For questions about or involving the absolute value function.

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2
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2answers
66 views

Prove triangle inequality using the properties of absolute value

So I was given the task of proving the following variant of the triangle inequality using only the properties of the absolute value: $\vert\lvert x\rvert -\lvert y \rvert \rvert \leq \lvert ...
2
votes
0answers
45 views

Simplifying an expression with absolute values

I am trying to simplify the function $D(\alpha,\beta)$ shown below (with $\alpha,\beta>0):$ $$ D(\alpha,\beta)=\frac{1+\alpha+2\beta}{2} + \frac{|\alpha-1|}{2} - 2 ...
2
votes
2answers
18 views

Build the graph of a function with absolute value.

The function is: And my idea of graphic (i did it using two graphs and deleting some parts) Is it correct?
0
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0answers
34 views

How to solve an equation with absolute value and x as exponent.

The inequation is: I thought that the solution would be the same as if there was no x as exponent, but in Microsoft Math it says it has no solution and about the equation it said it has solutions. ...
1
vote
1answer
24 views

Exercise on inequalities in bounded derivatives (from Spivak)

Suppose $f$ is two times differentiable in $(0,\infty)$ and that: $|f(x)| \leq M_{0}, \forall x>0$; $|f''(x)| \leq M_{2}, \forall x>0$. a) Show that $$|f'(x)| \leq ...
0
votes
1answer
21 views

A general method for solving inequations with absolute values

I've been asked to find which $b$ satisfy $|a + b| = |a| + |b|$ for $a \geq 0$. I'm familiar with the method described here and I tried to apply it but I'm confused about what I should do with the ...
0
votes
0answers
27 views

equivalence of norms vs equivalence of absolute values

Consider a field, $k$. A norm on $k$ is a map, $|-| : k \to \mathbb{R}$ such that $(1)$ $|v|= \iff v=0$ $(2)$ $|vw| = |v||w|$ $(3)$ $|v+w| \leq |v|+|w|$ and we say two norms $|-|_1, |-|_2$ are ...
4
votes
5answers
133 views

Proving $|a-1|+|a-2|+|a-3| \ge 2$

I need to prove the following sentence for $a\in\mathbb{R}$: $$ |a-1|+|a-2|+|a-3| \ge 2$$ Breaking the equation into cases it does work, i.e. for $a\le 1$: $$-a+1-a+2-a+3\ge 2$$ $$-3a \ge -4$$ $$a ...
0
votes
0answers
18 views

When does $\overline{U(0,1)}=B(0,1)$ hold?

Given $R$ an absolute valued ring (with unit), sometimes $\overline{U(0,1)}=B(0,1)$ (for example, $\mathbb{Q},\mathbb{R},\mathbb{C},\mathbb{H}$) and sometimes $\overline{U(0,1)}\neq B(0,1)$ (for ...
1
vote
4answers
27 views

Absolute value function inequality

I need to find the values of x that satisfy the inequality x|x| > x I know the possible outcomes are ...
0
votes
1answer
29 views

Modulus Inequalities Proof

Need to prove that: $$|x-1|+|x-5| \geq 6$$ I've tried squaring but I'm not sure if I'm doing it correctly? Thank you in advance Note: x is real and does not equal 1 or 5
4
votes
1answer
69 views

Evaluate $\int_{-1}^1 \frac {1}{\sqrt{|x|}} \text{d}x$

I need some help to solve this integral with absolute value. I'm not sure how to do these types of integrals. $$\int_{-1}^1 \frac {1}{\sqrt{|x|}} \text{d}x$$ Thank you
2
votes
3answers
73 views

Absolute value:$ |x-a| \leq \frac{|a|}{2} \Rightarrow |x| \geq \frac{|a|}{2}$

Prove that $$|x-a| \leq \frac{|a|}{2} \Rightarrow |x| \geq \frac{|a|}{2}$$ I've tried by direct proof and contradicion but nothing worked. I would like a hint or a tip of what should I do. Thanks ...
2
votes
2answers
71 views

For real numbers a and b, when is the equation |a + b| = |a – b| true?

I put that the statement was true only when a = 0 and b = 0 but the correct answer was that it only held true for a = 0 OR b = 0. With 'and' I figured |0 + 0| = 0 and |0 - 0| = 0. Could someone ...
0
votes
2answers
37 views

Finding functions max and min (abs value)

I have the function $$g(x)=|x^2-x-2|$$ which is defined on $$-\frac{3}{2}\leq x\leq \frac{3}{2}$$ I am struggeling with that g(x) has absolute values wrapped around. I taught that I just draw the ...
1
vote
1answer
23 views

What is this equal to? : $|A+B|^2$ where $A = P e^{ia}$ and $B = Q e^{ib}$

$A$ and $B$ are two complex numbers: $A = P e^{ia}$ $B = Q e^{ib}$ I would like to know what is this equal to? : $|A+B|^2$ Please also give a small proof if possible.
0
votes
1answer
16 views

Deploy the absolute inequality formua

Please help me derive that equation: $$f(x) = \begin{cases} 1, & \text{if $(x-a)^2-(x-b)^2>0$} \\ -1, & \text{otherwise} \end{cases}$$ where $x,a,b \ge 0$ Thank you in advance This is ...
2
votes
3answers
416 views

A simple inequality looking for a more elegant proof…

I have been given the following statement by a professor: Let $m\leq x\leq M$ and define $K=\max(|m|,|M|)$. Then $|x|\leq K$. Now I can clearly see that this is true, and working through ...
2
votes
3answers
89 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
27 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
25 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
33 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
43 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
26 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
34 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
91 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
65 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
52 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
85 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
26 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
78 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
46 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
85 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
73 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
31 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
27 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
181 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
10 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
54 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
59 views

Absolute value in exponential, signal energy?

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