For questions about or involving the absolute value function.

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1answer
32 views

Covariance of absolute values of random vaiables

How would I go about calculating $\operatorname{cov}(|X|,|Y|)$, if I know $f_{X,Y}(x,y)$ and $\operatorname{cov}(X,Y)$ ?
0
votes
2answers
31 views

Absolute value of a complex number proof

Ok, so I have the following proof. Let $z$ and $w$ be complex numbers. Prove $\lvert z+w \rvert ^2 + \lvert z-w \rvert^2 = 2[\lvert z \rvert^2 + \vert w \rvert^2]$. Using $\vert z \rvert^2=z\bar{z}$, ...
0
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1answer
39 views

Calculate the conjugate of a complex number

ok, so I have to calculate the conjuage of ${(8-2i)^4\over(4+3i)^5}$ using the properties such as $\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\bar z_1}{\bar z_2}$ and $\overline{(z_1z_2)}=\bar ...
0
votes
1answer
127 views

Absolute Value Equivalence relation inequality Question

I'm having trouble understanding what exactly to do to see if the following relation is symmetric and transitive. I've already determined that it is reflexive. Could someone please help me? For $a, b ...
-3
votes
5answers
49 views

What is the set of real solutions (x,y) that satisfy this absolute value equation?

How many real solutions (x,y) from |x-y| + |x+y| = 1 ? I really wonder how to find it. My attempt: I think I need to separate this problem into some cases: First case: for |x-y| >0 we got: x-y + ...
1
vote
5answers
71 views

How to solve this absolute value equation?

Consider the absolute value equation: |x| + |x-2| +|x-4|= 6 How to find the solution(s)? My attempt: For |x|, we got x, for x>=0 and -x, for x <0 For |x-2|, we got x-2, for x >= 0 and -(x-2), ...
2
votes
2answers
86 views

Triangle inequality problem with equality

How does one prove that, for any reals $x,y$ , there holds the equality $$|x|+|y|+||x|-|y|| = |x-y|+|x+y|\quad?$$ I have tried this using both the reverse and triangle inequalities, but I cannot get ...
3
votes
2answers
72 views

Let $f$ be a holomorphic in $D(0,1)$, with Re$\,f(z) >0$ and $f(0)=1.$ Then $\lvert\, f'(0)\rvert\leq 2$

Let $f:D(0,1) \to \mathbb{C}$ be a holomorphic function, such that $$ \mathrm{Re} \,f(z) >0\quad \text{and}\quad f(0)=1. $$ How to prove $\lvert\, f'(0)\rvert\leq 2 \ ?$ This is now a ...
1
vote
1answer
105 views

Expanding Binomial with Absolute Value

I want to expand the least-squares formula $\sum |a-b|^2$, but I can't follow the reasoning behind what I've heard is the answer: $|a-b|^2 = |a|^2 - 2|ab| + |b|^2$ Instructions or a link would be ...
1
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1answer
36 views

rule for the power of absolute value expressions

Is $|x^n|=|x|^n$ for any rational $n$ and for any real number $x$? If the above is true, what is the proof?
0
votes
1answer
63 views

how to write the absolute value of $| x_n - x$ | separately

$| x_n - x$ | = | $x_n$ | - | $x$ | is this right or is it less than or equal to. the equality is in fact $| x_n - x$ | less than or equal to 3 how does this mean that $| x_n |$ smaller or equal ...
0
votes
1answer
55 views

System of equations with parameter

I have been trying to solve this problem for a week now. It goes like this: Find all values of $a$ for which the system $$ \begin{cases} x^2-2x+y^2 = 1 \\[1ex] \dfrac{x+|x|}{y-a}=2 \end{cases} $$ has ...
0
votes
3answers
30 views

How to determine different absolute value equation cases?

This is a question from this post. From: $$ |3x|=\left\{ \begin{align} 3x & \text{ , if }x\geq 0 \\ -3x & \text{ , if }x <0 \end{align} \right\} $$ $$ |4x+1|=\left\{ \begin{align} ...
0
votes
0answers
40 views

Division Algorithm With Negative and Absolute Value

(a) Prove that $d \, |\, a$ implies that $d \,| (−a)$. (b) Prove that $d\, |\, a$ if and only if $d \,| (−a)$. (c) Prove that $d \,|\, a$ if and only if $d\, \Big|\, |a|$. I can see why these ...
0
votes
0answers
45 views

Using the negation of a statement to disprove original statement

Prove the following statement is false by first writing the negation, then proving the negation is true: For all sets, S, if S ⊆ ℕ, then there exists some t ∈ S such that |t| ≥ 1. So far, I've ...
6
votes
5answers
162 views

How to find $\int|\cos x|\,dx$?

