For questions about or involving the absolute value function.

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0answers
19 views

Absolute Value Graph Problem in Gelfand's Functions and Graphs

I am working through Gelfand's Functions and Graphs, where I am currently on the absolute value section. At the end of the chapter practice problems, Gelfand poses a set of problems regarding ...
2
votes
2answers
45 views

Calculation double Integral over Ball (optical size)

I hope that someone can help me with the following problem. I have to show that $$\int_{B_1(0)}\int_{B_1(0)}\frac{1}{|x-y|^2}dxdy=4\pi^2~,$$ with $B_1(0)\subset\mathbb{R}^3$. I have no idea how to ...
0
votes
1answer
40 views

Inequality with two absolute value

How can you tackle an inequality problem that has two absolute values? Example is the following $p + |k| > |p| + k$ and the questions is a quantitative comparison between A) $p $ B) $k$ The ...
0
votes
0answers
21 views

Is following a norm or absolute value of a vector?

I'm reading a paper regarding power minimization and came across with following equation: $g_{i,j}=|h_{i,j}|^2/d^\alpha$ Where $h_{i,j}$ is a complex vector of dimension $N$. I don't know and it ...
0
votes
2answers
83 views

Is there any clever way to solve inequalities like $|y^2-y-2|\geq 4 + |y^2+y-2|+|y+4|+|y|$?

I have to solve different types of inequalites of this type: $$|y^2-y-2|\geq 4 + |y^2+y-2|+|y+4|+|y|$$ I know the standard method for solving these inequalities, by finding the all zeros of the ...
2
votes
0answers
32 views

Definite integral of absolute values over a simplex

I'm attempting to evaluate the following integral (note that $v_n=1-v_1-v_2-\dots -v_{n-1}$, and assume $n$ is even): $$ I_n=(n-1)! \int_{v_1=0}^1 \int_{v_2=0}^{1-v_1} \cdots \int_{v_{n-1}=0}^{1-v_1-\...
-2
votes
1answer
38 views

Find $x$ where $|(x-3)^2-1|\geq |(x+2)^3+5|$ [closed]

Find all real $x$ such that the following is true: $|(x-3)^2-1|\geq |(x+2)^3+5|$
1
vote
4answers
34 views

Absolute Difference of Two Integers

New to math. I'm looking for an explanation (proof, rule, relationship or property) that explains that the absolute value of the difference between two integers $x$ and $y$ are equal regardless of: ...
2
votes
4answers
92 views

Prove $|x|=\max\{x,-x\}$

first time poster here, so please excuse my noobiness I'm going through some basic first year college math exercises, because i found out i still can't do some of the proofs, and I've encountered ...
0
votes
2answers
43 views

If $\left| x-2\right| < \frac{1}{100}$ Show That $\left| x^2 -4 \right| < \frac{1}{10}$

I am working through a problem involving absolute value inequalities, and I just can't seem to get on the right track. The problem is, if: $$\left| x-2 \right| < \frac{1}{100} \text{,} $$ show ...
1
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3answers
60 views

Simplify $\sqrt{a^6 + 2a^4b^2 + a^2b^4}$

Simplify $\sqrt{a^6 + 2a^4b^2 + a^2b^4}$ for $a < 0$ I've almost got it but I have a question about the answer. This is my solution: $$\sqrt{a^6 + 2a^4b^2 + a^2b^4}$$ $$\sqrt{a^2 (a^4 + 2a^2b^2 + ...
5
votes
3answers
129 views

Why there is no value for $x$ if $|x| = -1$? [duplicate]

According to the definition of absolute value negative values are forbidden. But what if I tried to solve a equation and the final result came like this: $|x|=-1$ One can say there is no value for $x$...
0
votes
1answer
83 views

Find $\theta$ and $\phi$ that maximize $\mid -2ia\sin\theta - 2ib\sin\phi + 2c(1-\cos\theta) +2d(1-\cos\phi)\mid$

