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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

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3answers
54 views

Find the general solution to the ODE $x\frac{dy}{dx}=y-\frac{1}{y}$

I have been working through an ODE finding the general solution and following the modulus through the equation has left me with four general solutions, as shown below. Online ODE solvers, however, ...
3
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2answers
42 views

Is absolute value a function or property? [closed]

Not the most groundbreaking curiosity, but entertaining my brain. In basic algebra, is absolute value a function, counting the distance between the value and zero? Or is it a property, that every ...
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2answers
30 views

$\int\limits^\infty_{-\infty} xe^{-|(x-u)|} dx = ?$

I'm trying to solve this integral with absolute values. Wolframalpha shows that $\int\limits^\infty_{-\infty} xe^{-|(x-u)|} dx = 2u$, but when I break the absolute value into two integrals I don't get ...
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0answers
14 views

Absolute operator in a constraint [duplicate]

I have a constraint of the following form $tr(A*X)+|(tr(A*X))|==0$ where A is constant hermitian matrix and X is a hermitian complex matrix variable. The condition on $X$ is that it should be ...
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2answers
169 views

Is $\int^{\pi}_0|e^{iRcos\theta-Rsin\theta}|d\theta=\int_0^{\pi}e^{-Rsin\theta}d\theta$?

I'm looking over my notes and I noticed a few dubious lines of reasoning ( I will take it to my lecturer as well for further clarification but I won't be able to do that for a few days ) ...
4
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2answers
81 views

How to show $\int_{0}^{\infty}\frac{x}{x^{2}+1}\log\left(\left|\frac{x^r+1}{x^r-1}\right|\right)dx = \frac{\pi^2}{4r}$?

After playing around with a few values of $r$, I have the following conjecture: $$\int_{0}^{\infty}\frac{x}{x^{2}+1}\log\left(\left|\frac{x^r+1}{x^r-1}\right|\right)dx = \frac{\pi^2}{4r}$$ for $r\gt 0$...
1
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1answer
59 views

Getting rid of absolute value bars to rewrite a linear program in standard form

I'm attempting to reformulate a linear program into standard form. Part of this problem requires me to eliminate absolute values from all of my variables. Specifically, I have the problem: \begin{...
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0answers
34 views

$\|a\| = \sup_{p \le \infty} |a|_p$ and prime number theorem

For $a \in \mathbb{Q}$, let $\|a\| = \displaystyle\sup_{v \le \infty} |a|_v$ where $|.|_\infty$ is the absolute value on $\mathbb{R}$ and $|.|_p$ is the absolute value on $\mathbb{Q}_p$ with the ...
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4answers
90 views

Prove $\mid$x-1$\mid$+$\mid$x+5$\mid$ $\ge$ 6 for all real numbers [duplicate]

I need to prove it with proof by cases. I graphed it out and it seems that I need to put into 2 cases which is when x$>$1 and x$>$-5. But this where my problem is, how do plug that back into the ...
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2answers
120 views

Proof of complex inequality: $|z-1|\le|\sqrt{z^2-1}|\le|z+1|$ for $\operatorname{Re}(z)>0$

Hello I'm trying to prove some inequality of complex number but I'm getting stuck: the inequality is as follows: $$|z-1|\le|\sqrt{z^2-1}|\le|z+1|\quad\mbox{ for } \operatorname{Re}(z)>0$$ Now ...
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3answers
46 views

Logarithmic Equation with absolute value

I am trying to solve the following logarithmic equation: $$\log _{x^{2}}\left | 5x+2 \right |-\frac{1}{2}=\log _{x^{4}}9$$ Surprisingly, the absolute value is not the problem, I have created this ...
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2answers
64 views

Absolute Value Proof By Cases

I'm currently working through D. Velleman's How to Prove it. I have a question regarding an absolute value proof by cases (#10; section 3.5). The question asked is to prove that: $$ \forall x\in\...
2
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1answer
43 views

