For questions about or involving the absolute value function.

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14
votes
4answers
525 views

How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$

let $a,b,c,d,e\in R$,and such $$a^2+b^2+c^2+d^2+e^2=1$$ find this value $$A=\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$$ I use computer have this $$A=\dfrac{2}{\sqrt{10}}$$ ...
12
votes
6answers
897 views

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
10
votes
7answers
924 views

what does $|x-2| < 1$ mean?

I am studying some inequality properties of absolute values and I bumped into some expressions like $|x-2| < 1$ that I just can't get the meaning of them. Lets say I have this expression $$ ...
10
votes
6answers
1k views

How to calculate with absolute value.

Calculate:$$\frac{ \left| x \right| }{2}= \frac{1}{x^2+1}$$ How do I write the whole process so it will be correct? I need some suggestions. Thank you!
10
votes
5answers
461 views

Is -5 bigger than -1?

In everyday language people often mix up "less than" and "smaller than" and in most situations it doesn't matter but when dealing with negative numbers this can lead to confusion. I am a mathematics ...
10
votes
1answer
302 views

How to determine all valuations of the field $\mathbb{Q(\sqrt[n]{2})}$?

This is an exercise from the book Algebraic Number Theory by Jurgen Neukirch, on page 166. And, after solving several previous exercises, I found this to be particularly difficult to solve. I am ...
7
votes
2answers
353 views

How does one DERIVE the formula for the maximum of two numbers

I want to derive (not prove that this is true) the formula $\max (x,y) = \dfrac{x + y + |y-x|}{2}$ I was reading a proof (which they have the result ahead of time already) that we do cases and then ...
7
votes
2answers
164 views

Maximum of the difference

What is the maximum value of $f(… f(f(f(x_{1} – x_{2}) – x_{3})-x_{4}) … – x_{2012})$ where $x_{1}, x_{2}, … , x_{2012}$ are distinct integers in the set ${1, 2, 3, …, 2012}$ and $f$ is the absolute ...
6
votes
6answers
126 views

why minimum of these functions happen at a special place?

why minimum of these functions happen at a special place? how to use derivative to find the minimum of these functions? $$|x-1| + |x-2| + \dots + |x-9|$$ minimum is for $x = 5$ $$|x-1| + |x-2| + \dots ...
6
votes
3answers
512 views

How prove this inequality: $\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|$? [duplicate]

Let $x_{1},x_{2},\cdots,x_{n}$ be real numbers. Show that $$\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|.$$ I think this problem may be solved using nice methods, but I can't find ...
6
votes
5answers
3k views

Inequality with two absolute values

I'm new here, and I was wondering if any of you could help me out with this little problem that is already getting on my nerves since I've been trying to solve it for hours. Studying for my next ...
6
votes
4answers
265 views

Inequality for absolute values

How do you show either of the equivalent inequalities: $$2(|a|+|b|+|c|)\leq |a+b+c|+|a+b-c|+|a-b+c|+|a-b-c|$$ or $$|x+y|+|x+z|+|y+z|\leq |x|+|y|+|z|+|x+y+z|$$ Hold for complex numbers or in $n$ ...
6
votes
2answers
1k views

Question regarding usage of absolute value within natural log in solution of differential equation

The problem from the book. $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6 -y$ I understand the solution till this part. $\ln \vert 6 - y \vert = x + C$ The solution in the book is $6 - Ce^{-x}$ ...
6
votes
2answers
2k views

How to use triangle inequality to establish Reverse triangle inequality

I need to use $|a+b| \leq |a|+|b|$ to show that $||a|-|b|| \leq |a-b|$. I have tried to represent $||a|-|b||$ as $||a|+(-|b|)|$, and then get $||a|+(-|b|)| \leq |a|+|-|b||$, but that isn't leading ...
6
votes
2answers
193 views

Sum of two absolute values in complex plane

I'm trying to find out all $z \in C$ that satisfy the following condition: $|z+1|+|z-i|=3$ I understand that $|z|=r$ represents a circle with a radius of $r$. I also understand that $|z+1|=r$ can ...
5
votes
5answers
172 views

Evaluating the following integral: $ \int \frac{x^2}{\sqrt{x^2 - 1}} \text{ d}x$

For this indefinite integral, I decided to use the substitution $x = \cosh u$ and I've ended up with a $| \sinh u |$ term in the denominator which I'm unsure about dealing with: $$\int ...
5
votes
3answers
128 views

How would I prove $|x + y| \le |x| + |y|$?