How can you find for what values of the $\theta$ and $\phi$ angles the following modulus will assume its greatest possible value? $$\mid -2ia\sin(\theta) - 2ib\sin(\phi) + 2c(1-\cos(\theta)) +2d(1-\...
0
votes
0answers
22 views

Equation with absolute values and a parameter

I know how to solve equations with absolute values but without a parameter in the absolute value. If "a" is a real and positive parameter |2x - 3a| + |a + 1 - x| = |x + 1| But how can to approach ...
2
votes
2answers
73 views

Evaluate $\lim_{x \to 0^-} \left( \frac{1}{x} - \frac{1}{|x|} \right)$ (possible textbook mistake - James Stewart 7th)

I was working on a few problems from James Stewart's Calculus book (seventh edition) and I found the following: Find $$\lim_{x \to 0^-} \left( \frac{1}{x} - \frac{1}{|x|} \right)$$ Since ...
0
votes
0answers
59 views

How to solve $y + |y| = \cdots$

I want to calculate the equipotential lines for $f(x, y) = x + y + |x| + |y|$. The domain is $ℝ^2$ and range $[0, \infty)$. I started like this: $$ x + y + |x| + |y| = c \ge 0 \\ y + |y| = c - x - |x|...
2
votes
4answers
63 views

How to solve this inequality with absolute value: $ \frac{\left|x-3\right|}{\left|x+2\right|}\le 3 $

Good morning to everyone. I have an inequality that I don't know how to solve: $$ \frac{\left|x-3\right|}{\left|x+2\right|}\le 3 $$ I tried to solve it in this way: $$ \frac{\left|x-3\right|}{\left|x+...
1
vote
2answers
52 views

Why are absolute values involved in functions of random variables?

From a textbook: If $X$ is a continuous random variable, then so too is the new random variable $Y = Y (X)$. The probability that $Y$ lies in the range $y$ to $y + dy$ is given by $$g(y)=\int_{...
0
votes
1answer
49 views

A question about the absolute value in integrals

I do really understand why we put the absolute value when integrating functions leading to $\log$ function for example: $$ \int{\dfrac{\mathrm dx}x}=\log\lvert x\rvert + C$$ , it is very common in ...
1
vote
2answers
53 views

Express the function $ f $ without using absolute value signs $\left|\frac{x-2}{x+3}\right|e^{\left|x-2\right|}$?

Good evening to everyone: This is the equation $$ f(x) = \left|\frac{x-2}{x+3}\right|e^{\left|x-2\right|} $$ What I've tried is: $$ \frac{x-2}{x+3}\ge 0 => x-2 \ge 0 => x \ge 2$$ Then $$ \frac{-...
-4
votes
5answers
60 views

On the equation $|x|^2+|x|-6=0$

Which of the following are true for $$|x|^2+|x|-6=0$$ 1. It has $4$ roots 2. The sum of the roots is $-1$ 3. The product of the roots is $-4$ 4. The product of the roots is $-6$ Only one of the ...
0
votes
1answer
22 views

applying exponents in an absolute value brackets

So part of the problem I'm trying to solve is this: $|2-9|^{3{^3}}$ being the exponent, to the power of 3. Do I have to apply the exponent to everything in the bracket? 2*2*2 and 9*9*9 or does the ...
1
vote
2answers
41 views

Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$

Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$, if $z = a+ib$ then, I tried to do that with $|z| = (a+ib)(a-ib)$ then i multipled it $z$ with $z^-$ and then I got stuck. answer is $|z| = 5$
1
vote
4answers
65 views

How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$?

How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$ ? I've solved equations like $|a|+|b|=|a+b|$ where the condition must be that $a$, $b$ must be of same sign. But in case of three terms ...
1
vote
1answer
28 views

Is there a derivative for $|x|$ at $0$ specifically “in the direction” of positive $x$?

