# Tagged Questions

For questions about or involving the absolute value function.

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### On the equation $|x|^2+|x|-6=0$

Which of the following are true for $$|x|^2+|x|-6=0$$ 1. It has $4$ roots 2. The sum of the roots is $-1$ 3. The product of the roots is $-4$ 4. The product of the roots is $-6$ Only one of the ...
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### applying exponents in an absolute value brackets

So part of the problem I'm trying to solve is this: $|2-9|^{3{^3}}$ being the exponent, to the power of 3. Do I have to apply the exponent to everything in the bracket? 2*2*2 and 9*9*9 or does the ...
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### Steps to Graph Exponential Equations & Absolute Value

how to sketch: $-e^{|-x-1|} + 2$ Can someone clarify: $|f(x)|:$ we draw $f(x)$ and then reflect the ($-y$ parts) in the $x$-axis $f|(x)|:$ we draw $f(x)$ and then reflect the ($-x$ parts) in the $y$...
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### Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$

Compute $|z|$ , $z = \frac{(2+i)^7(1-2i)^3}{(1+2i)^8}$, if $z = a+ib$ then, I tried to do that with $|z| = (a+ib)(a-ib)$ then i multipled it $z$ with $z^-$ and then I got stuck. answer is $|z| = 5$
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### How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$?

How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$ ? I've solved equations like $|a|+|b|=|a+b|$ where the condition must be that $a$, $b$ must be of same sign. But in case of three terms ...
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### Intersection of a point and absolute value function contained within a circle

I'm attempting some crazy ideas while programming a game and ran into the following math problem that has been bugging me for a few days: Given a unit circle and a random point $P$ within the circle, ...
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### Confirmation on a monotonicity formula?

After a long series of difficult problems (which are completely irrelevant) I found myself experimenting with a way to convert a graphed function to a purely monotonic form (I hope I didn't butcher ...
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### Is there a derivative for $|x|$ at $0$ specifically “in the direction” of positive $x$?

I know that $|x|$ is not differentiable at $x=0$ because there is potentially an infinite number of tangent lines going through that point. But let's say we were interested in the motion of an object ...
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### Integrate a periodic absolute value function

$$\int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor$$ I ...
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### Compare difference between mean and actual

My problem is: I have two sets of numbers as follows: $X = {x_1, x_2, ..., x_n}; Y = {y_1, y_2, ..., y_m}$. Where $r$ is the actual value. $x^*$ is the mean of set X, $y^*$ is the mean of set Y, (n!=m)...
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### Could we invent a new number with $|p|=-1$?

We know that how a single definition $i^2=-1$ revolutionized our mathematics and solved many many problems. I wonder whether the definition $|p|=-1$ could have the potential of creating a new ...
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### Automorphisms of local field

(1)Suppose that $K$ is a local field but not $\mathbb C$. Then show that any automorphism of $K$ is continuous. (We can assume that $K$ is $\mathbb R$ with classical absolute value or $K$ is a finite ...
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### Real Analysis Absolute values [closed]

Someone please help me with detailed explanation on how to solve this problem. For all $a, b \in \Bbb R$, show that; $$| a - b | \geq | a | - | b |$$
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### Absolute value graph sketching

Where would you start if you were told to plot: $$||x-1|-1|$$ I looked at just $f(x) = |x-1|$ and noticed that the two equations are: $\pm (x-1)$ for $x \geq 1$ and $x < 1$. Extrapolating then: \$\...
Let a(1)< a(2) < ..< a(m) and b(1)< b(2)<..< b(n) be real numbers such that $$\sum_{i=1}^m |a(i)-x| = \sum_{j=1}^n |b(j)-x|$$ for all x belonging to R. Show that m=n and a(i)=b(i),...