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Questions tagged [absolute-value]

For questions about or involving the absolute value function also known as modulus function.

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Solving $(x^2-1)^2-({x^*}^2-1)^2\geq{0}$

Find $x^*\in[-2,2]$ such that for all $x\in[-2,2],$ the inequality $$(x^2-1)^2-({x^*}^2-1)^2\geq{0}$$ holds. The given solution is $-1$ and $+1.$ While solving I come across the inequality, $|{x}^2-1|...
Sumit Gupta's user avatar
3 votes
2 answers
98 views

Square root definition

Perhaps this is silly, but here it is: If we search for the square root definition it goes as follows (taken from Wikipedia): "In mathematics, a square root of a number x is a number y such that $...
Bruno de Paula Assunção's user avatar
1 vote
1 answer
39 views

Finding the minimum of the sum of Chebyshev distances

Given the vectors $(x_1, \dots, x_n)$ with $x_i = (x_{11}, \dots, x_{ik})$, I am trying to minimize the following function $$ s(a) = \sum_{i=1}^n \max_{j\in\{1, \dots, k\}} |a_j - x_{ij}|$$ with ...
Matt's user avatar
  • 11
4 votes
1 answer
208 views

How to find the double integral of an absolute value trig function?

I have this double integral: $$\int_{-\pi/2}^{\pi/2} \int_{-\pi/2}^{\pi/2} |\sin(y-x)|dydx$$ And I'm struggling to solve it. I have reached the answer $4/\pi^2$ however my answer isn't what the ...
Daranoker's user avatar
11 votes
9 answers
2k views

Doubts regarding absolute value

My fiancée is a teacher in a secondary school. She asked me a question connected to absolute value that I can't answer. Let's consider the following problem $$ \lvert x - 2 \rvert < \lvert x + 4 \...
Hendrra's user avatar
  • 3,000
4 votes
0 answers
103 views

Proof of absolute value of complex logarithm

I would like to prove that for a complex number z: $$\left| \text{Log}(-z) \right| \le \log \frac{1+|z|+|1+z|}{1+|z|-|1+z|}$$ Through numerical examples I already found that the equality only holds ...
leck's user avatar
  • 41
6 votes
6 answers
621 views

How to Solve a Linear System of Equations with Absolute Values

I have encountered a system of linear equations that involves absolute values: \begin{align} |x + y| &= 1 \\ |x| + |y| &= 1 \end{align} I am having trouble finding resources or methods to ...
Kuly14's user avatar
  • 161
2 votes
3 answers
74 views

Question about the property $0 < |x - c| < \delta$ if and only if either $c - \delta < x < c$ or $c < x < c + \delta$

I saw in a proof that used this property of absolute values: $0 < |x - c| < \delta$ if and only if either $c - \delta < x < c$ or $c < x < c + \delta.$ And I am confused why the ...
Sarah Anderson's user avatar
0 votes
1 answer
22 views

How to more formally prove this inequality

This is a simple problem I came up with while doing another problem: Given: $n < (n + \frac{1}{2}) < y < (n + 1)$ Prove: $|y - n| > |y - (n + 1)|$ So how I proved it was simply using the ...
Bob Marley's user avatar
0 votes
2 answers
41 views

Having trouble with this series involving absolute value

"let x, y, and z be real numbers such that $|x-y|+|y-z|+|z-x|=100$. What is the maximum value of $|x-y|$?" At first, I thought it might be 100, since that's the literal maximum value of the ...
Petunia's user avatar
0 votes
1 answer
46 views

Show that for all numbers $z\ge 0$, $|z-1|\ge \delta$ implies $|z^2-1|\ge \max\{\delta,\delta^2\}$

How to prove the following result: fix $\delta\ge 0$, for all numbers $z\ge 0$, $|z-1|\ge \delta$ implies $|z^2-1|\ge \max\{\delta,\delta^2\}$. I did not see why do we need to discuss $0\le \delta \...
Hermi's user avatar
  • 1,494
2 votes
1 answer
47 views

Local extreme points of $f(x)=\frac{5}{4} x^{\frac{4}{5}} - |x-2|$

Consider the real function $f(x)=\frac{5}{4} x^{\frac{4}{5}} - |x-2|$. $f$ is a continuous function and it is differentiable on $\mathbb{R}$ except at $\{0,2\}$. Indeed, $f$ is as follows: (i) If $x \...
user237522's user avatar
  • 6,593
4 votes
3 answers
351 views

Proving that a function is one-to-one (injective)

The question is to show that the function $ f(x) = 2x + |\cos x|$ is both one-one and onto. I have managed to show that it is onto but got stuck at proving that it is also one-to-one. To prove this I ...
FundamentalTheorem's user avatar
1 vote
0 answers
51 views

How do I find purely-real expressions for the absolute value of this complex function?

