# Tagged Questions

For questions about or involving the absolute value function.

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### Soft absolute value

I'm looking for a "soft absolute value" function that is numerically stable. What I mean by that is that the function should have $\mp x$ asymptotes at $\mp\infty$ and behave smoothly in $[-1,1]$. ...
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### Solve differential equation $y' = |1.1 - y| + 1$

How can the following differential equation be solved analytically? \begin{equation*} y' = |1.1 - y| + 1, \\ y(0) = 1. \end{equation*} I guess one must rewrite the differential equation piecewise ...
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### Online calculator for $p$-adic valuations and absolute values.

Does anyone know a website where I can enter a prime base and a rational and then get the $p$-adic valuation and the $p$-adic absolute value? For sure I know how to do it by hand, but I want to ...
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### How do I upperbound this expression?

With a given condition such as $$|x|^2 > |y|^2$$ Is there any way I can upper bound the following expression $$\log\left(1+\big||y|-x\big|^2\right) \leq \,\,\, ?$$ Thank you
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### Solve this inequality for B

I am working on a program that is supposed to qualify a value as "in range" and I have come up with the expression: $$\lvert a-b\rvert \leq c$$ to determine the value. Plugging in test numbers ...
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### How to graph $|z-1| <2$

Am I correct to rearrange this to $(z-1)^2 < 4$, and hence just graph as a circle or am I completely off?
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### Modular inequality of sequential terms: $|x_ny_n-xy| \le |x||y_n-y|+|y_n||x_n-x|$

How can I prove that $|x_ny_n-xy| \leq |x||y_n-y|+|y_n||x_n-x|$ ?
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### Decomposing absolute value terms

I have something like the following term: 7x1 + 9x2 + | 10 - 7x1 | + | 15 - 11x2 | I want to make it into something like this: Ax1 + Bx2 , where A and B are constants For two values x1 and x2, ...
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### Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$

Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$.What will be the easiest techique to solve this sum ? Just wanted to share a special type of equation and the fastest way to solve it.I am not asking ...
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### Prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$

I'm trying to prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$ I've first written down $-5\lt a-b \lt5$ and have tried to add different things from all sides of the ...
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### A difficult trigonometric integral involving absolute value

$$\int_{0}^{2\pi}\lvert\sin(x)\rvert\cos(nx)\,dx= -\frac{4\cos^2\bigl(\frac{\pi n}{2}\bigr)\cos(\pi n)}{n^2-1}$$ I'm not actually trying to solve this myself. The answer appears in my lecture notes ...
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### Prove that for $x\in\Bbb{R}$, $|x|\lt 3\implies |x^2-2x-15|\lt 8|x+3|$.

The problem I have is: Prove that for real numbers $x$, $|x|\lt 3\implies |x^2-2x-15|\lt 8|x+3|$. Since there aren't really any similar examples in my book, I've been unsure how to first approach ...
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### how to calculate $\int (|1+x|-|1-x|) dx$ and $\int$ max {${1-x^2,0}$}?

I'm used to integrate normal functions, but here I got quiet confused because these integrals : $\int (|1+x|-|1-x|) dx$ $\int$ max {${1-x^2,0}$} Include absolute value, and and option to choose ...
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### How do you algebraically derive “x <= 0” from “-x = | x |”

A = "-x = | x |" B = "x <= 0" If A, then B. By plugging in numbers or testing ranges less than zero, greater than zero, and equal to zero, I can verify that A ...
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Using the Mean Value Theorem, prove that $|\sin{a} - \sin{b}| \leq |a - b|$ $\forall a, b \in \mathbb{R}$. I'm working towards figuring out an approach for finding that $|\sin{a} - \sin{b}| \leq ... 3answers 510 views ### A unique solution Find the sum of all values of k so that the system $$y=|x+23|+|x−5|+|x−48|$$$$y=2x+k$$ has exactly one solution in real numbers. If the system has one solution, then one of the three$x's$, should be ... 5answers 530 views ### How to write an expression in an equivalent form without absolute values? The question I have in front of me is the very first problem in Trench's Introduction to Real Analysis: Write the following expression in equivalent form not involving absolute values:$a+b+|a-b|$... 2answers 38 views ### Inequality with absolution value for complex number How to show that inequality:$|1-\bar{\alpha} z| \ge |z-\alpha|z$and$\alpha$are complex number,$\alpha$is constans and$|z|<1$,$| \alpha| < 1$I can proof that by using substition ... 4answers 81 views ### is the following true:$|a-b| = ||a|-|b||$? is$|a-b| = \bigg||a|-|b|\bigg|$? I have tried a few examples and they seems to come out true, but I can't find any rule stating it. Is it true for all$a$and$b$? Or am I missing something? ... 2answers 65 views ### How to start proof of triangular inequality? [duplicate] $$\left| {\left| a \right| - \left| b \right|} \right| \le \left| {a \pm b} \right| \le \left| a \right| + \left| b \right|$$ 3answers 87 views ### Solve$|3-x|=x-3$. Solve:$|3 - x| = x - 3$. Answer:$|u| = -u$when and only when$u \le 0$. So,$|3 - x| = x - 3$when and only when$3 - x \le 0$; that is,$3 \le x$. Hi! I'm new here. I'm working out of ... 0answers 123 views ### two dimensional Gaussian integral with complex exponent of an absolute value I am trying to solve the following two dimensional integral: $$\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}{e^{ia(\left|x\right|-\left|y\right|)} \frac{1}{2\pi\sqrt{1-\rho^2}}e^{-\frac{x^2+y^2-2\rho ... 2answers 272 views ### Prove triangle inequality using the properties of absolute value So I was given the task of proving the following variant of the triangle inequality using only the properties of the absolute value: \vert\lvert x\rvert -\lvert y \rvert \rvert \leq \lvert ... 0answers 49 views ### Simplifying an expression with absolute values I am trying to simplify the function D(\alpha,\beta) shown below (with \alpha,\beta>0):$$ D(\alpha,\beta)=\frac{1+\alpha+2\beta}{2} + \frac{|\alpha-1|}{2} - 2 ... 2answers 21 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? 0answers 122 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. ... 1answer 29 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 $$|f'(x)| \leq ... 1answer 35 views ### A general method for solving inequations with absolute values I've been asked to find which b satisfy |a + b| = |a| + |b| for a \geq 0. I'm familiar with the method described here and I tried to apply it but I'm confused about what I should do with the ... 5answers 143 views ### Proving |a-1|+|a-2|+|a-3| \ge 2 I need to prove the following sentence for a\in\mathbb{R}:$$ |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 -4a ... 0answers 21 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 ...
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