For questions about or involving the absolute value function.

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2answers
70 views

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 ...
3
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3answers
117 views

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 ...
1
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2answers
55 views

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 $B = [-4, 2]$?
6
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3answers
666 views

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 ...
1
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3answers
44 views

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 $f(x)=x^2$
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1answer
35 views

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 $$|x-3|=\begin{cases} x-3, & x\ge 3\\ -x+3, & x>3 \end{cases}$$ ...
3
votes
1answer
75 views

Is there such a norm on any totally disconnected local field?

Let's set this definition of local field: Let $\mathbb{K}$ be a field and a topological space (non-discrete and totally disconnected). Then $\mathbb{K}$ is called a local field if both ...
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1answer
22 views

How do I see that $|Re^{it}((Re^{it})^2+4)| \ge R(R^2 - 4)$ for $R \in \mathbb R$ sufficiently large?

How do I see that $|Re^{it}((Re^{it})^2+4)| \ge R(R^2 - 4)$ for $R \in \mathbb R$ sufficiently large ? I've been trying to use triangle inequality and other ways, but I've not come to a conclusion. ...
1
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2answers
57 views

Proof of absolute inequality

I am new to proofs and would like some help understanding how to prove the following abs inequality. $$| -x-y | \leq |x| + |y|.$$ I think I should take out the negative in the left absolute value ...
1
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4answers
66 views

How to solve inequality for : $|7x - 9| \ge x +3$

How to solve inequality for : $|7x - 9| \ge x + 3$ There is a $x$ on both side that's make me confused...
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4answers
37 views

Integers $a$ for which the equation $\big\lvert 6\lvert x\rvert -8\big\rvert = a+x$ has the most solutions

$$\big\lvert 6\lvert x\rvert -8\big\rvert = a+x$$ I know this should be done graphically, looking at each case and seeing for which $a$ will it intersect the $x$-axis the most times, but I can't seem ...
3
votes
1answer
35 views

Give an example of a continuous function $f: (-1, 1) \rightarrow \mathbb{R}$ which attains a maximum at 0, but is not differentiable at 0

I need a little help with this exercise: Give an example of a continuous function $f: (-1, 1) \rightarrow \mathbb{R}$ which attains a maximum at 0, but is not differentiable at 0 I thought of the ...
3
votes
3answers
42 views

Plotting $|x - y| \leq 1/4$

I'm trying to plot $|x - y| \leq 1/4$ I've reduced this to $|x - y| \leq 1/4$, and then deduced via plugging in points that there should be two lines, one with y-intercept 0.25 and another with y ...
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votes
1answer
56 views

How to integrate a function with a nested absolute value: $|x^2 - 2|x||$? [closed]

I need help with this problem, $$\int_0^4|x^2 - 2|x||dx$$ what should I do with $2|x|$ ?
2
votes
1answer
45 views

Is this a correct way to use triangle inequality

If I have: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq f(x^*)$$ Can I proceed to say: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq |g_1(x) - g_2(x)| - |(g_1(a) - g_2(a))|$$ $$ \implies |g_1(x) - ...
2
votes
1answer
19 views

Proof that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$

I have included a bolded comment in a step in the part of Gouvea's proof of Ostrowski's theorem where he shows that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$ (the ...
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votes
2answers
67 views

Can every (convex) polygon be described by a single inequality (involving absolute values)?

For example, $$ |x| + |2x + y| + |x + 2y| + |y| + |x+y| < 4 $$ describes an octagon. I'm wondering whether an equation of this form always exist for any convex polygon, and if so, whether there ...
0
votes
1answer
27 views

What would the multifunctional inverse of $F(x)=|x|$ be?

What would the multifunctional inverse of $F(x)=|x|$ be, assuming $x$ is on the complex plane. Also, how would this usually be represented? Note that this won't be a 'true' function. (But assume a ...
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1answer
40 views

Doubt with Intervals and Inequalities

This doubt has been bothering me for ages. I would be truly grateful for any help. Problem 1: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals Solution: $x\in(2,4)\cup(4,6)$ ...
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6answers
149 views

Proof of $ |a-b| = |b-a| $

While working out some intriguing qualities of absolute values for my studies of calculus, I frequently used the formula below. I know that the formula below is clearly correct but how would I prove ...
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3answers
49 views

find the derivative of the integral

Prove that the following integral $F(x)$ is differentiable for every $x \in \mathbb{R}$ and calculate its derivative. $$F(x) = \int\limits_0^1 e^{|x-y|} \mathrm{d}y$$ I don't know how to get rid of ...
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4answers
50 views

Solve $|1 + x| < 1$

I'm trying to solve $|1 + x| < 1$. The answer should be $ -2 < x < 0$ which wolframalpha.com agrees with. My approach is to devide the equation to: $1+x < 1$ and $1-x < 1$ and then ...
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1answer
24 views

Has this question on Absolute Value's been asked wrong?

