Questions tagged [absolute-value]

For questions about or involving the absolute value function.

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2answers
5k views

Log laws and modulus

If you have the log of a modulus, (like after integration), how do the log laws work? So if you have $a\ln\left|2x-3\right|$ does it become: $\ln\left|(2x-3)^a\right|$ or $\ln(\left|2x-3\right|)^a$, ...
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1answer
888 views

Absolute function continuous implies function piecewise continuous?

I have a simple true/false question that I am not sure on how to prove it. If $|f(x)|$ is continuous in $]a,b[$ then $f(x)$ is piecewise continuous in $]a,b[$ Anyone that can point me in the ...
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2answers
454 views

True/false question: limit of absolute function

I have this true/false question that I think is true because I can not really find a counterexample but I find it hard to really prove it. I tried with the regular epsilon/delta definition of a limit ...
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1answer
3k views

double integral of an absolute function

I'm just a little unsure of how to tackle this one. I understand that typically you would separate the integral into two for where x is positive or negative, I'm just unsure of how to separate it for ...
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1answer
874 views

Separable first-order linear equation and absolute value removal

We can use the integral of $\frac{1}{x}$ in order to solve a separable first-order linear equation like this: $\frac{dy}{dt} + f(t) y = 0$ $ ln |y| = \left(-\int f(t)\,dt\right) + C $ and then: $...
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2answers
363 views

Why is the derivative of $\frac{|x|}{x}$ equal to $\emptyset$ at $x=0$?

I got a bit of a confusion here. If $\varphi(x)=\frac{|x|}{x}$, then $$ \varphi(x) = \left.\Bigg\{ \begin{array}{cc} 1 &if \ x>0\\ \emptyset & if \ x=0\\ -1 & if \ x <0 \end{array} \...
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1answer
100 views

whats the absolute time interval?

For the time interval $1< \lvert t+1 \rvert \leq 3 $ I am trying to solve for t to get my range to plot a function. I know $\lvert t+1 \rvert \leq 3 \iff -3\leq t+1 \leq 3 \iff -4 \leq t \leq 2$ ...
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1answer
236 views

Ring of integers in a field of fractions

Let $R$ be ring with complete non archimedian absolute value. Let $Q$ be the associated field of fractions with the extended absolute value. Does the ring $O_Q = \{x\in Q | |x|\leq 1\}$ is complete ?...
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2answers
177 views

Maximum of the difference

What is the maximum value of $f(… f(f(f(x_{1} – x_{2}) – x_{3})-x_{4}) … – x_{2012})$ where $x_{1}, x_{2}, … , x_{2012}$ are distinct integers in the set ${1, 2, 3, …, 2012}$ and $f$ is the absolute ...
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2answers
2k views

Why is the absolute value needed with the scaling property of fourier tranforms?

I understand how to prove the scaling property of Fourier Transforms, except the use of the absolute value: If I transform $f(at)$ then I get $F\{f(at)\}(w) = \int f(at) e^{-jwt} dt$ where I can ...
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2answers
147 views

Constructing new numbers from negative absolute value

Before the construction of the complex numbers, people thought you couldn't take the square root of a negative number. Then came along of the definition of the imaginary unit $$i^2 = -1$$ and now we'...
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4answers
941 views

double absolute values

I am having a little bit of problem with an inequality with nested absolute values: $$|z^2-1| \ge |z+|1-z^2||$$ I've tried solving it by making three cases, $z\ge1$, $z\le-1$ and $z$ between $1$ and ...
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2answers
654 views

When does a absolute value equation have one unique solution?

Find $m \in \mathbb R$ for which the equation $|x-1|+|x+1|=mx+1$ has only one unique solution. When does a absolute value equation have only 1 solution? I solved for $x$ in all 4 cases and got $x=\...
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2answers
89 views

Confusion solving $\sqrt{4m^2-4m+1}+|1-2m|\leq2$, weird solution.

I am trying to solve $\sqrt{4m^2-4m+1}+|1-2m|\leq2$. Since i know $|1-2m| = \pm(1-2m)$ i tough solving $\sqrt{4m^2-4m+1}+1-2m\leq2$ and $\sqrt{4m^2-4m+1}-1+2m\leq2$. As solutions i get $0\leq2$ and $m\...
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1answer
111 views

CDF for random variable $X(\omega) = 2(1-|2\omega - 1|)$

I don't know how to calculate this cdf, the modulus is very annoying, because the cdf definition is $P(X< x)$ in my case $P(\omega < x)$. But in the modulus equality I get this $P(-\omega < (...
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2answers
568 views

Calculating the absolute value of a complex number - am I right?