How would I write a detailed structured proof for: for all real numbers $x$ and $y$, $|x + y| \le |x| + |y|$ I'm planning on breaking it up into four cases, where both $x,y < 0$, $x \ge 0$ ...
5
votes
1answer
121 views

An absolute value problem

Let $a$ and $b$ in $\mathbb{R}$ 1) Show that $||a|-|b||\leq|a+b|\leq|a|+|b|$. 2) Prove that the one or the other of the two inequalities is an equality. It's fine whit the 1st question but i can't ...
5
votes
1answer
965 views

Prove variant of triangle inequality containing p-th power for 0 < p < 1

Sorry if this is a trivial question, but I am kind of stuck with proving the following inequality and have been searching for a while: $\rho \left( \sum\limits_i^n d_i \right) \leq \sum\limits_i^n ...
5
votes
1answer
91 views

Solving equation with absolute value signs

Can someone see why there is only get one solution when solving following equation in this way: The equation $|x+1|+|2x-3|=|x-5| $ $$|x+1|+|2x-3|=|x-5| $$ $$\pm (x+1) \pm(2x-3)=\pm(x-5)$$ $$\pm x ...
5
votes
2answers
553 views

Why is the absolute value needed with the scaling property of fourier tranforms?

I understand how to prove the scaling property of Fourier Transforms, except the use of the absolute value: If I transform $f(at)$ then I get $F\{f(at)\}(w) = \int f(at) e^{-jwt} dt$ where I can ...
5
votes
1answer
48 views

Maximum value problem

A function $\hspace{0.1cm}$$f:[0,1]\to[-1,1]$$\hspace{0.1cm}$ satisfying$\hspace{0.1cm}$ $|f(x)|\leq x$$\hspace{0.1cm}$ $\forall x\in[0,1]$. Then find the maximum value of: ...
5
votes
2answers
187 views

Absolute values in $\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$

in my math class we were given a list of indefinite integrals, and one of them was: $$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$$ My working: $$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}=\int ...
5
votes
2answers
120 views

Prove $||a| - |b|| \leq |a - b|$ [duplicate]

I'm trying to prove that $||a| - |b|| \leq |a - b|$. So far, by using the triangle inequality, I've got: $$|a| = |\left(a - b\right) + b| \leq |a - b| + |b|$$ Subtracting $|b|$ from both sides yields, ...
5
votes
2answers
328 views

Is there a lower-bound version of the triangle inequality for more than two terms?

The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$ Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need ...
5
votes
1answer
5k views

Derivatives of functions involving absolute value

I noticed that if the absolute value definition $\lvert{x}\rvert=\sqrt{x^2}$ is used then we can get derivatives of functions with absolute value, without having to redefine them as piece-wise. For ...
5
votes
1answer
148 views

Ring of integers in a field of fractions

Let $R$ be ring with complete non archimedian absolute value. Let $Q$ be the associated field of fractions with the extended absolute value. Does the ring $O_Q = \{x\in Q | |x|\leq 1\}$ is complete ...
4
votes
5answers
339 views

Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$

Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$ example1: $|x-1|+|x-2|\geq 1$ my solution:(substitution) $x-1=t,x-2=t-1,|t|+|t-1|\geq 1,|t-1|\geq 1-|t|,$ square, $t^2-2t+1\geq ...
4
votes
2answers
156 views

How do we know that $|i!| = \sqrt{\pi \operatorname{csch} \pi}$?

(Source: Wolfram Alpha) Or, to write it out in full, $$|i!| = \sqrt{\frac{2\pi e^\pi}{e^{2\pi} - 1}}$$ How is this identity derived? Also, knowing this, could we find the exact values for the real ...
4
votes
1answer
683 views

integral from 0 to $2\pi$ of $|\cos x|\operatorname{d}x$ not integrating as I'd expect

I drew a rough sketch of $|\cos x|$ and would guess the correct answer to this integral is $4$ because I know the area under the curve of $\cos x$ from $0$ to $\pi/2$ is $1$, and there are $4$ such ...
4
votes
3answers
164 views

Is there an alternate definition for $\{ z \in \mathbb{C} \colon \vert z \vert \leq 1 \} $.

Is there a method of constructing a subset of a reasonably arbitrary ring so that when the construction is applied the $\mathbb{C}$ the result is $B = \{ z \in \mathbb{C} \colon |z| \leq 1 \} $? My ...
4
votes
2answers
207 views

Why isn't this square root $+$ or $-$?

I was tasked with proving the identity $\tan(\frac x 2) = \dfrac {\sin(x)}{1+\cos(x)}$ I used the quotient identity for tangent and the half angle identities for sine and cosine to get $ \pm \dfrac ...
4
votes
4answers
105 views

Let $x$ be in the set of real numbers $\mathbb{R}$ and let $f(x)=|2x-1|-3|2x+4|+7$ be a function, write $f(x)$ without the absolute value.

Let $x$ be in the set of real numbers $\mathbb{R}$ and let $f(x)=|2x-1|-3|2x+4|+7$ be a function, write $f(x)$ without the absolute value. I thought of it this way: $$f(x)=\begin{cases}2x-1-3(2x+4)+7 ...
4
votes
3answers
748 views

Proving two integral inequalities

Can anyone help me to prove that these integral inequalities hold? Here $x$ is a real value: $$ \left| \int_a^b\ f(x) dx \right| \leq \int_a^b\ |f(x)| dx $$ Here $z$ is a complex value: $$ \left| ...
4
votes
3answers
486 views

Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?