I am working on a project that involves the squares of the absolute values of some two-real-input-variable complex functions, and I want to find purely-real expressions for those absolute values in ...
Lawton's user avatar
  • 1,861
1 vote
4 answers
118 views

How does $|a+bi| = \sqrt{a^2+b^2}$? If I use algebra and break it down I just get to $1=-1$.

I have gotten to a chapter in my pre-calculus textbook where it mentions this equation: $|a+bi| = \sqrt{a^2+b^2}$ It doesn't really explain why that equation is true, and it just sort of moves on ...
Darkturtle999's user avatar
1 vote
1 answer
68 views

The sum of a sequence where any term is equal to the absolute value of the difference between the prior two terms.

In a sequence, any term is equal to the absolute value of the difference between the prior two terms. Will the sum of the sequence converge? It's known that if the first two given terms are rational, ...
718281828's user avatar
2 votes
2 answers
44 views

Determining Parameter Values for a Set of Solutions Involving Absolute Value Equations

The problem in the textbook asks the following: At what values of the parameter "$a$" does the set of solutions to the equation $|x - 1| + |x - a| = 1 - a$ consist of three integers? I ...
curioushuman's user avatar
1 vote
0 answers
44 views

Understanding Robert's Proof of extending of the p-adic absolute value to field extensions

I am trying to understand a proof that the extension of the p-adic absolute value to a field extension $K$, given by $|x| = \sqrt[n] {|\text{N}_{K/\mathbb{Q}_p}(x)}|$ is an absolute value. Here $N_{K/...
ikey's user avatar
  • 133
2 votes
2 answers
133 views

Non-negativity constraints in linear-programming formulation of $L_1$ norm minimization

My goal is to solve the following minimization problem: $$ \min_{x}|Ax - b|_1 \tag{1}, $$ where, for simplicity, $A$ is an $n \times n$ matrix, and $x,b\in\mathbb{R}^n$. One can solve this by ...
Solarflare0's user avatar
9 votes
1 answer
466 views

Inequality involving an absolute value

Given the numbers $x_1, x_2,...,x_n$ in the interval $[-4;4]$, such that $ \sum_{i=1}^{n} x_i=0$, prove that: $$|\sum_{i=1}^{n} x_i^3|\le 16n$$ I have tried using various known inequalities, namely ...
fikooo's user avatar
  • 409
0 votes
0 answers
38 views

What's the importance of the Approximation Theorem - Artin-Whaples Approximation Theorem

The above pictures present the statement of the approximation theorem by Artin-Whaples and its corollary in their paper. I do understand that the theorem implies that we can use one element to ...
Z Wu's user avatar
  • 1,765
0 votes
1 answer
41 views

maximum value of complex numbers [closed]

Question- If $z_0,z$ are complex numbers s.t. $|z-i|\leq2 $ & $z_0=5+3i$, find $|iz+z_0|_{max}$ My Approach - by Triangle Inequality, we know $|z_1+z_2|\leq|z_1|+|z_2|\implies|z_1+z_2|_{max}=|z_1|...
Anmol's user avatar
  • 3
2 votes
1 answer
79 views

How do you prove $\lvert x - y \rvert < 1$ then $\lvert x\rvert<\lvert y\rvert +1 $?

How do you prove $\lvert x - y \rvert < 1$ then $\lvert x\rvert<\lvert y\rvert +1 $? I know this proof has the form of the triangle inequality, but I can't seem to figure it out. This is from ...
nnabahi's user avatar
  • 83
1 vote
0 answers
26 views

How do you present the absolute value as a norm and derive results?

I asked myself a question while reading the post: Absolute value of complex numbers $|a+bi|$ . Let us take Euclidean space $\mathbb R^2$, with dot product noted $\langle .,.\rangle:\mathbb R^2 \times \...
Stéphane Jaouen's user avatar
1 vote
4 answers
111 views

Absolute value of complex numbers $|a+bi|$

I didn't understand why the absolute value of $(a+bi)$ is equal to $\sqrt{a^2+b^2}$ but not $\sqrt{(a+bi)^2}$ like $|x|=\sqrt{x^2}$ if $|x|=\sqrt{x^2}$ is right and if we give $x = a+bi$ it should be $...
Egemen Yalın's user avatar
2 votes
1 answer
144 views

The upper bound for the max of absolute values of Gaussian distribution [closed]