To calculate the absolute value of a complex number u must use the following formular $(a^2+b^2)^½$=|a+bi| So for instance with -4-5i would have the absolute value $((-4)^2+(-5)^2)^½$=$(16+25)^½$=$\...
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2answers
2k views

Absolute value and sign of an elasticity

In my microeconomics book, I read that when we have $1+\dfrac{1}{\eta}$ where $\eta$ is an elasticity coefficient, we can write $1-\dfrac{ 1}{|\eta|}$ "to avoid ambiguities stemming from the negative ...
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1answer
59 views

how to prove if $a|b$ and $b\neq 0$, then $|a|\leq|b|$

where the conditions are: $a \neq 0$, $b \neq 0$ and $a$ and $b$ are integers. maybe i'm missing something very basic about the properties of an absolute values. My approach was to supposed, on the ...
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3answers
195 views

Is there an alternate definition for $\{ z \in \mathbb{C} \colon \vert z \vert \leq 1 \} $.

Is there a method of constructing a subset of a reasonably arbitrary ring so that when the construction is applied the $\mathbb{C}$ the result is $B = \{ z \in \mathbb{C} \colon |z| \leq 1 \} $? My ...
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1answer
420 views

The maximum absolute value of DFT of window vector

Let x=[1, ⋯ ,1, 0, ⋯ ,0] be a window vector of length N, which consists of B consecutive 1s and the remaining N-B consecutive 0s. I took the N-point DFT on x and got X=[X_0, X_1, ⋯, X_(N-1)] which is ...
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2answers
891 views

Sum of two absolute values in complex plane

I'm trying to find out all $z \in C$ that satisfy the following condition: $|z+1|+|z-i|=3$ I understand that $|z|=r$ represents a circle with a radius of $r$. I also understand that $|z+1|=r$ can ...
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1answer
160 views

An absolute value problem

Let $a$ and $b$ in $\mathbb{R}$ 1) Show that $||a|-|b||\leq|a+b|\leq|a|+|b|$. 2) Prove that the one or the other of the two inequalities is an equality. It's fine whit the 1st question but i can't ...
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4answers
2k views

Double module in a inequality

Can somebody explain to me (or link me a site which does) how to solve this? $$ ||x+1| -1| \geq 3 $$ I have no idea how to work out this double absolute value sign.
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2answers
871 views

Smoothing of absolute value and sign functions for numerical integration

I'm doing Numerical integration of ODEs. for a special system that has an always positive coordinate s and a conjugated momentum ...
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4answers
2k views

Is it possible to take the absolute value of both sides of an equation?

I have a problem that says: Suppose $3x^2+bx+7 > 0$ for every number $x$, Show that $|b|<2\sqrt21$. Since the quadratic is greater than 0, I assume that there are no real solutions since $y = ...
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2answers
272 views

Simplify $\ln(|x-x^2|) - \ln(|x-1|)$

As the topic says, I need to simplify: $$\ln |x-x^2| - \ln |x-1| $$ I don't know how to approach the problem at all. I'm not asking for the answer, but something to maybe get me going.
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3answers
211 views

absolute value inequality

I would like to know how to solve the inequality $$|x^2-y^2|\leq 2x+2y-4xy.$$ I have tried to solve it by myself and searched in the internet, but didn't come up with an answer. Thanks in advance. ...
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2answers
2k views

Simplifying expression with absolute value and unknown

This is probably a really easy task for the people of this site judging by what is normally discussed here. (I'm amazed by the knowledge here!) I have this expression that I need to simplify: $$\...
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2answers
129 views

Two solutions for $x$

I am trying to solve this. I already got the answer but my doubt is if I am doing it right. There are two solutions for $x$ in the equation, to 2 decimal places. What is the value of the greater ...
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4answers
481 views

Calculate absolute values with unknown constant

I am to calculate all $x$ if $f(x) = g(x)$ and if $$f(x)= |2x+2| + |3-2x|$$ $$g(x)= x + 14$$ How do I mix regular numbers with absolute values in such a sense? I thought I could calculate it like ...
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2answers
396 views

Spivak Calculus 3rd ed. $|a + b| \leq |a| + |b|$

I'm working through the first chapter of Michael Spivak's Calculus 3rd ed. Towards the end of the chapter he proves $ |a + b| ≤ |a| + |b| $ using the observation that $|a|= \sqrt{ a^2 }$ when $a$ ...
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1answer
136 views

Simplifying $|a+b|^2 + |a-b|^2$

I want to simplify $|a+b|^2 + |a-b|^2$ where $a, b \in \mathbb{C}$. I've used Wolfram Alpha to get $$ |a+b|^2 + |a-b|^2 = 2\left(|a|^2 + |b|^2\right) $$ I'm trying to understand the steps involved in ...
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3answers
2k views

Quadratic equation with absolute value

Prepping for the GMAT, I came across the following question: What is the product of all solutions of: $$x^2 - 4x + 6 = 3 - |x - 1|?$$ First, I set up two equations, ie: $$x^2 - 4x + 6 = 3 -...
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2answers
251 views

Why isn't this square root $+$ or $-$?