About the existence of tamely ramified extensions

I'm trying to understand the proof of the existence of tamely ramified extensions. For this, the theorem from my book says: Let $K$ be a complete field with respect to a discrete valuation, and let $\...
1
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2answers
27 views

Prove that $\sum_{j=0}^{\infty}|c_1r_1^j + c_2r_2^j| < \infty$, where $|r_1|, |r_2| < 1$

I'm trying to prove that: $$ \sum_{j=0}^{\infty}|c_1r_1^j + c_2r_2^j| < \infty $$ where $|r_1|, |r_2| < 1$ and $c_1, c_2$ are some arbitraty real numbers ($r_1, r_2$ are also real numbers). If ...
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2answers
49 views

Solving the equation $|z^2-z+1|=|x^2-1|$

$z \in \mathbb{U}$, let $x= |z-1|$, show that :$|z^2-z+1|=|x^2-1|$ I tried from both sides but nothing worked for me, any ideas ?
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1answer
23 views

I'm having trouble following this spherical coordinates integration.

Photo of the integration here It's basically a triple integral in spherical coordinates. I understand how they did the integration wrt $\phi$ and $\theta$, but I don't understand the "trick" for ...
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0answers
125 views

maximizing absolute value in linear programming

I know that this question has been answered several times, and based on the answers, I attempted something. But I simply do not get the right results. The question is as follows. I wish to solve the ...
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2answers
29 views

How to analyze absolute value functions

Can someone provide a systematic way to break ||x| - 1| into its different parts without using a graphical approach? It would be greatly appreciated if the restrictions on each part of the piece wise ...
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2answers
36 views

A problem regarding the triangle inequality

I wonder how can i solve this problem? I know it uses the triangle inequality and adding and subtracting the same variable. Yet i can't seem to get the proof right. Question: Let $ \epsilon \gt 0 $ ...
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1answer
28 views

Absolute value raised to a power expand and represent in matrix form

Consider an equation of the form $$\ddot{x}_i(t) = A*|y_N(t)-y_i(t)|^P+B*|x_E(t)-x_i(t)|^P+C*|y_S(t)-y_i(t)|^P+D*|x_W(t)-x_i(t)|^P+K,A,B,C,D,K,P \in \mathbb{R}$$. I wish to represent this in the form ...
2
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0answers
106 views

On the maximal unramified extension of $\mathbb{Q}_{p}$ of a given degree.

I'm stucked with a theorem withouth proof saw in a Book about G-Functions by Dwork, I will appreciate any hint, also I provide a ''proof'' of that theorem, but a feel that is too ''bla bla'' and I ...
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2answers
167 views

Proving that the absolute value function is elementary.

My proof is: The absolute value function f is defined as x when $x\geq 0 $, hence it is a polynomail hence elementary function and it is defined as -x when $x < 0$ hence it is also a polynomial ...
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1answer
25 views

Need help with my first college math class - multiple absolute value equation

There's an equation that we got assigned to solve in our first college math class. I was alright at math in high school, but I've never seen an absolute value equation similar to this one. |||||x|+...
2
votes
3answers
52 views

Is it allowed to solve this inequality $x|x-1|>-3$ by dividing each member with $x$?

Is it allowed to solve this inequality $x|x-1|>-3$ by dividing each member with $x$? What if $x$ is negative? My textbook provides the following solution: Divide both sides by $x: $ $\frac { x ...
1
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4answers
267 views

Solve the equation : $x^2 − 6 |x − 2| − 28 = 0$

The following is an absolute value quadratic equation that I want to solve: $$x^2 − 6 |x − 2| − 28 = 0$$ Here is what I did : $x^2 − 6 |x − 2| − 28 = 0$ $x^2 − 6 |x − 2| − 28 = 0$ $-6|x-2|=28-x^2$...
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2answers
36 views

Square root of absolute difference is greater than absolute of square root difference

For $x,y \ge 0, |\sqrt{x} - \sqrt{y}| \le \sqrt{|x - y|}$. How to prove this. I have tried by squaring both sides, but failed. Can you help me out.
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1answer
53 views

