Questions tagged [absolute-value]

For questions about or involving the absolute value function.

2,019 questions
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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, ...
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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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$\int\limits^\infty_{-\infty} xe^{-|(x-u)|} dx = ?$

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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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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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 ) ...
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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 $r\gt 0$...
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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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$\|a\| = \sup_{p \le \infty} |a|_p$ 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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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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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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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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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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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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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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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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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 ...
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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 ...
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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 ...
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 ...