For questions about or involving the absolute value function.

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2answers
26 views

Solve the equation $\Bbb|x-\sqrt{x}|=\lfloor x\rfloor-\sqrt{x}$ [on hold]

$$\Bbb|x-\sqrt{x}|=\lfloor x\rfloor-\sqrt{x}$$ With a great desire for you
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1answer
23 views

solve and skecth $\log{|z|}=-2\arg(z)$

Ive asked this question a week ago, but nobody managed to answer but it is doing my heading from then. I know usually You demand some initial work done on the question but I just dont know how to ...
0
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1answer
20 views

Double integral over the set with an absolute value of $y$

I need to calculate an integral over the set: $$D \colon 0\leq x\leq \pi\text{ and }|y|\leq x$$ from the set (definite integral) $D \int \cos(y)dA$ I don't understand what $|y| \leq x$ means. Can ...
1
vote
1answer
39 views

Double integration over function with absolute values

I have having difficulty in how to solve the following double integral problem involving absolute values and the assumption that $\alpha > 1$: $\iint_{-\infty}^{+\infty} \frac{1}{1+|x|^\alpha} ...
0
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1answer
29 views

Does every non-archimedean absolute value on field take value in $\mathbb{Q}$

Let $K$ be a field, a non-archimedean absolute value is defined to be a map $K\to \mathbb{R}$ satisfying $|x|=0\Rightarrow x=0$, $|x|\cdot|y|=|xy|$ and $|x+y|\leq\max(|x|,|y|)$. Is there an example ...
6
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5answers
311 views

Absolute value and max/min function: why $a + b + |a - b|=2\max(a,b)$? [duplicate]

I am being told that $a + b + |a - b|$ is equal to $2\max(a,b)$. What is the reasoning behind this?
3
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2answers
53 views

Prove $|x|^2$ = $x^2$

My first attempt at this proof divided into 2 cases, one where $x^2$ is greater than or equal to 0, and another where $x^2$ is less than 0. For the first case, I said that the definition of absolute ...
1
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5answers
62 views

$\left|x\right| < \left|\tan(x)\right|$ close to $0$

I was trying to prove this inequality $\left|x\right| < \left|\tan(x)\right|$ in a neighborhood of $0$. I tried splitting into the four cases opening the modulus but still wasnt able to solve it. I ...
0
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1answer
19 views

Why/how do I ignore these absolute values while using Variation of Parameters?

I'm given the initial value problem: $x' = \frac{3}{t}x + e^{3t}$, $x(1) = 2$ Using the variation of parameters formula, I end up with: $2e^{3 \ln|t|} + e^{3 \ln|t|} \int^{t}_{1}e^{-3\ln|s|} e^{3s} ...
4
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3answers
84 views

What is the fastest way to find the range of functions having modulus: $f(x) = |x+3| - |x+1| - |x-1| + |x-3|$

While solving problems I saw a question in which I was supposed to find the range of a function $$f(x) = |x+3| - |x+1| - |x-1| + |x-3|$$ I know the way in which I can take different cases of $x$ ...
0
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1answer
35 views

Complex numbers converge if their absolute values and arguments converge

Let the sequence $\{z_n\}_{n>0}$ and $w \not=0$ be such that $|z_n| \to |w|$ and $\operatorname{Arg}(z_n) \to \operatorname{Arg}(w)$. Show that $z_n \to w$. My proof: $z_n= |z_n|e^{i \arg(z_n)} ...
3
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2answers
41 views

Prove $||a| - |b||$ is less than or equal to $|a-b|$

I was given the hint to split it into two cases ($|a| - |b|$ being positive and negative) and then use the triangle inequality. However, since the triangle inequality says that $|a+b|$ is less than or ...
5
votes
1answer
52 views

Prove that $|x|^2$ = $x^2$.

This is what I did, but I'm not sure if it's a good enough proof: Since $|x|$ is equal to $x$ when $x$ is greater than or equal to 0, and is equal to $-x$ when $x$ is less than 0, I said that $|x|^2$ ...
0
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2answers
37 views

Absolute Value Rational Inequalities

Ok so I have the following two inequalities: \begin{equation} \left| \frac{x+6}{x-2}\right| \leq 4 \end{equation} and \begin{equation} \frac{x^2-1}{\left| x+2\right|} \leq 3(1-x) ...
-2
votes
1answer
102 views

How to prove that $\lim_{(x, y) \to (0, 0)} \left(\frac{1}{|x|} + \frac{1}{|y|}\right) = 0$? [closed]

Prove that $$\lim_{(x, y) \to (0, 0)} \left(\frac{1}{|x|} + \frac{1}{|y|}\right) = 0$$ I couldn't prove this. Please suggest a solution.
1
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4answers
61 views

Prove $\lvert x\rvert$ = $\lvert-x\rvert$ for all real numbers $x$ [closed]

Been at this one for a long time. I'm trying to use the fact that $|x|$ = $x$ if $x$ is greater than or equal to 0, and $|x|$ = $-x$ if $x$ is less than 0. Then I want to split the proof into these 2 ...
0
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2answers
27 views

Piecewise from Rational Absolute Value Function

How would one separate a function like the following into piecewise? $$f(x)={\left|4-x\right|\over{\left|x-4\right|}}$$ I've been taught that with a rational function with an absolute value in the ...
4
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3answers
64 views

Proving $\max$ of $a, b$.