Prove that $E[\max_{1\leq i\leq n}\vert X_i\vert]\leq C\sqrt{\log(2n)}\sigma$ for some constant $C \in R$, where $X_1, ..., X_n\sim \mathcal{N}(0,\sigma)$ are i.i.d. Hint: Use Jensen's inequality: $f(...
Nick Y's user avatar
  • 57
1 vote
5 answers
96 views

How to solve $|x-5| + |x-4| ≥ 3$

I was given the following question to solve for homework. The solution $S$ I got was $$S = \{x : x ≥ 6 \text{ or } x\leq3\}$$ I checked my solution with the answers provided and it was correct. My ...
Oofy2000's user avatar
  • 618
0 votes
1 answer
52 views

Morse and Fesbach identity $\sum_{n=0}^{\infty} e^{-(n+\frac{1}{2})|u|} \; P_n(x) =(2\cosh u - 2x)^{-1/2}$

In book called Methods of theoretical physics from Morse and Feshbach, there is identity, which I wanted to prove ($P_n(x)$ are Legendre polynomials): $$\sum_{n=0}^{\infty} e^{-(n+\frac{1}{2})|u|} \; ...
Edward Henry Brenner's user avatar
1 vote
1 answer
57 views

Statistics Question about the Probability of Absolute Values

I was tutoring a student today and was given a question that stated the following: Find $P(|z| > -0.29)$. Seems simple enough. But this doesn't really make sense in concept. Firstly, absolute ...
PrestonALewis's user avatar
0 votes
0 answers
27 views

Searching for a more concise solution to $|x - 1| + |x + 1| < 2$ [duplicate]

I came up with what I think is the solution to exercise 11. (v) on chapter 1 of the third edition of book Calculus by Michael Spivak. Find all numbers $x$ for which $|x - 1| + |x + 1| < 2$. ...
Approxiz's user avatar
  • 463
0 votes
1 answer
120 views

Find recurrence relation for sum of absolute deviations in a sequence

Given an ordered sequence $S_n = s_1, s_2, ..., s_n$ we define its cost as the sum of absolute deviations from any median: $$ cost(S_n) = \sum_{i=1}^{n} \left| s_i - median(S_n) \right|\text{, where } ...
Andrei Onoie's user avatar
1 vote
2 answers
158 views

Proving a function has a compact support

I'm trying to understand Willie Wong's answer which shows that for each $t\in \mathbb{R}$, $g(\tau , t - \tau)$ has a compact support, i.e., the set $A_t = \{ \tau \in \mathbb{R}: g(\tau , t - \tau) = ...
S.H.W's user avatar
  • 4,359
0 votes
0 answers
16 views

The value of the vertex in a absolute value graph

In the Thomas Calculus Early Transcendentals (Fourteenth Edition in SI units), the vertex of an absolute-value function is included in the domain of the segment with an increasing gradient. For ...
J Herson's user avatar
7 votes
2 answers
127 views

How to compute $\displaystyle\lim_{n\to\infty} \frac1{n+1}\sum_{k=1}^n \left|X+\frac kn\right|-\left|X-\frac kn\right|$?

I was just playing around with the real absolute value, trying to build something smooth (for no particular reason). After some experimentation I got to the sequence $(f_n)_n$ given by $$f_n(X) := \...
Alma Arjuna's user avatar
  • 3,851
1 vote
1 answer
60 views

Is $\lceil x \rceil=\frac{1}{2}(1+ \sum_{n=-\infty}^{+\infty}(\operatorname{sgn}(x-n)-|\operatorname{sgn}(x-n)|+1+\operatorname{sgn}(n)))$ true?

I'm looking for a function that defines $\lceil x \rceil$ and I found the following: $$\lceil x \rceil=\frac{1}{2}(1+ \sum_{n=-\infty}^{+\infty}(\operatorname{sgn}(x-n)-|\operatorname{sgn}(x-n)|+1+\...
orenstep's user avatar
0 votes
0 answers
33 views

Absolute value in integrals leading to logs

I know that when working with real numbers, the input for a logarithm cannot be negativea, and hence when taking the integral of $\frac{1}{x}$, we take the absolute value: $$ \int{ \frac{1}{x}dx}=ln|x|...
Starlight's user avatar
  • 1,794
0 votes
0 answers
30 views

Finding local maximum of a function containing absolute value

I found an online answer to this question, but I think there is something wrong with it Question: $f(x) = |x|^m |x - 1|^n, \forall x \in \mathbb{R}$ (m) and (n) are natural numbers (positive integers)....
Ali Helal's user avatar
1 vote
2 answers
157 views

Why can't $|x| = x$ be solved the conventional way?

