For questions about or involving the absolute value function.

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Absolute value proof with epsilon

I'm having trouble with this proof. any hints would be greatly appreciated! If $x$ is a positive real number, show that for some $\epsilon$ $>0, $ then $y\in \Bbb{R}$ is positive if $|(x-y)|< $ ...
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1answer
35 views

between what two disjoint sections we can do a unification in order to get this group of solutions?

between what two disjoint sections we can do a unification in order to get this group of solutions? $0<|x+6|\leq{0.4}$ in other words, in what values should I fill the blankets: (____,____) ...
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1answer
71 views

Tricky logarithm problem

I having a problem in this logarithm problem involving modulus- Solve for x |x-1|^((log(x))^2-2log(x))=|x-1|^3 Bases same so powers equal. If I take log x as a then I get the following quadratic- ...
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2answers
52 views

limit of absolute value

$$ \lim_{x \to 0} \frac{\lvert2x-1\rvert - \lvert2x+1\rvert}{x} $$ Defining the function piecewise reveals the limit is in fact, continuous about 0 However when I go to solve it in a normal ...
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4answers
58 views

How to solve a convoluted absolute value inequality?

$$ \lvert \lvert x-2\rvert -3\rvert \lt 5 $$ How can I attack this the best way? I see that both sides are positive. Squaring yields: $$ \lvert x-2\rvert ^2 -6 \lvert x-2\rvert +9\lt 25 $$ $$ ...
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1answer
26 views

Solving absolute value equation, different methods.

I'm interested to know how people solve absolute value equations differently and how many methods there are out there. For example, say I wish to solve $|x-2|-|x-3|=|x+4|$. How would you solve it ...
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1answer
73 views

How can the integral of $|\sin(x)|$ be $-\cos(x)\text{sgn}(\sin(x))$?

Wolfram|Alpha tells me that $\int|\sin(x)| = -\cos(x)\text{sgn}(\sin(x))$ (which happens to also be its derivative), but I don't understand how this is possible, because the resulting function jumps ...
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1answer
32 views

Inequalities with more than one absolute value

I saw a question which asked to find all the solutions to: $|x+2|+|x-5|=7$ For $x\leq -2$, the answer is $-2$. For $-2< x <5$, the answer is $R$. For $x>5$, the answer is $5$. First I ...
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1answer
33 views

Finding $\lim_{t\to 0}\frac{|t-2|}{t}$ and $\lim_{t\to \infty}\frac{|t-2|}{t}$

Find $$\lim_{t\to 0}\frac{|t-2|}{t}$$ and $$\lim_{t\to\infty}\frac{|t-2|}{t}$$ Usually I would simply the top and bottom but I'm not sure what to do for absolute values. Any help would be ...
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1answer
26 views

Let $G$ be a group of order $36$ and $H$ be a subgroup of $G$ with order 4. Then which is/are true?

Let $G$ be a group of order $36$ and $H$ be a subgroup of $G$ with order 4. Then (1) $H\subset Z(G)$ (2) $H=Z(G)$ (3) $H$ is normal in $Z(G)$ (4) $H$ is abelian group Can I tell $H$ abelian, ...
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1answer
46 views

Trying to prove an absolute value inequality $\left | a\sqrt{2} -b \right | > \frac{1}{2(a+b)}$

I am trying to prove that: $$\left | a\sqrt{2} -b \right | > \frac{1}{2(a+b)}$$ I was given that $a$ and $b$ are any positive integers. Can someone please help me? Thanks.
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2answers
53 views

Show that if $a,b \in \Bbb R$ then [duplicate]

$\max\{a,b\} = \frac12(a+b+|a-b|)$ and $\min\{a,b\} = \frac12(a+b-|a-b|)$ how would you go about solving this? I started with suppose $a \leq b$ Also, show min{a,b,c} = min{min{a,b},c}. How would ...
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2answers
16 views

Which conditions imply $\sup_n |\ln x_n| < \infty$?