How do I find closed form for $\int|\cos x|\,dx$ for all real $x$? It can be expressed as incomplete elliptic integral of the second kind: $$\int|\cos x|\,dx=\int\sqrt{1-1^2\sin^2x}\,dx=E(x,1)$$ ...
2
votes
3answers
37 views

How to solve $\lim _{k\rightarrow 1}\dfrac {1+\ln k}{\left| \ln \left( \ln k\right) \right| }$

How to solve $\lim _{k\rightarrow 1}\dfrac {1+\ln k}{\left| \ln \left( \ln k\right) \right| }$ I stucked at the denominator.
0
votes
2answers
29 views

Limit of |x-2| as x approaches -2

I believe that it equals -4. In the epsilon-delta definition, we can set delta equal epsilon and I become this satisfies the definition. The problem is I can't seem to prove based on this that 0 less ...
0
votes
2answers
29 views

How to simplify abs(x)/x

I've been trying to find a way to simplify $\frac{|x|}{x}$ if $x$ is real and $\neq{0}$. The two possible outcomes to this are $\pm{1}$ but I believe there is one required answer. I've noticed that if ...
3
votes
4answers
562 views

Finding the definite integral of a function that contains an absolute value

The integral in question is this: $\int_{-2\pi}^{2\pi}xe^{-|x|}$ My attempt: Since there is a modulus, we split it up into cases. I'm not really sure which cases to split it into, do I just ...
1
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5answers
65 views

For what real number $c$, this equation has exactly three solutions?

For what real number c does the equation $|x^2 + 12x + 34| = c$ has exactly three solutions?
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3answers
34 views

Finding the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$.

So I am trying to find the integral of $\int_{-\infty}^{\infty}e^{-|4x|}$. I know the integral converges, and I know the answer as well, but I am confused on how to get the correct answer. My problem ...
1
vote
3answers
43 views

Limit-related inequalities with absolute values

Recently I decided to learn calculus on my own and I stumbled across something which I cannot figure why is correct. Let $f$ be some function for which you know only that if $0<|x-3|<1$, then ...
0
votes
1answer
41 views

Is an absolute value acting on complex numbers a linear operator?

I just have to prove that it isn't with O(A+B)=O(A)+O(B) and O(kA)=k(OA) where O is the linear operator (i.e the absolute value), A+B and A would be a complex number, and k is some real constant. I ...
0
votes
0answers
26 views

Definition of the absolute value of a polynomial

I am having a hard time verifying that this is the definition of the absolute value of a polynomial: Given a polynomial with (possibly) complex coefficients: $p(z) = a_0 + a_1 z + a_2 z^2 + ... + ...
1
vote
1answer
28 views

Absolute values and inequalities

So I've been trying to solve this one for a few hours and am now out of ideas on how to approach this problem. Here are the inequalities: $$\text{show that if}$$ $$z,w \in \Bbb C$$ $$|z| < ...
1
vote
3answers
168 views

Is there a function whose derivative is $|x|$?

Is there a function $y=f(x)$ such that $$\frac{df}{dx}|_{x=a} =|a|$$ for all $a\in \mathbb R$? I'm in a debate with my friend over it and we are stuck
0
votes
1answer
51 views

On complex numbers and absolute values

Exercise 1.31 of Analysis by Apostol states: Given three complex numbers $z_1,z_2,z_3$ such that $|z_1| = |z_2| = |z_3| = 1$ and $z_1 + z_2 +z_3 = 0$. Show that these numbers are vertices of an ...
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4answers
45 views

Showing an Absolute Value Inequality Problem Proof

I tried solving this question but it does not works for me. Q.) Show that $\left|x + \frac1{x}\right| \ge 2$ for all $x \ne 0$ There are two ways to do. One is squaring and other is to use absolute ...
0
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1answer
33 views

Simplify the equation $\left | \frac{4-3m_3}{3+4m_3} \right |= \left | \frac{-3-4m_3}{4-3m_3} \right |$

$$\left | \frac{4-3m_3}{3+4m_3} \right |= \left | \frac{-3-4m_3}{4-3m_3} \right |$$ I am always confused when it comes to modulus. I know if there is modulus any one of the side then when we remove ...
1
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3answers
49 views

What are the steps to solving |3x + 1| > |2x - 7| with the given answer as $(-∞,-8)\cup(6/5,∞)$?

What are the steps to solving $|3x + 1| > |2x - 7|$ with the given answer as $(-∞,-8)\cup(6/5,∞)$? I am having difficulty with understanding inequalities with absolute value functions on both ...
2
votes
2answers
52 views

Finding the limit of $F(x)=\frac{x^2-4}{|x+2|}$

Let $F(x)=\dfrac{x^2-4}{|x+2|}$ and find the following limits $(a) \; \; \lim_{x \to -2^-}F(x)=$ $(b) \; \; \lim_{x \to -2^+}F(x)=-4$ $(c) \; \; \lim_{x \to -2}F(x)=DNE$ I substituted $-2$ to find ...
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votes
1answer
30 views

Can one apply absolute value to both sides of an inequality?

