# Tagged Questions

For questions about or involving the absolute value function.

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### $2\log ^2_{4}(|x+1|)+\log_4(|x^2-1|)+\log_{\frac{1}{4}}(|x-1|)=0$

Find the sum of solutions to: $$2\log^2_{4}(|x+1|)+\log_4(|x^2-1|)+\log_{\frac{1}{4}}(|x-1|)=0$$ I'm not sure about what to do with the absolute values, how can I get rid of them? Should I solve ...
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### How $|x|<a\implies a>0$

The title is not exactly what I'm asking, so sorry for that. I was doing a problem in my mathematics text book. It is given that $|x|<a$, I thought if $a=2$ then we can put $x=1$ but what if ...
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### Proof that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$

I have included a bolded comment in a step in the part of Gouvea's proof of Ostrowski's theorem where he shows that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$ (the ...
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### Can every (convex) polygon be described by a single inequality (involving absolute values)?

For example, $$|x| + |2x + y| + |x + 2y| + |y| + |x+y| < 4$$ describes an octagon. I'm wondering whether an equation of this form always exist for any convex polygon, and if so, whether there ...
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### What would the multifunctional inverse of $F(x)=|x|$ be?

What would the multifunctional inverse of $F(x)=|x|$ be, assuming $x$ is on the complex plane. Also, how would this usually be represented? Note that this won't be a 'true' function. (But assume a ...
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### Doubt with Intervals and Inequalities

This doubt has been bothering me for ages. I would be truly grateful for any help. Problem 1: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals Solution: $x\in(2,4)\cup(4,6)$ ...
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### Proof of $|a-b| = |b-a|$

While working out some intriguing qualities of absolute values for my studies of calculus, I frequently used the formula below. I know that the formula below is clearly correct but how would I prove ...
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### find the derivative of the integral

Prove that the following integral $F(x)$ is differentiable for every $x \in \mathbb{R}$ and calculate its derivative. $$F(x) = \int\limits_0^1 e^{|x-y|} \mathrm{d}y$$ I don't know how to get rid of ...
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### Solve $|1 + x| < 1$

I'm trying to solve $|1 + x| < 1$. The answer should be $-2 < x < 0$ which wolframalpha.com agrees with. My approach is to devide the equation to: $1+x < 1$ and $1-x < 1$ and then ...
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### Has this question on Absolute Value's been asked wrong?

I was going over some basics on Khan's Academy in preparation for a test. To my surprise I got this wrong: Has this been worded wrong? Surely the person farthest from sea level is Howard? This ...
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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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### Evaluating $\int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y$

I'm always having the wrong result from the following: $$\int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y$$ I would really appreciate some guidance on how to go ...
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### Any tips for solving $\frac{|4x-2|}{|2x+1|} \le 1$ as succinctly as possible?

$\frac{|4x-2|}{|2x+1|} \le 1$ So as I currently see it, I have two choices: 1) Attempt to solve algebraically but that has led me down some long paths when I believe the question should be solvable ...
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### A quadratic polynomial bounded by another

Suppose $p(x)$ and $q(x)$ are two quadratic polynomials in real coefficients such that: $$\lvert p(x) \rvert \leq \lvert q(x) \rvert ~ ~ ~ \text{for all} ~ x \in \mathbb{R} \tag{1}$$ Is the above a ...
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### “p-adic absolute value” in polynomial ring

I'm working with Gouvea's book on p-adic numbers. In problem 34 I'm asked to give a "p-adic" (I put it in quotes as in my understanding its just a p-adic-like) valuation and absolute value for an ...
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### Given $f\in L^1(\mathbb{R})$ with $||f||_1 < \infty$, is it true that $\int_{\mathbb{R}} ||f||_1 - f(x) \, dx = 0$?

According to my intuition so far, the answer should be yes, hinging very important on the assumption that $||f||_1 < \infty$. To speak very roughly, if the $L^1$ norm of $f$ is finite, it seems ...
I noticed the following equality in some material regarding limits and infinite series. $$\left |\frac{x}{x+1} - 1 \right| = \left |\frac{-1}{x+1} \right|$$ And I'm honestly stumped (and slightly ...