For questions about or involving the absolute value function.

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4answers
54 views

What are all values of $x$ in $\mathbb{R}$ that satisfy $4 < |x+2| + |x-1| < 5$?

I am having some problems getting started with this problem, as I never had to deal with an inequality that was between two values with absolute values. Any help is appreciated. The problem is find ...
0
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1answer
17 views

Prove that positive and negative numbers with an absolute value with equations with a variable in bars, too, having two solutions.

I've read this and it's known that positive and negative numbers with an absolute value such as $|9|$ and $|-2|$ in an equation with a variable also in those bars on the other side have two solutions ...
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0answers
17 views

Time derivative of a function involving absolute value

I have a functional looking like this $$ L[u] = \int \int_{\Omega} k \left| \nabla u \right|^2$$ , for which I want to take the time derivative $\dot{L}$. I am not sure that I handled the time ...
3
votes
1answer
42 views

Name of Inequality

Let $x_i, y_i$ be complex numbers for all $i$. Is there a name for the following inequality? $$\left| \sum_{i=1}^n x_i \right| \leq \sum_{j=1}^n |x_j| $$ In particular, is it a special case of this ...
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2answers
15 views

How to simplify an expression with absolute and log functions?

I'm confused with regard to simplifying this expression: $$ |x| - |x-A| > \ln(\Gamma) $$ I was thinking of taking square on both ends, and that's basically where I got confused. Should I square ...
2
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2answers
27 views

Expectation value of absolute value of difference of two random variables

I do not really know how to prove the following statement: If E(|X-Y|)=0 then P(X=Y)=1. The main problem is how to handle the absolute value |X-Y|. If I say that |X-Y| >= 0 it follows that ...
0
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0answers
6 views

Derivative gradient power metric

I use the the following definition of gradient power metric of an image $I$ $M(I)=\sum_{i,j} \left|\frac{||I|*[-1, 1]|}{\sum_{i,j} ||I|*[-1, 1]|} \right|$ (I take $|I|$ bacause $I$ may have complex ...
2
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1answer
50 views

How to deal with x/|x| in an equation?

How do I solve the following for x? $$ 0 = x-b+\lambda\frac{x}{|x|} $$ I'm trying to minimize $$f(x) = \frac{1}{2}(x-b)^2 + \lambda|x|$$ I took the derivative and now I'm trying to set it to $0$ and ...
1
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2answers
32 views

My proof of: $|x - y| < \varepsilon \Leftrightarrow y - \varepsilon < x < y + \varepsilon$

Is it reasonable to prove the following (trivial) theorem? If yes, is there a better way to do it? Let $x, y \in \mathbb{R}$. Let $\varepsilon \in \mathbb{R}$ with $\varepsilon > 0$. ...
0
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2answers
42 views

Absolute Value Theorem

When trying to prove the inequality $$ |a +b| \leq |a| + |b| \text{, for any real numbers a and b} $$ I manage to use the absolute value definition to get to following inequality: $$ ...
1
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1answer
28 views

Absolute value in exponential, signal energy?

How can this give this result? Isn't the absolute of $(e^(-2*t))$ always 1?
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4answers
56 views

Why is $ \sum_{n=0}^{k}|m-n|=\sum_{n=0}^{m}(m-n)+\sum_{n=m}^{k}(n-m)$? [closed]

The problem is: Why is $$ \sum_{n=0}^{k}|m-n|=\sum_{n=0}^{m}(m-n)+\sum_{n=m}^{k}(n-m)\;?$$
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0answers
19 views

Sudoku and absolute value equation

I know there is many mathematical way to reformulate the Sudoku problem. I'm wondering if there is a way to reformulate this problem as an absolute value equation : \begin{equation} Ax + B|x|=b ...
4
votes
5answers
80 views

Solution to $\sqrt{x^2-5}+3>|x-1|$

I tried many ways to solve this but I just can't figure it out... $$\sqrt{x^2-5}+3>|x-1|$$
1
vote
1answer
42 views

Classification of Discrete Subrings of $\mathbb C$

I am interesting in classifying the subrings of $\mathbb C$ which are discrete with respect to the standard topology (that is, the topology induced by the standard absolute value). Here, I am using ...
0
votes
1answer
28 views

Finding a non-piece wise function that gives us the $i$'th largest number.

A friend of mine was asked to find a non-piece wise function on four variables $i,a,b,c$ such that $f(i,a,b,c)$ is the $i$'th largest number among $\{a,b,c\}$. Using max and min or defining the ...
0
votes
1answer
56 views

Why is $A = \{x \mid 1 < |x| < 2\}$ connected?