Can I claim this: $$c>b \implies |c|>|b|$$ with $c>0$? I ask because I want to use it in a proof but I am not sure. Thank You.
0
votes
1answer
33 views

Maclaurin series for $\frac{1}{|1+x|}$

I believe that there is no Maclaurin Series for $\frac{1}{|1+x|}$ as the latter is not differentiable at $x=-1$. However, would it be appropriate for me to refer $\frac{1}{|1+x|}$ as 'not a smooth' ...
4
votes
2answers
70 views

Piecewise linear function and absolute value

While writing a solution to homeworks for my students, I had to write the function $$f(x)=\left\{\begin{array}{ll} \frac{x+2}{2}, & x\leqslant -4\\ \frac{x}{4}, & -4\leqslant x\leqslant 4 \\ ...
12
votes
2answers
116 views

Finding all solutions to the equation $|||||x|-1|-1|-1|-1|=0$

I was presented this question by a student I was tutoring: Suppose $x \in \mathbb{R}$. Find all solutions of the equation $$|||||x|-1|-1|-1|-1|=0.$$ What I explained to the student: Given ...
2
votes
6answers
93 views

Adding $2$ absolute values together: $|x+2| + |x-3| =5.$ [duplicate]

I came across a very basic absolute value question $|x+2| + |x-3| =5.$ Initially, I thought the answer was $x=-2$ and $x=3$ because I let each absolute values be either positive and negative and ...
0
votes
2answers
51 views

How to find roots for $y = ||x^2-x-20|-8|$

$$ y = ||x^2-x-20|-8| $$ After I set $y = 0$, I do not know how to deal with multiple absolute values.
1
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1answer
66 views

Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
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9answers
2k views

What's wrong with solving absolute value equations in this way?

Say I have $3x-2 = |x|$. Why can't I just do this: $3x - 2 = -x$ and $3x - 2 = x$ and then get two values for $x$: $1$ and $0.5$? I know the answer $0.5$ doesn't work if you plug this in. However, I ...
0
votes
2answers
577 views

Expressing absolute value equations as piecewise functions

I'm not sure how to express this function in piecewise form without using absolute values: $$ f(x) = 3|x-2| - |x+1|$$ I know how to do it when there is just one absolute value, such as: $$g(x) = ...
1
vote
1answer
92 views

Inequality: $\left|x^3-y^3\right|<|x|^3+|y|^3$

Could anyone show me why $$\left|x^3-y^3\right|<|x|^3+|y|^3$$ for all real numbers (x,y) except 0? I'm thinking of whether of how to remove the modulus sign on the left hand side of the ...
3
votes
2answers
140 views

Does my proof of $|x+y| \le |x| + |y|$ make sense? How do I conclude a proof?

Thank you for reading it. I know I made a lot of mistakes. This is my first ever proof that I have attempted. Another note is that I only have been studying proofs for about a week. Any advice will be ...
2
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2answers
29 views

Taking root from absolute expression

Why is the following true? (Where all terms are positive) $$|x-y| < \epsilon^2 \implies |\sqrt x - \sqrt y| < \epsilon$$
3
votes
1answer
68 views

Geometric idea behind equations of the form $|x-a|\pm|x-b|=c$

So let's say I want to solve $$|x-a|\pm|x-b|=c$$ Using the classic multiple cases approach, one can show that the solutions are given by $$x=\frac{a+b\pm c}2 $$ But how can one make sense of this ...
3
votes
4answers
3k views

How to solve inequalities with absolute values on both sides?

If you have an inequality that has two absolute value bars like $|4x+1|<|3x|$, how do you go about doing this? I know that if $4x+1<3x$, then those $x$'s will work but what else do I do? I think ...
3
votes
1answer
55 views

Why is the value of $\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|\,dx$ independent of integer parameter $n$?

I am not able to find an easy solution for the following formula $$\int_0^{2\pi}|2\cos(nx)+\sqrt{3}|dx=4+\frac{4}{3}\pi\sqrt{3}.$$ Please help me prove it. Why it does not depend on the (positive) ...
3
votes
0answers
39 views

Nested absolute operations

The question is: are the following two functions equivalent? And if yes, what properties of the absolute value should I use to prove it? $f_1(x,y,z)$ = $|\, x + |y+z| \,|$ $f_2(x,y,z)$ = $| \,|x+y| ...
4
votes
2answers
116 views

Calculating the best match between two sets

I’m a PHP developer and I have a problem calculating the perfect match between two different data sets. I have data sets from companies, where each company defines the requirements for a specific ...
0
votes
2answers
57 views

Graphs for mod functions

Can someone please teach me how to obtain graphs for the following types of functions: $2+3|x-1|$ $|x-1|+|x|+|x+1|$ $|x-1|-|x|-|x+1|$ $|x-1|^2$ Thanks.