Claim: If f is integrable, $\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$ Proof (attempt): We know $-|f|\le f \le|f|$, so $\int-|f| \le \int f \le \int|f|$.* Since, if $-b<a<b$, we say ...
4
votes
1answer
25k views

Integral of an absolute value function

How do I find the definite integral of an absolute value function? For instance: $f(x) = |-2x^3 + 24x|$ from $x=1$ to $x=4$
4
votes
1answer
57 views

Does the triangle inequality follow from the rest of the properties of a subfield-valued absolute value?

(This is a much more specific version of my earlier question from over a year ago.) Let $F$ be a field, let $E$ be an ordered subfield of $F$, and let $\;\; |\hspace{-0.03 in}\cdot\hspace{-0.03 in}| ...
4
votes
2answers
351 views

Spivak Calculus 3rd ed. $|a + b| \leq |a| + |b|$

I'm working through the first chapter of Michael Spivak's Calculus 3rd ed. Towards the end of the chapter he proves $ |a + b| ≤ |a| + |b| $ using the observation that $|a|= \sqrt{ a^2 }$ when $a$ ...
4
votes
1answer
241 views

intersection of two graph

i would like to clarify some questions from GRE,which at first seems a little difficult to understand,suppose that we have some function $f(x)=|2*x|+4$ and graph of this function is given ...
3
votes
4answers
237 views

inequality method of solution

Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$ I've tried first determining when the absolute value ...
3
votes
4answers
162 views

Show that $|z+1|\le|z+1|^2 +|z|$ for all $z \in \mathbb{C}$

Question: Show that $|z+1|\le|z+1|^2 +|z|$ for all $z \in \mathbb{C}$ So far I have, Suppose $1\le|z+1|$ $|z+1|\le|z+1|^2$ $|z+1|\le|z+1|^2+|z|$ Now I must show $|z+1|<1$ but this is where ...
3
votes
3answers
116 views

Solving $ \left| \frac{-2x-6}{4} \right| \le 5$ for $x$

Say I have a statement like: $$ \left| \frac{-2x-6}{4} \right| \le 5. $$ And I want to find the closed interval form of $x$. i.e. I want to know what the maximum and minimum $x$ can be. How do I ...
3
votes
2answers
187 views

Complex number with z to the power of 4

I have to find all $z\in C$ for which BOTH of the following is true: 1) $|z|=1$ 2) $|z^4+1| = 1$ I understand that the 1) is a unit circle, but I can't find out what would be the 2). Calculating ...
3
votes
2answers
103 views

Two solutions for $x$

I am trying to solve this. I already got the answer but my doubt is if I am doing it right. There are two solutions for $x$ in the equation, to 2 decimal places. What is the value of the greater ...
3
votes
6answers
333 views

Evaluating $\int |x|^3 \; dx $

$$\int |x|^3 \; dx $$ In my module it is suggest to use integration by parts, $$ \text{ Set }I = \int (|x|^3 \cdot 1) \; dx = |x|^3 \cdot x - \int \color{red}{\frac {x^3}{|x|^3}3x^2}\cdot x \; dx$$ ...
3
votes
3answers
77 views

Please help with absolute value $|x^2 - 3x| = 28$

Just a question about solving an absolute value equation: $$|x^2 - 3x| = 28$$ Do I just solve this as if the absolute value brackets weren't even there? $$x^2 - 3x - 28 = 0$$ $$(x+4)(x-7) = 0$$ ...
3
votes
2answers
99 views

How does the triangle inequality work for $|x-y|$?

I know that $|x+y|\leq |x|+|y|$... But is it similar for $|x-y|$? That is, is $|x-y|\leq |x|+|y|$? I ask because of the following: $x-y=x+(-y)$, so $|x+(-y)|\leq |x|+|-y|=|x|+|y|$ Is it possible ...
3
votes
3answers
282 views

Exposition On An Integral Of An Absolute Value Function

At the moment, I am trying to work on a simple integral, involving an absolute value function. However, I am not just trying to merely solve it; I am undertaking to write, in detail, of everything I ...
3
votes
3answers
317 views

How does one calculate the integral of the sum of two absolute values?

I know how to find the integral of just one absolute value, but this problem presents the integral of the sum of two absolute values. Help! I want to evaluate: $$ \int_a^b{(|x-1| + |x+1|) dx} $$
3
votes
1answer
2k views

General Proof for the triangle inequality

I am trying to prove: $P(n): |x_1| + \cdots + |x_n| \leq |x_1 + \cdots +x_n|$ for all natural numbers $n$. The $x_i$ are real numbers. Base: Let $n =1$: we have $|x_1| \leq |x_1|$ which is clearly ...