IMO 1959 - Problem 2 - Simplification of square-root values

I have a question on how the simplification process goes for squaring square-roots and how absolute values interact with one another. Question is from IMO Math-Olympiad 1959 #2 $A = \sqrt{x + \sqrt{...
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2answers
5k views

Finding first and second derivative of an function with an absolute value

Given the equation $f(x)= |x^2-9|$ where $-4\le x\le 5$, I must find the extremes, as well as the concavities. This I know how to do. The issue is I'm unfamiliar on how to find the first and second ...
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2answers
22 views

Proving that changing signs inside absolute value is valid for any two real numbers: $\forall x,y\in\mathbb{R}[\rvert x-y \lvert = \rvert y-x \lvert]$

I am just beginning learning about proofs, which is pretty cool, and I wanted to prove that $\forall x,y\in\mathbb{R}[\rvert x-y \lvert = \rvert y-x \lvert]$. I was wondering if my proof is correct: ...
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1answer
49 views

How to prove $|x_i - x_j| \geq |y_i - y_j| - 2 ||x - y||_\infty $ [closed]

Let $x$ and $y$ be two real vectors of length $n$. Let subscripts $i$ and $j$ denote the $i$-th and $j$-th elements of a vector. How can we prove $$|x_i - x_j| \geq |y_i - y_j| - 2 ||x - y||_\infty $...
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2answers
83 views

Clueless on how to prove $\|(a,b)\| \le |a|+|b|$ [duplicate]

Basically, the inequality says that the norm of a vector is always less than or equal to the sum of the absolute value of its components. And I know that the norm is defined as: $$\|(a,b)\|=\sqrt{a^...
3
votes
4answers
58 views

Why is $|x|$ defined as $\sqrt{x^2}$ instead of $(\sqrt{x})^2$?

I can't seem to understand this even though it might be utterly simple for some people. For me, saying $|x|=\sqrt{x^2}$ is a bit weird since $\sqrt{x^2}$ doesn't force positivity as there are always ...
2
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0answers
57 views

Calculating integral of absolute integrand

I want to calculate the integral $$ \int_0^1\int_0^1\int_0^1 {\rm d}r_1{\rm d}r_2{\rm d}r_3 \, r_1 r_2 r_3\int_0^\pi \int_0^\pi \int_0^\pi {\rm d}\phi_1{\rm d}\phi_2{\rm d}\phi_3 \\ \left| r_1r_2\sin(\...
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1answer
27 views

Rewriting $\vec x_1\cdot \vec x_2 |\vec x_3|+\vec x_2\cdot \vec x_3 |\vec x_1|+\vec x_3\cdot \vec x_1 |\vec x_2|$?

Given three arbitrary vectors $\vec x_1, \vec x_2,\vec x_3$ in a (three dimensional) vector space, consider: $\vec x_1\cdot \vec x_2 |\vec x_3|+\vec x_2\cdot \vec x_3 |\vec x_1|+\vec x_3\cdot \vec ...
4
votes
2answers
141 views

Prove that this is a metric

$d:\Bbb C \times \Bbb C \to \Bbb R$ Defined by $$d(z,w) := 2\frac{|z-w|}{\sqrt{(1+|z|^2)(1+|w|^2) }},$$ prove that $d$ is metric in $\Bbb C$. I had proved $d$ satisfies the two conditions to be ...
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1answer
57 views

there are at most finitely many points

I have the formula $\frac{2a_{i}}{a^{2}_{i}+1}=-b_{i}$ and I want to show that $|a_{i}|=1$ only when $|b_{i}|\geq1$ which is easy to prove but my question is how I can prove the statement " there are ...
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2answers
82 views

Which are differentiable for all x?