$A$ is $(-1, -2) \cup (1, 2)$, and these are two disjoint sets whose union makes up $A$, so it fits the definition of disconnected but the book says that $A$ is a domain (it is open and connected). ...
4
votes
1answer
43 views

All $a$ that equation has at least one root. $a^2+7|x+1|+5 \sqrt{x^2+2x+5}=2a+3|x-4a+1|$

Find all $a$ such that the equation has at least one root. $$a^2+7|x+1|+5 \sqrt{x^2+2x+5}=2a+3|x-4a+1|$$ What have I done: substitution $t=x+1$ and some rearrangements ...
1
vote
1answer
25 views

Integral $\int_0^\infty |x-c|e^{-2x}dx$

I have to evaluate the integral: $$\int_0^\infty |x-c|e^{-2x}dx$$ with c $\in \mathbb{R}$. I would evaluate the integral this way: http://math.ucr.edu/~jmd/9B_S14_AbsInt.pdf. This would give me one ...
5
votes
4answers
640 views

Definition of abs() function

Let $\text{abs}(a)$ denote the absolute value of $a$. Is it true that $\text{abs}(a)\geq{-a}$? I suppose that $\text{abs}(a)>{-a}$, but my math book says the other way. Please help me to understand ...
2
votes
3answers
62 views

Derivative of absolute value of $|x^5|$

Differentiate $|x^5|$. I know the formula for the derivative of absolute value but I can't seem to apply it to get $5x|x^3|$.
1
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4answers
59 views

The interval determined by absolute value inequality $|1-2x| < |1+x|$

I'm working on a series convergence problem and am stuck on this part: The series converges when $|1-2x| < |1+x|$. How can I proceed from here to pick the values of $x$ that satisfy this ...
3
votes
5answers
96 views

Find integral of absolute values by splitting integrals, $\int_{-1}^{4} (3-|2-x|)\, dx$

I have trouble splitting the integral $$\int_{-1}^{6} (5-|2-x|)\, dx$$ Tried so far: Split the 3 and the absolute value to two separate integrals. Draw absolute value graph. Integrate both. I ...
0
votes
1answer
47 views

Modulus of exponential function with real and complex arguments

Can anyone please explain why $$|e^{\frac12 \sin(2x) }|\le e^{1/2}$$ for all real $x$, while $$|e^{-i\sin(x)^{2}}|=1$$ for all real $x$?
2
votes
1answer
48 views

Ordered Field: $|x|\le y$ iff $-y\le x\le y$

I had a question regarding this part of a theorem that describes the inequalities of the absolute value function for order field $\mathbb{F}.$ Here is the theorem: Theorem: Let $\mathbb{F}$ be an ...
2
votes
2answers
40 views

Forcing an absolute value of x after a square root operation

Given the following two equations: $$ f(x) = x \\ g^2(x) = 2x $$ I need to find the $(x,y)$ coordinates for when they meet. So after performing the square root operation, we have: $$ f(x) = x \\ g(x) ...
2
votes
3answers
74 views

Find the set of complex numbers $z$ which satisfy: $\left\lvert\frac{z-3}{z+3}\right\rvert=2$

Find the set of complex numbers $z$ which satisfy $$\left\lvert\frac{z-3}{z+3}\right\rvert=2\text.$$ I need help on that one. Thank you.
1
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1answer
20 views

Using the triangle inequality to show that if $|x| < 4$ then $|x^2-2x+3| < 27$

I'm starting school soon and doing some review problems to prep for Calculus. I'm a bit stuck on this problem: Show that if $|x| < 4$ then $|x^2-2x+3| < 27$. I know that I have to use the ...
0
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0answers
52 views

Proof of nearest integer equality

Let $N(n)$ be the nearest-integer function undefined on half-integers. There are many valid ways to define $N(n)$, I like to choose $N(n) =\arg \min_{z \in \mathbb{Z}} |n-z|$. Consider the function ...
0
votes
2answers
37 views

differentiation of $g(x) = \lvert f(x)\rvert$ where $f(x)$ and $D(f(x)) = 0$

I'm really stumped on this problem and don't know how to go about it. It says $g(x)$ = $|f(x)|$ and to show that if $f(c) = 0$ and g is differentiable at c, then one must have $D(f)(c) = 0$. ...
5
votes
2answers
73 views

Integrate a periodic absolute value function [duplicate]

\begin{equation} \int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor \end{equation} I ...
0
votes
1answer
43 views

Inequality which involves complex numbers and absolute values

How can I solve the following inequality: $|\frac{(1+(1-\theta)z)}{1-\theta z}| \leq 1$ ? $z$ is a complex number. I have to find the values of $\theta$ for which the inequality is satisfied.
0
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1answer
35 views

how to solve absolute inequality functions

I have noticed in the past while solving inequality functions that when you want to change the inequality symbol you need to switch the $+$ or $-$ signs of the function itself. How do I solve this ...
0
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1answer
30 views

Finding an absolute value

Instruction says Find |3 - $\sqrt{10}$| Given answer is $\sqrt{10}$ - 3 I cannot give an explanation.
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0answers
13 views

X numbers that when subtracted will produce the same absolute value

Let's say I have X unique numbers and I choose one number y out of this set. Is it possible to create these X numbers such that the absolute difference between y and any other number in X will always ...
0
votes
3answers
43 views

Why does $-b < a < b \implies |a| < b$ (and also the converse)?