I was tasked with proving the identity $\tan(\frac x 2) = \dfrac {\sin(x)}{1+\cos(x)}$ I used the quotient identity for tangent and the half angle identities for sine and cosine to get $ \pm \dfrac {\...
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6answers
380 views

Evaluating $\int |x|^3 \; dx $

$$\int |x|^3 \; dx $$ In my module it is suggest to use integration by parts, $$ \text{ Set }I = \int (|x|^3 \cdot 1) \; dx = |x|^3 \cdot x - \int \color{red}{\frac {x^3}{|x|^3}3x^2}\cdot x \; dx$$ ...
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2answers
651 views

Expectation Values inside absolute value operator

first: are these equality true ? $$|E[Y]-E[X]|=|E[Y]|-|E[X]|.$$ $$|E[Y]-E[X]|^2=|E[Y]|^2-|E[X]|^2$$ second: what is result of this relation: $$\sum_{i=1}^{3}p_i.(X_i-\mu)^2=?$$ where the $\mu =\...
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4answers
393 views

the solution set of $\left | \frac{2x - 3}{2x + 3} \right |< 1$

what is the solution set of $\left | \frac{2x - 3}{2x + 3} \right |< 1$ ? I solved it by first assuming: $-1 < \frac{2x - 3}{2x + 3 } < 1$ ended with: $x > 0 > -3/2$ Is that a ...
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1answer
120 views

plotting the following set of points in the XY plane

Represent the following set of points in the XY plane : $$\{ ( x , y ) \; | \; |x| + |y| = 1 \}$$ What i got: 1) if $x > 0, y > 0 : x = 1 - y$ 2) if $x > 0, y < 0 : x = 1 + y$ 3) if $...
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3answers
2k views

Solve an absolute value equation simultaneously

My question is : Solve simultaneously $$\left\{\begin{align*}&|x-1|-|y-2|=1\\&y = 3-|x-1|\end{align*}\right.$$ What I did : $y=3 - |x-1|$ is given. Thus $y = 3-(x-1)$ or $y = 3-\left(-...
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2answers
466 views

Solving $|x-2| + |x-5|=3$ [duplicate]

Possible Duplicate: How could we solve $x$, in $|x+1|-|1-x|=2$? How should I solve: $|x-2| + |x-5|=3$ Please suggest a way that I could use in other problems of this genre too Any help to ...
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1answer
227 views

Absolute value of a real number

My question is: Solve: $|x-4|< a$, where $a$ belongs to the real numbers. Solve this by considering various cases depending upon whether $a$ is negative, positive or zero. What I have tried so ...
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2answers
426 views

Equality with absolute values - Is this a valid solution?

For this problem $|2 - |x-2|| = 2$ I've found the values $x = -2$ and $x = 2$. However, an third solution was presented to me, which I can't seem to find by myself: $x = -6$. Is this solution ...
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1answer
914 views

How to manage the absolute value on a differential equation $|T(x)'+A(T,x)+B(T,x)| = f(T,x)$

Hi everyone I need to solve an equation of this type: $|T(x)'+A(T,x)+B(T,x)| = f(T,x)$ with boundaries conditions. The absolute value is my problem. Of course without it, the solution of these is ...
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2answers
240 views

Math Database For Problem Descriptions In An App.

I am developing an app for kids and they will have a variety of problems from percentage problems, absolute value problems, negative number problems, fraction problems, etc. I was hoping to have a ...
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1answer
758 views

Finding the absolute extreme values for a multivariable function

Find the absolute extreme values taken by $f(x,y) = x^2 + 4y^2 + x - 2y$ on the closed region enclosed by the ellipse $1\over4$$x^2 + y^2 = 1$. I know this might be a basic question but could ...
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1answer
317 views

The relationship between the derivative of $f(x)$ and $|f(x)|$

I have seen it in an exercise book. I don't know how to do it. If $f(x)$ is continuous at $x=a,$ and $|f(x)|$ can be differentiated at $x=a,$ then $f(x)$ is differentiable at $x=a.$
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2answers
187 views

I can't find a absolute value function that have [-1,1] range

I want a function $f:\mathbb{R}\to[-1,1]$ with absolute value like $f(x)=|a-x|\ldots$ that have $[-1,1]$ range. Can anybody help me?
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5answers
225 views

The solution set of the equation $|2x - 3| = - (2x - 3)$

The solution set of the equation $\left | 2x-3 \right | = -(2x-3)$ is $A)$ {$0$ , $\frac{3}{2}$} $B)$ The empty set $C)$ (-$\infty$ , $\frac{3}{2}$] $D)$ [$\frac{3}{2}$, $\infty$ ) $E)$ All real ...
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1answer
122 views

When is possible to ignore an absolute value

$e^{\frac{2}{x-1}\log\left|x-1\right|}+1\neq 0$ Since that this is an exponential function, this equation is verified $\forall x \in \mathbb{R}$? Or I have to consider the absolute value of the ...
1
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2answers
334 views

How to set up the existence condition of an absolute value

$$ \frac{\sqrt{4 + \arccos\left|\frac{2-x}{x+3}\right|}}{\sqrt{x^2 - 4x + 5} - 3} $$ I'm trying to find the natural domain of the function above. I set up this conditions: $$ \begin{cases}\sqrt{x^2 ...