname{Var}[f(X)]?$$ This was stated without proof in a research ...