# Questions tagged [absolute-value]

For questions about or involving the absolute value function.

2,020 questions
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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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### Solving an absolute-value inequality: $−|x|+2 \geq 8x$

How would I go about solving the domain of this inequality? $$−|x|+2 \geq 8x$$ I can't combine the $x$'s so I don't know what to do. Could I say: $$-x + 2 ≥ 8x$$ and $$x - 2 ≥ 8x$$ and solve the ...
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### What does it mean to have an absolute value equal an absolute value?

I have no problem reading absolute value equations such as $|x -2| = 2$. I know this means that the distance of some real number is $2$ away from the origin. Because the origin splits the number ...
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### Veach's thesis, projected solid angle sanity check

Here's an equivalence from Veach's thesis on light transport (page 88, 3.16): $$|\cos(\theta)|\sin(\theta)d(\theta)d(\phi) \equiv \\ \sin(\theta)d(\sin(\theta))d(\phi)$$ This seems wrong in the ...
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### Absolute value of a negative number

I was reading 'The method of Coordinates - Gelfand' and in the section about the absolute value of a number, it is stated what follows : if x > 0, then |x| = x, if x < 0, then |x| = -x, if x = 0, ...
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### On Cauchy sequences over finitely generated modules over complete DVR

I have a question on Cauchy sequences. Let $R$ be a complete DVR with a valuation $v$, $\pi$ a uniformizer, $V$ a finitely generated $R$-module. Since $R$ is PID, $V$ is the direct sum of cyclic ...
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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 ...