How do I prove that $$\max{\{a, b\}} = \frac{a + b + \left | a - b \right |}{2}$$ I have no idea how to even start the proof, any idea / intuition that can get me started is greatly appreciated. ...
0
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3answers
40 views

Square divided by absolute value

First time posting on Math SE, with kind of a basic algebra question. Question Does the relation: $$\dfrac{(ab)^2}{|ab|} = \left|ab\right|$$ with $a,b \in \mathbb{R_{\ne 0}}$ always hold? It seems ...
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0answers
19 views

Function with absolute value in denominator - limits

f(x)=(x-1)/(|2-x|-1) |2-x|= { |2-x|; x < +2} {-|2-x|; x >= +2} State domain, range and the equations of the asymptotes. D(f)= {x | x > 3 or x < 3} R(f)= {y | y > 1 or y <= -1} ...
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0answers
35 views

Show that absolute value satisfies triangle inequality, how? [duplicate]

I wish to show that given $a,b,c \in \mathbb{R}$, the following holds: $|a - c| \leq |a-b| + |b-c|$ Using the definition $| x | = \max\{x, -x\}$ I can't seem to be able to show that $|a - c| \leq ...
2
votes
6answers
89 views

Solving the absolute value inequality $\big| \frac{x}{x + 4} \big| < 4$

I was given this question and asked to find $x$: $$\left| \frac{x}{x+4} \right|<4$$ I broke this into three pieces: $$ \left| \frac{x}{x+4} \right| = \left\{ \begin{array}{ll} ...
2
votes
2answers
59 views

Evaluating the Fourier coefficients of $abs(x)$

Let's get started: $$\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi} |x|e^{-inx} dx$$ since $|x|$ is an even function: $$= \frac{1}{\pi}\int_0^{\pi} xe^{-inx} dx$$ Integration by parts yields: ...
0
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0answers
23 views

Continuity properties of an example function $f:\mathbb{R}^n\to\mathbb{R}$

Consider the function $f:\mathbb{R}^n\to\mathbb{R}$ defined as follows: $$ f(x)=\begin{cases} ||x||^2 & \text{if $||x||\le 1$,}\\ 1/||x||^2 & \text{if $||x||> 1$,} \end{cases} $$ where ...
0
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3answers
41 views

Nested absolute-value inequality

I try to solve a problem in two ways, but the results are not the same. Method 1. $$\lvert \lvert x \rvert + x \rvert \le 2$$ For $x < 0$, we have $\lvert x \rvert = -x$. Therefore: $$\lvert ...
0
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1answer
25 views

Modulus of Two Complex Numbers, Squared

I have a very silly question to ask! I have $|z_{1} + z_{2}|^2 = |z_{1}|^2+|z_{2}|^2+2|z_{1}||z_{2}|\cos{\theta}$, where $z_{1}$ and $z_{2}$ are complex numbers. For the life of me I cannot ...
1
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1answer
56 views

Prove those inequalities are true [duplicate]

I want to prove that those inequalities are true for $a, b ∈ R$: $$ |a + b| ≤ |a| + |b| $$ $$ ||a| − |b|| ≤ |a − b| $$ $$ |a − b| ≤ |a − c| + |c − b| $$ Now I can see that they are true, and I could ...
0
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1answer
32 views

Proving inequality with absolute value [duplicate]

How can I show the following inequality for any real numbers $x,y,z$? $$\frac{|x-z|}{1+|x-z|}\le \frac{|x-y|}{1+|x-y|} + \frac{|y-z|}{1+|y-z|}.$$ The triangle inequality could be useful, but I am ...
0
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1answer
22 views

Which of the two following solutions is correct for absolute value of this expression?

I'm currently going through Spivak and ran across this problem, but i see a difference in my answer and the answer that i'm checking it again. The problem is to eliminate the absolute value signs in ...
5
votes
6answers
148 views

Prove that $|-x| = |x|$

Using only the definition of Absolute Value: $\left|x\right| = \begin{cases} x & x> 0 \\ -x & x < 0 \\ 0 & x = 0,\end{cases}$ Prove that $|-x| = |x|.$ This seems so simple, but I ...
3
votes
2answers
113 views

Derivative of $x\cdot|x|$ on $x=0$?

$$f(x) = x |x|$$ Wolfram Alpha says is: $$f'(x) = \frac{2x^2}{|x|}$$ and thus $f'(0)$ is indeterminate, while an HP48 says that: $$f'(x) = |x| + x \operatorname{sgn} x,$$ which would yield $f'(0) ...
1
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1answer
29 views

Integrating an integrand with an absolute value on exponential

This is one heck of an embarrassment but it is amazing how these bits of subtlety gets lost in the back of the head after the first year of undergraduate studies-with every computation chucked into ...
2
votes
3answers
69 views

Find all $z$ such that $\left|\tan z\right| = 1$

Find all z such that $$\left|\tan z\right| = 1$$ The first thing that came to my mind was to write tangent in terms of $e^z$ and take its modulus, but I couldn't solve it in this way.
0
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1answer
19 views

Normal distribution question involving absolute value

I don't quite understand how to do the following question. How I tried to do it is to imagine the normal distribution curve, with the highest peak at 4. I understand that |Q| means the absolute value ...
3
votes
3answers
57 views

Is finding the second derivative of $\sqrt[3]{\vert x\vert}$ the best method to determine if it is convex?