I was going over some basics on Khan's Academy in preparation for a test. To my surprise I got this wrong: Has this been worded wrong? Surely the person farthest from sea level is Howard? This ...
0
votes
4answers
34 views

Soft absolute value

I'm looking for a "soft absolute value" function that is numerically stable. What I mean by that is that the function should have $\mp x$ asymptotes at $\mp\infty$ and behave smoothly in $[-1,1]$. ...
2
votes
2answers
52 views

Solve differential equation $y' = |1.1 - y| + 1$

How can the following differential equation be solved analytically? \begin{equation*} y' = |1.1 - y| + 1, \\ y(0) = 1. \end{equation*} I guess one must rewrite the differential equation piecewise ...
6
votes
5answers
896 views

If three complex numbers $z_k$ have modulus $1$, then $|z_1+z_2+z_3| = |\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$

Our teacher gave us a hard question (according to her, it is pretty hard for our level): Given that $|z_1| = |z_2|= |z_3|=1,z \in\mathbb{C}$, prove that $|z_1+z_2+z_3| = ...
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4answers
119 views

Can some explain very quickly what $ |5 x + 20| = 5 $ actually means?

I don’t mean to bother the community with something so easy, but for the life of me, I can’t remember how to do these. I even forgot what they were called, and I’m referring to the use of the “$ | $” ...
3
votes
1answer
141 views

mean value theorem sin(b) - sin(a)

It's too much hassle to post it here as latex, to so here's the screenshot. I don't understand why |cos(c)| = 1 Why 1? Why not $\frac {\sqrt{3}}{2}$? Why absolute value assumes the max value a ...
0
votes
3answers
68 views

What is $\lim_{x\to 7^-} \frac{\left|x-7\right|}{x-7}\,$?

$\displaystyle \lim_{x\to 7^-} \frac{\left|x-7\right|}{x-7} = $ Writing absolute value as: $x-7 > 0$ $x > 7$ which means $x - 7$ when $x > 7$ then: $ -(x - 7) < 0$ $-x + 7 < 0$ ...
2
votes
1answer
28 views

non-archimedean absolute value (Ostrowski's theorem)

I'm reading the proof of Ostrowski's theorem in Gouvea's book on p-adic numbers and there is one step that I don't understand. Let $|\cdot|$ be a non-archimedean absolute value and $n=rp+s$ where ...
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2answers
30 views

Evaluating $\int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y$

I'm always having the wrong result from the following: $$ \int_{-5}^5\int_{-5}^5-\frac{3}{2}|x+y|-\frac{3}{2}|x-y|+15\,\mathrm{d}x\,\mathrm{d}y $$ I would really appreciate some guidance on how to go ...
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5answers
70 views

Any tips for solving $\frac{|4x-2|}{|2x+1|} \le 1$ as succinctly as possible?

$\frac{|4x-2|}{|2x+1|} \le 1$ So as I currently see it, I have two choices: 1) Attempt to solve algebraically but that has led me down some long paths when I believe the question should be solvable ...
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1answer
35 views

A quadratic polynomial bounded by another

Suppose $p(x)$ and $q(x)$ are two quadratic polynomials in real coefficients such that: $$\lvert p(x) \rvert \leq \lvert q(x) \rvert ~ ~ ~ \text{for all} ~ x \in \mathbb{R} \tag{1}$$ Is the above a ...
1
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1answer
34 views

“p-adic absolute value” in polynomial ring

I'm working with Gouvea's book on p-adic numbers. In problem 34 I'm asked to give a "p-adic" (I put it in quotes as in my understanding its just a p-adic-like) valuation and absolute value for an ...
3
votes
2answers
39 views

Is every Pisot-like integer the product of a Pisot integer and a root of unity?