I know that $|x|$ is not differentiable at $x=0$ because there is potentially an infinite number of tangent lines going through that point. But let's say we were interested in the motion of an object ...
3
votes
3answers
34 views

reciprocal factor of absolute value when evaluating a square root expression

Learning with an old russian math book, i found the following evaluation for the function $f(x)=\sqrt{1+x^2}$: $f(\frac1x)=\vert x \vert^{-1}\sqrt{1+x^2}$ My evaluation gave me $\sqrt{1+\dfrac1{x^...
0
votes
3answers
51 views

Why is $\sqrt{x/x^{-1}}$ OR $\sqrt{x/{1/x}}$ = $\lvert x\rvert$ and not just x

I have this task: Find equal expression to square root of fraction of x and its inverted value (this is translated from my mother tongue so I'm sorry if I've used incorrect terms). Anyway the starting ...
1
vote
2answers
47 views

Rejecting a Solution to a Modulus Question

Why is the solution of $|1+3x|<6x$ only $x>1/3$? After applying the properties of modulus, I get $-6x<1+3x<6x$. And after solving each inequality, I get $x>-1/9$ and $x>1/3$, but why ...
0
votes
1answer
25 views

Find the solution set of $ \left\lvert 100\left(\frac{x-y}{y}\right) \right\rvert - \left\lvert 100\left(\frac{y-x}{x}\right) \right\rvert < 1 $

I apologize if this is a rather trivial question however I was wondering if I can have a bit of guidance in tackling this problem: $$ \left\lvert 100\left(\frac{x-y}{y}\right) \right\rvert - \left\...
1
vote
1answer
31 views

Integrals of a function and its absolute value

Is the following proposition true? Let $f(x)$ be a real-valued function defined on $[a,b] \subset \mathbb{R}$, and suppose that the integral, $$ I = \int_a^b f(x) dx, $$ exists in the sense of ...
0
votes
1answer
40 views

Joint pdf of X and Y with absolute value

Question. Joint probability function of continuous probability X, Y is here : $f_{X,Y}(x,y) = k(|x|-|y|) \ \ \ \ \ \ \ \ \ \ (-1< y< x< 2)$ Then what is k? I mean how can I differentiate ...
1
vote
1answer
105 views

How to minimize $|Ax+By + C|$ given that $x \geq 0$ and $y\geq 0$ [duplicate]

I am trying to solve problem related to absolute value function, i.e given $Z(x,y) = |Ax + By + C|$ , what is the minimum value of $Z$, if $x \geq 0$ and $y\geq 0$ and x,y belongs to integers
2
votes
4answers
112 views

Prove that $||a|-|b||$ is smaller or equal to $|a-b|$

I am stuck with this question: Show that $\vert \vert a\vert - \vert b\vert\vert \le \vert a-b\vert$ I had tried proving this using the following method below: $\vert a\vert+\vert b\vert \ge \vert ...
2
votes
3answers
59 views

Solve the equation $|2x-1| -|x+5| = 3$

Problem : Solve the equation $|2x-1| - |x+5| = 3$ In my attempt to solve the problem, I only manage to get one of the solutions. Attempted Solution $$\begin{equation} \begin{split} |x|-|y| & \...
0
votes
0answers
24 views

On the Arrangement of Intermediate Subgroups

I am trying to find a journal paper 'On the Arrangement of Intermediate Subgroups' by M.S. Bah and Z.I. Borevich appearing Rings and Linear Groups, Krasnodar (1988), 14-41. This is a Russian text and ...
1
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1answer
39 views

what is the result for the following Integral?

I would like to find the result for the following integral $$ \int_{-\infty}^\infty x e^{-|x|/a}\cdot e^{-|x-y|/b} \, dx $$ where $a$ and $b$ are constants. $x$ and $y$ are variables
0
votes
1answer
21 views

Finding the set of values for k of a modulus function.

"Find the set of values of k for which |(x-4)(x+2)| = k has four solutions." EDIT: Ok so I thought I'd start with setting the modulus function equal to k and -k to get the two set of results. Doing ...
0
votes
1answer
28 views

I need to find the following Integral?

