# Questions tagged [absolute-value]

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

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $|x - 1| + |x + 1| < 2$. ...
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### 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)....
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $37, 42$ ...
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### 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 $x+b>0$, ...
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### 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 $\epsilon>0$, ...
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