# Questions tagged [absolute-value]

For questions about or involving the absolute value function.

2,019 questions
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### Absolute value squared - Meaning

I'm wondering if it's correct to say that the absolute value squared means that the values above $1$ are magnified, whereas the values below $1$ are damped (I'm considering only positive values ...
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### When will equality holds in reverse triangle inequality?

Prove the reverse triangle inequality :$|z\pm w|\ge||z|-|w||$ for all $z, w \in \mathbb C$, with equality holds if and only if either $z$ or $w$ is a real multiple of another. I have proved the ...
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### Difference between the equation inequalities and absolute value inequalities

Using symbol lab I put this in $|x+4|\le |2x+10|$ and the answer I get is $x \le -6$ or $x\ge -14/3$, but when I manually worked out it was $\;x \ge -6\;$ or $\;x\ge -14/3$. My working out is in the ...
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### limit of absolute value

$$\lim_{x \to 0} \frac{\lvert2x-1\rvert - \lvert2x+1\rvert}{x}$$ Defining the function piecewise reveals the limit is in fact, continuous about 0 However when I go to solve it in a normal ...
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### How do I solve $|-2x^2+1+e^x+\sin x| = |2x^2-1|+e^x+|\sin x|$ where x belongs to [0,2π]?

How do I solve $|-2x^2+1+e^x+\sin x| = |2x^2-1|+e^x+|\sin x|,$ where $x$ belongs to [0,2π]? My book solves it in this way: since RHS is positive, it concludes that $1- 2x^2 \ge 0$ and $\sin x \ge 0$....
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### On the Newton Polygon for $p-$adic Power series

I'm studyng a Book about $p-$adic numbers, and I have troubles with a "degenerate" case of a Newton polygon. Let $f(X)=\sum a_{i}X^{i}\in\mathbb{Q}_{p}[\![X]\!]$, we define the Newton poligon of $f$ ...
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### Absolute value and credit card balance

I'm embarrassed to ask this question, but my child has the following homework question: "Use absolute value to describe the relationship between a negative credit card balance and the amount owed." ...
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### What does $\|u\|^2_2$ mean?

Given a vector $u = (x, y, z)$ what is $\|u\|_2^2$ ?
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### Solve the equation |x-1|=x-1

Solve the equation:$|x-1|=x-1$ My solution: Case 1 :$x\ge1$, Hence $x-1=x-1$, therefore infinite solution Case 2 :$x<1$, Hence $1-x=x-1$,$x=1$, hence no solution But the solution i saw ...
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### For which values of $a$ we will get two different roots?

In given the following system of equations: $$|x-1| > 2x+2$$ $$x^2 + ax + a -1 = 0$$ For which values of $a$ we will get two different roots?
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### taking the absolute value of complex numbers to an arbitrary power [closed]

I need $|\frac{i^{n}}{n}|$ and I have seen the problem simplified to $\frac{|i^{n}|}{n}$ and I am confused by this as isn't $\frac{1}{n}$ the coefficient of i so we could just square it and take the ...
### How can I understand how to draw the graph of the function $|x-a|$? [closed]
How can I draw the graph of this function in related to $a$ ? $$f(x) = |x-a|$$