I would like to ask if any one can find the following integral $$ \int_c^d x . e^{-{\frac{\vert x \vert}a}}.e^{-{\frac{\vert x-y \vert}b}} dx $$
4
votes
4answers
73 views

Show that $f(x)=f(y)$ then $|x|=|y|$, where $f(x )=\frac{1+|x|}{x}$

Let $f: \mathbb{R}^{*}\to \mathbb{R}$ function definied by $f(x )=\dfrac{1+|x|}{x}$ Show that $f(x)=f(y)$ then $|x|=|y|$ Indeed, $$f(x)=f(y)\\ \iff \\\dfrac{1+|x|}{x}=\dfrac{1+|y|}{y} \\ \iff \\ \...
1
vote
1answer
29 views

How do I represent $f(x) = \int_{-1}^{0} |x + t| dt, 0 \leq x \leq 2$ in an integrated form?

Given the following function, how do I define it without the integral symbol? $$f(x) = \int_{-1}^{0} |x + t| dt, 0 \leq x \leq 2$$ I don't understand how I determine when $x + t$ is positive and ...
1
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2answers
53 views

build absolute value equations know solution

We have absolute value equations with unknown coefficients: $$|x + a| = b$$ and we know the solutions: $$x = 11 \text{ and } x = 5$$ We need to find $a$ and $b$. From $$11 + a = b \\ 5 + a = -b$$ we ...
-1
votes
2answers
38 views

Normal distribution, probability and modulus question [closed]

Say $X$ is a random variable which is normally distributed with mean $0$ and variance $1$. How do I find $k$ such that $$\mathbb{P}(|X-k| < |X+k|) = 0.7$$
1
vote
0answers
37 views

Find the product all real numbers in an equation

What is an easy and fast way to solve the problem without going through all these possibilities: a) $n^2-9n+20>0, 16-n^2>0$, b) $n^2-9n+20>0, 16-n^2<0$, c) $n^2-9n+20<0, 16-n^2>0$, ...
1
vote
1answer
41 views

Is there an easy way to solve this absolute values problem?

This is a simple problem. What I want to know is whether there is an easy and fast way to solve the problem. I solved this problem by considering four situations: a) $x>1$, b) $0<x<1$, c) $x&...
0
votes
3answers
37 views

How to solve this problem on absolute value function?

If $a,b\in \mathbb R$ and be distinct numbers satisfying $$|a-1|+|b-1|=|a|+|b|=|a+1|+|b+1|$$ then the minimum value of $|a-b|$ is ? ($|...|$ represents absolute value) I tried solving the equalities ...
1
vote
2answers
40 views

If $a\le b\le c$, then $|b-d|\le\max\{|a-d|,|c-d|\}$

I am trying to find a simple way to show the fact that if $a\le b\le c$, then $|b-d|\le\max\{|a-d|,|c-d|\}$ for any number $d$. Is there a way to do this besides breaking it up into the cases 1) $...
1
vote
0answers
42 views

Uniform convergence fourier series |sin(x)|

As an exercise we have to calculate the fourier series of |sin(x)| (was no problem) and after that we are meant to show that this series converges uniformly towards |sin(x)|. After thinking about it ...
0
votes
0answers
27 views

Is complex multiplication the only multiplication operation on $\mathbb{R}^2$ that works with the Euclidean norm?

What I'm asking is: viewing complex multiplication as binary operation on $\mathbb{R}^2$, is usual multiplication of complex numbers the only operation $\otimes$ on two vectors $\vec{u}$ and $\vec{v} \...
2
votes
1answer
26 views

why is $\left|xy\ log(\left|x\right|+\left|y\right|)\right|\leq\left|(\left|x\right|+\left|y\right|)log(\left|x\right|+\left|y\right|)\right|$?

I should note that this was used by my book in order to show that the limit of $xy\ log(\left|x\right|+\left|y\right|)$ at $(0,0)$ is $0$. After several attempts in vain, I plotted the function $\...
3
votes
4answers
131 views

The Definition of the Absolute Value

The Absolute Value can be defined in many ways, but these are the two most common : 1. As a Piecewise Function $$ |x|= \begin{cases} -x&\text{if } x < 0\\ x&\text{if } x\geq 0 \end{cases} ...