I want to find conditions which imply that $\sup_n|\ln x_n| < \infty$. Intuitively I think that $\inf_n x_n > 0$ and $\sup_n x_n < \infty$ should be enough, but I don't know how to write it ...
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1answer
47 views

Determine the symmetry of $y=|x-4|$

Determine whether the graph of $y = |x − 4|$ is symmetric with respect to the origin, the $x$-axis, or the $y$-axis. A. not symmetric with respect to the $x$-axis, not symmetric with respect to the ...
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3answers
169 views

Spivak's Calculus chapter 1 problem 12 v

I am having trouble proving $$|x|-|y|≤|x-y|.$$ In the solutions it says $$|x|=|y-(y-x)|≤|y|+|y-x|, \quad \text{so} \quad |x|-|y|≤|x-y|.$$ Am I missing something here? How did he get $|x-y|$ on the ...
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2answers
65 views

Equation with logarithms and absolute value

I have this equation: $$ \ln\frac{2-|y-1|}{1-|y|} = \ln x $$ which becomes $$ \ln(2-|y-1|)-\ln(1-|y|) = \ln x. $$ Can the first term in LHS be written as ...
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1answer
92 views

Is there a simple closed form of $|\alpha(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor) + \beta(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor)|$?

Let $d_n(x)$ denote the $n$'th digit after the decimal point in $x$. Examples: $d_8(e) = 2,\;d_5(\pi) = 9$ $\alpha(x)$ and $\beta(x)$ are defined this way: $$d_n(\alpha(x)) = \left\{ ...
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2answers
95 views

Proof related to absolute value

I was trying to prove $|x||y| = |x\cdot y|$ but do not have a clue to start. I have seen examples of |x|+|y| >= |x+y| but could not translate it to my problem. Please my fellow math geniuses, help a ...
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0answers
33 views

Why the plus-minus sign within a pseudo-Riemannian-manifold arc length integral?

Deep with the Wikipedia page on arc length, there exists the following puzzling excerpt (mathematics further marked up by yours truly for readability): Generalization to (pseudo-)Riemannian ...
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1answer
33 views

Differential Inequalities involving Absolute Values

I have to show that $|f '(x)| \leq 1, \ \forall x\in R$. The information I have been given is $|f(x)-f(y)|\leq |x-y|$ ... cauchy schwarz inequality. This is for calculus. Thanks so much.
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1answer
20 views

How to combine OR linear inequality with absolute value

I have x < -10 OR x > 15 How do I turn it into a single inequality using an absolute value? Like a < |x+b|. What are ...
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3answers
47 views

Injectivity in function $f(x)=x\cdot|x|+1$

I want to prove that $f(x) = x\cdot|x|+1$ is injective, and if it is; find the inverse of the function. $f(a) = f(b) \iff a|a|+1 = b|b|+1 \iff a|a| = b|b|$ $\begin{cases} -a^2 = b^2 \quad undefined ...
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1answer
35 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 $\operatorname{cov}(X,Y)$ ?
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2answers
44 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 $\vert z \rvert^2=z\bar{z}$, ...
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1answer
40 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 ...
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1answer
211 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 ...
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5answers
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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 + ...
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5answers
74 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), ...
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2answers
88 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 ...
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2answers
81 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 ...
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1answer
170 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 ...
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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?
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1answer
64 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 ...
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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 ...
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3answers
31 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} ...
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0answers
51 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 ...
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5answers
164 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: $$\int|\cos x|\,dx=\int\sqrt{1-1^2\sin^2x}\,dx=E(x,1)$$ ...
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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.
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2answers
30 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 ...
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2answers
32 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 ...
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4answers
1k 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 ...
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5answers
66 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
35 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 ...
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3answers
47 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 ...
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1answer
53 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 ...
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0answers
36 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 + ... + ...
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1answer
30 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| < ...
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3answers
186 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
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1answer
53 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
51 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 ...