I don't have any intuitiom for this because it's just something I memorized. I only understand that $|a| = a$ if $a$ is already positive (or $0$), and $|a| = -a$ if $a$ is negative since we want to ...
0
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0answers
28 views

integrate an absolute value periodic function

$$ \int_{-\frac{\pi}{2}}^{t} |\cos{t}|dt = \sin(t-\pi\lfloor(\frac{t}{\pi}+\frac{1}{2})\rfloor)+ 2\lfloor(\frac{t}{\pi}+\frac{1}{2})\rfloor $$ In know that this integral holds. It can be obtained by ...
0
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1answer
70 views

Value of Pi derivation

Derive the value of Pi. I want with explanation. Is there any possible way? How do scientists calculate it?
0
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6answers
46 views

Compute: $|z + 2| = |z − 3i|$

Find all complex numbers that solve this equation: $|z + 2| = |z − 3i|$ How would I go on about solving this one? 4 times? Like this? $I. z+2=z-3i$ $II. z+2 = -(z-3i)$ $III. -(z+2) = z-3i$ $IV. ...
2
votes
1answer
28 views

Find image of two variables function

I have problem with proving using double inclusion that it's an image of function where we have open interval for instance: Find image $f[A]$ where $A=(0,2) \times (1,3)$ of $f(x,y)=|x-y|$. My try: ...
0
votes
1answer
18 views

Absolute Value proof involving epsilon and delta

The question is: If $x,z$ $ϵ$ $R$, show that for every $ϵ > 0$ there is a $ δ > 0$ such that if $y$ $ϵ$ $R$ satisfies $|y-x| < δ$ then $|zy-zx| < ϵ.$ I honestly have no idea how to go ...
1
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2answers
39 views

How to show that $\int_0^x |z|\,dz = \frac12 x|x|$?

Say you are integrating a simple $|z|dx$ from $0$ to $x$. How do you go about solving to get $.5 x|x|$?
2
votes
1answer
75 views

Solving messy integral with modulus and trigonometry.

If $$a\in \mathbb R,\int_{a-\pi}^{3\pi+a}|x-a-\pi|\sin(x/2)dx=-16$$ then a can be? I tried substituting $x-a=u$ and then breaking into two integrals removing modulus then used $\int \sin x=-\cos ...
1
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2answers
26 views

Absolute value of addition of positive real numbers great than that of subtraction?

$$∀a,b ∈ R+, |a + b| > |a - b|$$ I'm wondering if this is true? I'm not sure exactly how I could check or prove it to myself with the absolute value there. I thought I might be able to do ...
3
votes
2answers
39 views

Graph of $f(x)$ given, find graph of $f(|x|)$

I know the graph of $f(x)=x^2-2x$. Google calculator https://www.google.com/#q=graph+of+x%5E2-2x But how can I find the graph of $f(|x|)=|x|^2-2|x|$? What is the best method to approach here? ...
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0answers
27 views

Trivial absolute value

Let $K/L$ be a algebraic extension. Suppose that $\left|\cdot\right|$ is a absolute value in $K$ such that is trivially in $L$. Then is trivially in $K$. Thanks for anny suggestion. If is trivially ...
0
votes
1answer
21 views

Absolute value of the sume of two complex number

I have a question about the following. $|A+B|^2$, where $A, B $ is complex number. The question is , when can $|A+B|^2$ be equal to $|A|^2 + |B|^2$?
1
vote
1answer
20 views

Show that $|a|>|b|/2$ knowing that $|a−b|<|b|/2$.

I'm working on a proof and I need to show that $|a|>|b|/2$ knowing that $|a-b|<|b|/2$. I would like to do it without enumerating the different cases. Thanks for you ideas.
1
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0answers
17 views

Signal-extraction knowing both the sum and the sum of the absolute values of normally distributed variables

I have two normally distributed variables $X∼N(μ_{x},σ_{x}²)$ and $Y∼N(μ_{y},σ_{y}²)$. I can observe both the sum of their values and the sum of their absolute values, i.e. $Z₁=X+Y$ and $Z₂=|X|+|Y|$. ...
0
votes
2answers
70 views

Find the limit $\lim_{x\to 0^-}| \left( 1+x^{3} \right)^{1/2}-1-x^{5} |/(\sin x-x)$

I am studying for my first calculus exam (well, it's half an exam), and of course we have to solve limits, without using L'Hospital rule, and using asymptotic analysis. I can't solve this one ...