# Tagged Questions

For questions about or involving the absolute value function.

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### Jargon for maximum/minumum absolute value in a set

Given a group of numbers $-5,-3,1,2$, the maximum is 2, the minimum is -5. What is the mathematical jargon for the maximum and minimum in absolute terms (i.e. -5 and 1 respectively)? Basically, I ...
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### Distribution of minimum absolute value

Consider $K$ independent Laplace variables $X_k, k=1,\ldots,K$, with mean 0 and scale $\lambda$ (so that their PDF is $f(x)=\frac{1}{2\lambda}e^{-\frac{|x|}{\lambda}}$. Let $Y$ be the variable taking ...
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### Why is $\max(x, x')$ equivalent to $\frac{1}{2}( x + x' + |x - x' |)$?

Why is it that $$\max(x, x') = \frac{1}{2}( x + x' + |x - x'|)$$ is true? Is it supposed to be obvious? Because it seems to come out of thin air for me. Anyway, I've verified this by plotting it in ...
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Problem: Find all $x$ such that $|x^2+6x+6|=|x^2+4x+9|+|2x-3|$ I can't understand how to get started with this. I thought of squaring both sides of the equation to get rid of the modulus sign, ...
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I've recently started to think about this, and I'm sure a couple of you out there have, too. In Algebra, we learned that $|x|\geq0$, no matter what number you plug in for $x$. For example: $$|-5|=5\... 2answers 54 views ### Solve x^2-|5x-3|-x<2,\ \ x\in \mathbb{R}  Solve x^2-|5x-3|-x<2,\ \ x\in \mathbb{R}  I tried x^2-|5x-3|-x<2 , case 1 , x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\  x^2-(... 0answers 40 views ### Integral of |\cos(ax))|\times e^{-x^2/b} I can compute the following integral very easily (a and b are real and positive):$$\int_{-\infty}^{\infty} \cos(ax)\times \frac{1}{\sqrt{\pi b}}\cdot e^{-\frac{x^2}{b}}\,dx = e^{-\frac{a^2b}{4}}$... 1answer 186 views ### Lower bound on absolute value of determinant of sum of matrices I needed to find a lower bound on$|\det(A+B)|$where$|.|$is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But$||\det(A)|-|\...
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Problem: Find all $x$ such that $|x^2-3x+1|<1$ I can't understand how to get started with this. I've never tried to solve quadratic Inequalities before. At first I thought of working with the ...
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### Doubt with Absolute Value Inequality

Problem: Find all values of $x$ for which $\dfrac{|x-2|}{x-2}>0$ My incorrect attempt: Using the definition the Modulus, $|x-2|=x-2$ for all $x\ge2$ and $|x-2|=-x+2$ for all $x\le2.$ ...
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### Find $\int_a^b \sin |x| \, \mathrm{d}x$

How to find the integral $$\int_a^b \sin |x| \, \mathrm{d}x \,?$$ I'm able to obtain definite integral of form $\int_a^b \lvert\sin x \rvert \, \mathrm{d}x$ but not when the modulus operator is ...
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### Why is $\sqrt{x^2}= |x|$ rather than $\pm x$? [duplicate]

Shouldn't the square root of a number have both a negative and positive root? According to Barron's, $\displaystyle \sqrt{x^2} = |x|$. I don't understand how.
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### Proving that $|a-b|≤|a|+|b|$ [closed]

Can someone prove this to me: $$|a-b|≤|a|+|b|$$ I am in 8th grade and I have this for my homework. Thanks for helping.
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### How to solve equations containing multiple $|x|$s?

Suppose I have an equation which looks like: $$|x-2| + |2x+1| = 3$$ or, $$|x-1| + |x-3| - |5x-1| = 2$$ How should I solve such problems? What i do is generally a kind of "hit-and-trial" ...
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### Graph $y=|x+8|+|x-8|$

Graph $y=|x+8|+|x-8|$ I tried to simply this with $$y=(x+8)+(x-8) \implies y=2x,x>0\\ y=(-x+8)+(-x-8) \implies y=-2x,x<0$$ But this looks quite different from the original. I look ...
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### Basic absolute value property

Hello all I am wondering if anyone has the correct proof that I should use for Spivak calculus ( chapter 1, question 12 ) that says $$|xy|=|x| \cdot |y|$$ from past times I know it is true , but I ...
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### Question on valuation axioms - Relating to $\mathbb{R}$

In an earlier thread, I asked if there was a standard generalization of the absolute value of $\mathbb{C}$ that could be placed on a field, but might not take values in $R_{\geq 0}$. What somebody ...
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### Solutions for $|x^2-5x+2|=4$

Problem: Find all values of $x$ such that $|x^2-5x+2|=4$ The only way I can see to solve this would be to square both sides of the equation so as to eliminate the modulus sign. However, that ...
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### Absolute inequality derivation

I have been trying to prove an inequality that I am not even sure if it is even true or not. However I am experiencing great difficulties with this proof. I have an intuition that it is true and have ...
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### Why if $B = \{x : |x+1| ≤ 3 \}$ then $B$ equals $[ -4, \infty )$?

I really don't understand why $B$ is from $-4$ to infinity because $x+1 ≤ 3$ $x ≤ 2$ and $-3 ≤ x+1$ $-4 ≤ x$ Shouldn't it be $B = [-4, 2]$?
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### Arranging problem: 4 couples, 8 seats in a row… Am I making this too simple?

I am in a prob and stats course... haven't taken one in awhile and would like some help on these two problems. I think I am probably making these a little two simple. Four married couples have ...
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### Rotate the graph of a function?

How do I rotate a graph of a function around a point, and show it in the related equation? An example could be $f(x)=\lvert x\rvert$ (absolute Value) and $f(x)=x^2$
### Examine the continuity of function $f(x)=\frac{2x^2-4x}{|x+1|+|x-3|-2}$
Using the definition of absolute value for $$|x+1|=\begin{cases} x+1, & x\ge -1\\ -x-1, & x>-1 \end{cases}$$ and $$|x-3|=\begin{cases} x-3, & x\ge 3\\ -x+3, & x>3 \end{cases}$$ ...