For lack of better terminology, let's call an algebraic integer $\beta$ Pisot-like if $|\beta|_{\mathbf{v}} > 1$ for the place $\mathbf{v}$ of $\Bbb{Q}(\beta)$ corresponding to the embedding $\beta ...
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1answer
32 views

Online calculator for $ p $-adic valuations and absolute values.

Does anyone know a website where I can enter a prime base and a rational and then get the $ p $-adic valuation and the $ p $-adic absolute value? For sure I know how to do it by hand, but I want to ...
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2answers
29 views

How do I upperbound this expression?

With a given condition such as $$|x|^2 > |y|^2$$ Is there any way I can upper bound the following expression $$\log\left(1+\big||y|-x\big|^2\right) \leq \,\,\, ? $$ Thank you
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votes
2answers
79 views

For X,Y random variables, with pdfs that are symmetric around 0, does $V(X)\geq V(Y) \Rightarrow E(|X|)\geq E(|Y|)$?

I need to show the following thing. Consider two continuous random variables $X,Y$ which take values in $[-1,1]$ and are have pdf's that are symmetric around zero. How can I show that $V(X)\geq V(Y) ...
3
votes
3answers
86 views

Prove that $g(x):=|f(x)|$ is differentiable iff $f'(a)=0$

Suppose that $f(a)=0$. Prove that $g(x):=|f(x)|$ is differentiable iff $f'(a)=0$ Not sure how to go about this at all. The limit definition that I am working with is $$ g'(a)=\lim_{x \rightarrow ...
2
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1answer
46 views

An expression with gcd and abs is transformed magically!

There's a problem to calculate $\sum^{n}_{i=1}\sum^{m}_{j=1}\frac{|i-j|}{\gcd(i,j)}$, whose tutorial gives the following transformation I really don't understand. ...
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1answer
27 views

Given $f\in L^1(\mathbb{R})$ with $||f||_1 < \infty$, is it true that $\int_{\mathbb{R}} ||f||_1 - f(x) \, dx = 0$?

According to my intuition so far, the answer should be yes, hinging very important on the assumption that $||f||_1 < \infty$. To speak very roughly, if the $L^1$ norm of $f$ is finite, it seems ...
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2answers
27 views

Absolute Value Algebra with inverses

I noticed the following equality in some material regarding limits and infinite series. $$ \left |\frac{x}{x+1} - 1 \right| = \left |\frac{-1}{x+1} \right| $$ And I'm honestly stumped (and slightly ...
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2answers
40 views

Solving Equations And Inequations Based On The Absolute Function [closed]

Today I came across some equations and inequations based on the absolute function. These were $|x^2+4x+3|+2x+6=0$ $|x^2+6x+7|=|x^2+4x+4|+|2x+3|$ $1\le |x-1|\le 3$ $\frac{2}{|x-4|}\gt 1$ $||x|-1|\le ...
2
votes
1answer
25 views

Differentiability of an absolute function.

Check the differentiability of $f(x)=x|x|$, $x$ is in $\mathbb{R}$. I know that it is differentiable when $x>0$ and $x<0$. I am not sure about the case when $x=0$. I found that as $$\lim ...
1
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0answers
34 views

Simplifying a function with absolute value and two variables

I am trying to simplify the following $ f(\alpha, \beta)= \big|1-(\frac{\alpha+\beta}{2}+\frac{|\beta-\alpha|}{2})\big|-2 \big|1-\alpha-(\frac{\alpha+\beta}{2}+\frac{|\beta-\alpha|}{2})\big|+ ...
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0answers
33 views

Solve this inequality for B

I am working on a program that is supposed to qualify a value as "in range" and I have come up with the expression: $$\lvert a-b\rvert \leq c$$ to determine the value. Plugging in test numbers ...
0
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1answer
37 views

How to graph $|z-1| <2$

Am I correct to rearrange this to $(z-1)^2 < 4$, and hence just graph as a circle or am I completely off?
0
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1answer
32 views
0
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1answer
37 views

Decomposing absolute value terms

I have something like the following term: 7x1 + 9x2 + | 10 - 7x1 | + | 15 - 11x2 | I want to make it into something like this: Ax1 + Bx2 , where A and B are constants For two values x1 and x2, ...
0
votes
3answers
93 views

Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$

Solve $\frac{|x|}{|x-1|}+|x|=\frac{x^2}{|x-1|}$.What will be the easiest techique to solve this sum ? Just wanted to share a special type of equation and the fastest way to solve it.I am not asking ...