I warn the reader beforehand that, perhaps, this question is laughably simple; yet I need assistance answering it. It is taught in schools how one can solve equations containing absolute values. The ...
Camelot823's user avatar
  • 1,465
0 votes
1 answer
82 views

Linearization of nested absolute value objective $|a-b-|c||$

I am trying to define an optimization problem that minimizes the distance between $a(x)$ and $b(x)$, where I need to adjust $b(x)$ downwards using the cost function $c(x)$ (hence, the cost must always ...
Jean-Paul's user avatar
  • 195
1 vote
0 answers
74 views

Property of vertices in random graphs

For a random graph $G\sim G\left(n,p\right)$ with probability $p$, for every vertex $v$, I need to prove that with high probability, $X=\deg\left(v\right)$ satisfies the condition $\left|X-np\right|\...
Ali AD's user avatar
  • 33
0 votes
0 answers
161 views

Problems with proving exponential distance function

I want to show when ${G(A,B)=\frac{1}{m}\sum_{i=1}^m}\frac{1}{\frac{1}{n}\sum_{j=1}^{n}e^{-\vert{a_i-b_j}\vert}}$, ${G(A,C)\leq G(A,B) \cdot G(B,C)}$. I think from $\vert a_i-b_k+b_k-c_j \vert \leq \...
Jmp3r's user avatar
  • 11
0 votes
0 answers
24 views

Approach/methodology for solving conditional (if, then) inequalities with absolute values

I am having trouble learning how to work with conditional inequalities (i.e. inequalities within an if statement), especially when it comes to solving for a variable that would make the if statement ...
Mark_Parker's user avatar
0 votes
0 answers
43 views

Can we get rid of the absolute value on this case?

I have a small misunderstanding about the correct solution of a differential equation. In fact, my problem is more about understanding the form of the solution rather than the equation itself. Say we ...
Waxler's user avatar
  • 105
0 votes
1 answer
39 views

What can differences of odd summation be?

Let's suppose that we're summing $m$ and $n$ many odd numbers starting from $1$ separately, i.e $$S_m = 1+3+\cdots +(2m-1)$$ $$S_n = 1+3+\cdots +(2n-1)$$ What can their differences be among $37, 42$ ...
user avatar
1 vote
2 answers
68 views

Number of solutions in different cases to a given equation

Consider an absolute value equation for $a, b\in \mathbb{R}$ and $c\in \mathbb{N}$, $$|x+a|+|x+b| = |c|$$ What can be said about the number of solutions to this equation? If $x+a<0$ and $x+b>0$, ...
user avatar
1 vote
1 answer
57 views

Expected value of absolute value of sum of independent variables (Convergence in probability)

Given $X_1, X_2,..., X_n$ independent random variables with $P(X_n=k^n)=P(X_n=-k^n)=1/2$ (assuming $k$ is an arbitrary constant). Let $S_n = X_1 + X_2 +... + X_n$. Determine for which $\epsilon>0$, ...
tt99999's user avatar
  • 11
1 vote
0 answers
42 views

How to evaluate the integral of ln abs (sinx)/(1+x²) from 0 to infinity? [closed]

This is a random integral equation I’ve thought suddenly.I know ln(sinx) has a Fourier series but I don’t know how to deal with the absolute value I’ve tried many times but I still can’t do it
Integral's user avatar
0 votes
2 answers
97 views

Why $|A_1 - A_2| + |B_1 - B_2|$ would not be greater than both $|A_1 - B_1| + |A_2 - B_2|$ and $|B_1 - A_2| + |A_1 - B_2|$ at the same time? [closed]

Here is an example: $A_1, A_2, B_1, B_2 = 5, 8, 0, 9$ respectively $|A_1 - A_2| + |B_1 - B_2| = 12$ $|A_1 - B_1| + |A_2 - B_2| = 6$ $|B_1 - A_2| + |A_1 - B_2| = 12$ For any $A_1, A_2, B_1, B_2 \in \...
Ahmet Yasir TUĞ's user avatar
1 vote
2 answers
132 views

Why does $|x|^2$ not equal $x^2$ [closed]

A friend told me that the two are unequal. I had a though that maybe one has two solutions on the same point, but from what I can tell that's not the case. Edit: LaTeX is hard :).
The Bison's user avatar
0 votes
1 answer
68 views

What are the intersections of $(x-10)^2+(y-10)^2=6^2$ and $y=-2|x-10|+19$?

The real intersection between these two equations is about $x=18.39$ and $x=11.81$. However, I keep coming up with imaginary solutions. Please explain how I can solve for those values. My work so far $...
eeidtyi's user avatar

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