I have an exercise where I have to tell on which intervals a function is concave or convex. I usually do it using second derivative, but I would like to know if there is a simpler way of doing so, ...
0
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5answers
79 views

Is the absolute value of zero positive or negative?

If I had $|x|$, then we know, for pretty much any $x$, that the following is true:$$|x|\ge0$$$$|0|=0?$$Which, by the nature of how we usually apply the absolute value, the solution is positive and ...
0
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2answers
29 views

One absolute value inside of another absolute value in the equation

Let's say we have an equation: $||x|-2| = |2|x|+4|$ How does one go solving it? Symbolab says that it currently doesn't support step by step explanation for this problem, so I would really ...
2
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2answers
35 views

Why is there $(2,3)$ corner coordinates on the graph, using this function: $f(x)=2x-1+|x-2|?$

Currently I am studying about absolute values and I had to consider this function and draw it's graph: $$f(x)=2x-1+|x-2|$$ I helped myself with symbolab, here is the link: ...
1
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1answer
41 views

How Does This Line Intersection Equation Work?

For the lines: $y = ax + b$ $y = cx + d$ The standard intersection equation $x_i = \frac{d - b}{a - c}$ $y_i = \frac{ad - bc}{a - c}$ If the points that I was given to find the line $y = ax + ...
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3answers
79 views

Integrating $f(x)=\int|\cos(x)|dx$ and then solving $f(x)=\frac {2x}{\pi}$?

I realised the other day that by applying absolute value signs to the cosine function and then integrating, I would get an almost sine function that doesn't have negative slope. And then I also ...
0
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1answer
23 views

Validity of inequalities using integrals and absolute value

This question is similar to this one but the only response was pointing out mistakes in the solution. My goal is to determine whether the operator $T: C[0,1] \to C[0,1]$ defined by $Tx = ...
0
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1answer
35 views

Integrating absolute terms

This is just to clarify my doubt regarding absolute values functions. Lets say there is a function $$f(x) = ax^{2} - \left|\frac{bx}{c}\right|$$ and we are asked to integrate this over $-\infty \to ...
3
votes
2answers
40 views

Absolute value equation with rational expression

I am to solve the equation: $|\frac{2x}{x^2 - 3} | < 1$ And so: 1. I rewrote it as $|2x| < |(x - \sqrt 3) | |(x + \sqrt 3)|$ And I tried to divide it into a few intervals For $ ...
1
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1answer
52 views

how to minimize a summation containing an absolute value

Thank you all in advance I have been having some trouble figure out the following problem. You are given a sample {$y_i$}, i=1,… N, from an unknown probability distribution p(y). I want to show the ...
1
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2answers
33 views

Integration of $|e^{-(2+j)t}|^2$

The integration of $|e^{-(2+j)t}|^2$ from zero to infinity is $1/4$ when I separate above as $|e^{-2t}|^2 \cdot |e^{-2jt}|^2$ and integrate. $|e^{-2jt}|$ was taken as $1$. But when I integrate the ...
5
votes
4answers
126 views

Why does $\sqrt{x^2}$ seem to equal $x$ and not $|x|$ when you multiply the exponents?

I understand that $\sqrt{x^2} = |x|$ because the principal square root is positive. But since $\sqrt x = x^{\frac{1}{2}}$ shouldn't $\sqrt{x^2} = (x^2)^{\frac{1}{2}} = x^{\frac{2}{2}} = x$ because of ...
5
votes
1answer
157 views

Verify integration of $ \int\frac{\sqrt{2-x-x^2}}{x^2}dx $

This is exercise 6.25.40 from Tom Apostol's Calculus I. I would like to ask someone to verify my solution, the result I got differs from the one provided in the book. Evaluate the following integral: ...
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vote
2answers
22 views

Find the points at which $f$ has an absolute maximum or minimum on $I$ without graphing

Assume $I=[0.9,3.1], f:I\rightarrow\mathbb{R}$ is defined by $f(x):=|x^2-4x+3|, x\in I$. Without sketching the graph of $f$ on $I$, find points at which $f$ has an absolute minimum on $I$ and points ...
1
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4answers
179 views

What is the name for this operation?

What is the name for this operation? Effectively, take a range and adjust its center point to 0 on a number line. In the case of the example above, I'm specifically looking for the name or ...
1
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1answer
15 views

Some equations involving multiple absolute values

Consider the following equation: $$|x+y^2|+|x-y^2|+|y+x^2|+|y-x^2|=a$$ I'm looking for the method for solving some problems regarding this equation, namely: 1) prove that if $a=2015$, then the ...