For questions about or involving the absolute value function.

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2
votes
3answers
79 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 ...
1
vote
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?
0
votes
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?
0
votes
1answer
22 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 ...
3
votes
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} ...
0
votes
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
29 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 ...
0
votes
2answers
57 views

How to sketch trigonometric functions?

I was given this as an assignment in Calculus for Life Sciences and I really would like to figure it out: sketch: y=sinx, y=cosx, y=tanx over -2x≤x≤2x
0
votes
1answer
34 views

Simplifying $i |x|$

Is there any way that one could condense the expression $$i |x|$$ where $i$ is the imaginary unitto get the $i$ inside of the absolute value? I have not been able to find any way to do so but I feel ...
0
votes
2answers
56 views

When is square root the inverse of the square?

For years I have been wondering about something, and this is the day when the problem shall be rectified forever, by the help of you, ofcourse! :) It seems no one I ask really know the answer. Today ...
0
votes
1answer
12 views

Absolute value inequality for Pettis integral

Let $f:[a,b]\rightarrow E$ be absolutely continuous and Pettis integrable, i.e. there exists $I_f\in E$ such that $x^*(I_f)=\int x^*\circ f$ for $x^*\in E^*$. Because $f$ is absolutely continuous, ...
0
votes
5answers
146 views

Prove that $\displaystyle \lim_{x \to 0} \dfrac{|x|}{x}$ does not exist

I thought absolute values were positive? Why is the there a negative $x$ in example $7$ in the attached picture. Can someone explain?
0
votes
3answers
89 views

Prove that $|x+y+z| \le |x|+|y|+|z|$

Prove the following: $$|x+y+z| \le |x|+|y|+|z|$$ It is so trivial that I do not have idea how to show it. Thus, how do I show it?
3
votes
5answers
62 views

Maximum formula: $\max\{x,y\}=\frac{1}{2} \left(x+y+|x-y|\right)$

Show that: $$ \max\{x,y\}=\frac{1}{2} \left(x+y+|x-y|\right) $$ I have no idea how to prove this; most likely it is trivial.
0
votes
1answer
49 views

How to obtain $||x|-|y||\le|x-y|$ from $|x|-|y|\le |x-y|$? [duplicate]

Having the following inequality: $$|x|-|y|\le |x-y|$$ does it imply that $||x|-|y||\le|x-y|$ if it does (i think it does) how to prove it?
0
votes
5answers
83 views

If $|a+b|≤1,$ then $ |a|≤|b|+1.$

How can I prove If $|a+b|≤1,$ then $|a|≤|b|+1$ in real analysis ? I try to use Triangle inequality
1
vote
2answers
25 views

what is the absolute value of an absolute value of x and what is the absolute value multiplied with the same absolute value

So, if I take the absolute value over the absolute value: $|x|$ does it become like this $||x||$ (I know it is something different than the Euclidian norm) or will it just remain like $|x|$ What I ...
0
votes
0answers
20 views

Proving a Comparison

If $\varrho(x,y)=min\{|x-y|,p-|x-y|\}$ ($p>0$ and $x,y\in[0,p)$) then prove that: $\varrho(x,y)\leq\varrho(x,z)+\varrho(z,y)$ This is how far I've gotten: If ...
1
vote
2answers
75 views

How do I prove this trigonometric integral inequality?

If f is integrable and monotone on [a,b] then $\left |\int^b_a f(x)\cos x\,dx\right | \le 2(|f(a)-f(b)|+|f(b)|).$ I've tried integration by parts and using the integral inequality property but I'm ...
1
vote
1answer
42 views

Solving natural logarithms with absolute value

Question from my text: $e^{4x-2014} - 7 = |-3|$. I've never seen this before and my text is useless! Thank you!
0
votes
1answer
70 views

Absolute value in double integral

Would be appreciated if anyone could shed some lights on how to solve the double integral with absolute value in it. \begin{align} \int_0^t\:du\int_0^\infty e^{-\mu\left|u-s\right|}Ae^{-\lambda ...
1
vote
4answers
70 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
votes
1answer
28 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 ...
0
votes
0answers
23 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
44 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 ...
1
vote
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
votes
2answers
114 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
votes
0answers
7 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
votes
1answer
51 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
vote
2answers
35 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
votes
2answers
52 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
vote
1answer
53 views

Absolute value in exponential, signal energy?

How can this give this result? Isn't the absolute of $(e^(-2*t))$ always 1?
0
votes
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 $$ \sum_{n=0}^{k}|m-n|=\sum_{n=0}^{m}(m-n)+\sum_{n=m}^{k}(n-m)\;?$$
1
vote
0answers
23 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
84 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
55 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
29 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
57 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
44 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
27 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
654 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
75 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
vote
4answers
65 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 ...
4
votes
2answers
89 views

Is this inequality always valid? $\left|\sum_{i=0}^{\infty}x_i\right|\leq \sum_{i=0}^{\infty}|x_i|$

Let $x_i\in\mathbb{R}$ for all $i\in\mathbb{N}.$ Is the following inequality always true? $$\left|\sum_{i=0}^{\infty}x_i\right|\leq \sum_{i=0}^{\infty}|x_i|$$
3
votes
5answers
107 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
67 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 $x$?
1
vote
1answer
53 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
81 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
vote
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 ...