$F_1:$ \begin{cases} x^2/|x|, & \text{$x \neq 0$} \\ 0, & \text{$x=0$} \end{cases} $F_2$:$$|\sin(x)|^2$$ $F_3$:$$|\cos(x)|$$ I know that the absolute value of $\cos (x)$ would not be ...
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2answers
55 views

$(1-x)^{3}|1-xe^{iy}|^{4}|1-xe^{2iy}|^{2}<1$ for real $x,y$ with $x\in(0,1)$

For real $x,y$ with $x\in(0,1)$ we have $$(1-x)^{3}|1-xe^{iy}|^{4}|1-xe^{2iy}|^{2}<1$$ My attempt : Obviously, we have the identity $(1-x)^{3}|1-xe^{iy}|^{4}|1-xe^{2iy}|^{2}=(1-x^{3})|1-x\cos y-...
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1answer
21 views

Question regarding absolute equalities vs. absolute inequalities

If I have $|3x| < x + 4$, I break it into two cases and get that $x<2$ and $x>-1$ and the question is done (I think). In my solutions I have another problem that asks me to solve $|2x+3| = 2-...
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1answer
50 views

Prove that for all $x\in\mathbb{R}\enspace \lvert x\rvert + \lvert x-6 \rvert\geq 6$

I am beginning proofs in analysis. I am reading Kane's book, but I am not sure if this proof counts as a proof in analysis. I have tried proving by contradiction, but I failed. I also tried using the ...
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3answers
34 views

Absolute value limits

How would I calculate the limit $$\lim_{x \to 1} \frac{|x^2-1|}{x-1}?$$ I really have no idea. I know that $$|x^2 - 1| = \begin{cases} x^2 - 1 & \text{if $x \leq -1$ or $x \geq 1$}\\ 1 - x^2 ...
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2answers
68 views

Proof of inequality $|x \sin \alpha + y \cos \alpha| \leq \sqrt{x^2 + y^2}$ [duplicate]

Just like in the title, I'm asking for any hints for proving (propably simple) inequality: $$ |x \sin \alpha + y \cos \alpha| \leq \sqrt{x^2 + y^2} $$
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1answer
33 views

Absolute Value Rules

Why can $\lvert \cfrac{3}{x} - 3 \rvert$ be turned into $\cfrac{3}{\lvert x \rvert} \lvert x-1 \rvert$? Where can I find more rules/tricks like this?
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2answers
29 views

Explaining The Shape Of An Absolute Value Graph

I don't understand why the shape of the graph would be like this, for the give equation. Why wouldn't the shape resemble that of a normal quadratic graph $(abs(x)+3)^2$?
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votes
4answers
2k 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| \...
0
votes
2answers
129 views

why would a function have an absolute minimum if the hessian matrix is 0?

I have that $f (x,y)=(y-x+1)^2$ And from what I've seen, there is a line of minimum values at $y=x-1$. Well, my teacher asks the following: given that the hessian matrix determinant is 0 in the ...
17
votes
3answers
299 views

Quadratic inequality puzzle: Prove$ |cx^2 + bx + a| ≤ 2$ given $|ax^2+bx+c| ≤ 1$

I came across this problem as part of a recreational mathematics challenge on university: Suppose $a, b, c$ are real numbers where for all $ -1 \le x \le 1 $ we have $|ax^2 + bx + c| \le 1$. Prove ...
2
votes
1answer
56 views

Solve a system of equation with absolute value and rotation

Solve $\begin{cases} \left|x_1-x_2\right|=\left|x_2-x_3\right|=...=\left|x_{2018}-x_1\right|,\\ x_1+x_2+...+x_{2018}=2018. \end{cases}$ I think there must be such a way to solve systems of equation ...
0
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0answers
30 views

Solving Modulus Inequalities: When to use and not use distributive property

I'm going to use examples to explain what I mean. Solve $|d| + 6 \leq 10$. \begin{align} |d| + 6 \leq 10 \implies d + 6 \leq 10 \implies d \leq 4 \end{align} or \begin{align} -d-6 \leq 10 \implies ...
0
votes
1answer
49 views

Normal distribution - absolute value solved

The random variable Y is normally distribution with mean = 8 and S.D = 5. Show that, P(|X−8|<6.2) = 0.785 What I did: (6.2 + 8)/5 = 2.84. The value of 2.84 is 0.9977 in the table. Normally, I ...