For questions about or involving the absolute value function.

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0
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1answer
76 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 ...
0
votes
2answers
110 views

solving absolute value equation 2

My question is : Solve simultaneously- $$\left\{\begin{align*}&|x-1|+|y-2|=1\\&y = 3-|x-1|\end{align*}\right.$$ I tried to solve this question by the method told by Marvis as I had ...
2
votes
3answers
331 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 = ...
0
votes
2answers
102 views

about solving: Absolute value

How to solve: $|\sqrt{x-1}-2| + |\sqrt{x-1}-3|=1$. I would like to know how to solve an absolute value equation when there is a square root sign inside.
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vote
2answers
130 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 ...
-1
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3answers
135 views

An inequality with absolute value and a parameter: $|x-4|>a$

Solve : $|x-4|>a$. Case 1: $a>0$; Case 2: $a<0$ Progress I am getting answers which look similar in both cases: Let $a>0$ so $x>4+a$ or $x<4-a$ , Let $a<0$ so ...
1
vote
1answer
171 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 ...
1
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2answers
179 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 ...
0
votes
1answer
386 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 ...
0
votes
2answers
86 views

Two Analysis Questions

1) Define : $\langle z\rangle := (1+|z|^2 ) ^\frac{1}{2} $ for all $z \in \mathbb{C} $. Prove : $\langle x+y\rangle \leq 2\langle x\rangle\langle y\rangle $ for all $x,y \in \mathbb{R} ^N$ . 2) ...
1
vote
1answer
439 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 ...
3
votes
6answers
334 views

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

$$\int |x|^3 \; dx $$ In my module it is suggest to use integration by parts, $$ \text{ Set }I = \int (|x|^3 \cdot 1) \; dx = |x|^3 \cdot x - \int \color{red}{\frac {x^3}{|x|^3}3x^2}\cdot x \; dx$$ ...
1
vote
1answer
283 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 $x=a.$
-1
votes
2answers
165 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?
2
votes
1answer
11k views

Reverse Triangle Inequality Proof

I've seen the full proof of the Triangle Inequality $|x+y|\le|x|+|y| $ However, I haven't seen the proof of the reverse triangle inequality: $||x|-|y||\le|x-y|$ Could you please prove this using ...
0
votes
2answers
207 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 ...
1
vote
2answers
341 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 ...
0
votes
5answers
183 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 ...
4
votes
1answer
25k views

Integral of an absolute value function

How do I find the definite integral of an absolute value function? For instance: $f(x) = |-2x^3 + 24x|$ from $x=1$ to $x=4$
0
votes
0answers
63 views

Name for $\max(x, \frac 1x)$, $x > 0$? [duplicate]

Possible Duplicate: Multiplicative Identity analog for absolute value In looking for the most extreme scaling, I'm comparing $f(a)$ and $f(b)$ where $f(x) = \max(x, \frac 1x)$, $x > 0$? ...
0
votes
1answer
79 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
vote
2answers
168 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 ...
1
vote
0answers
88 views

Fields with their own absolute value

Let $F\hspace{.02 in}$ be a field. $\:$ Let $E\hspace{.02 in}$ be a non-zero subring of $F$. Let $\hspace{.03 in}\leq\hspace{.03 in}$ be a total order on $E\hspace{.02 in}$ that makes $E\hspace{.02 ...
2
votes
1answer
528 views

How to manipulate absolute values when shifting parts in an inequality

I have the following inequality... $|4x - 2| \le 0.5$ I want to manipulate this so it is just $|x|$ on one side, and everything else on the other, but I'm not sure how the absolute value complicates ...
2
votes
2answers
501 views

How to minimize an equation with absolute values?

How would I go about minimizing the expression $\left(|z_1| + |z_2|\right) \times \left(|z_1 + z_2| + |z_1 - z_2|\right)$ subject to the constraint $|z_1|^2 + |z_2|^2 = 1$ given that $z_1$ and ...
-4
votes
1answer
448 views

Absolute Value theory [closed]

Could someone explain me the basics about absolute value? Not only how to solve it, but also the theory and the demonstration of theorems about it.
1
vote
6answers
393 views

How could we solve $x$, in $|x+1|-|1-x|=2$?

How could we solve $x$, in $|x+1|-|1-x|=2$? Please suggest a analytical way that I could use in other problems too like this $ |x+1|+|1-x|=2$ and of this genre. Thank you,
1
vote
1answer
211 views

How to solve equations with absolute value and using the Archimedean property

I'm trying to learn Real Analysis on my own, but I found that i'm a bit rusty with the elementary stuff. How do I solve equations like $|x| + |x+1| = 1$ and $|x-1| + |x+1| = 2$? I don't want the ...
0
votes
5answers
211 views

Solving an equation with absolute values

The equation I am trying to solve is this : $\newcommand\abs[1]{|#1|}\abs{3y+7}=\abs{2y-1}$. My conventional approach is to split this into three intervals with $1/2$ and $-7/3$ being the two "split" ...
0
votes
2answers
80 views

Domains and setbuilder notation

We're learning about domains and setbuilder notation in school at the moment, and I want to make sure what I did was right. My thought process: \begin{align*} -\frac12|4x - 8| - 1 &< -1 \\ ...
0
votes
1answer
777 views

derivative of absolute value of a complex function

If $f:U\subset\mathbb{C}\mapsto\mathbb{C}$, where $f(x+iy)=u(x,y)+iv(x,y)$ is a meromorphic function and if $f$, $f'$, and $f''$ are not zero in the strip $a<x<b$, can we get ...
4
votes
3answers
488 views

Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?

Claim: If f is integrable, $\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$ Proof (attempt): We know $-|f|\le f \le|f|$, so $\int-|f| \le \int f \le \int|f|$.* Since, if $-b<a<b$, we say ...
1
vote
1answer
6k views

Limit with absolute value

I found this limit within the Calculus Single Variable book from Thomas. $$ \lim _{x \to -2^-} (x+3) \frac{|x+2|}{(x+2)}$$ This is how I'm trying: First of all, we need to found where the absolute ...
5
votes
1answer
987 views

Prove variant of triangle inequality containing p-th power for 0 < p < 1

Sorry if this is a trivial question, but I am kind of stuck with proving the following inequality and have been searching for a while: $\rho \left( \sum\limits_i^n d_i \right) \leq \sum\limits_i^n ...
1
vote
1answer
555 views

Prove by contradiction or contrapositive? If $|x+y|<|x|+|y|$, then $x<0$ or $y<0$.

Prove: If $|x+y|<|x|+|y|$, then $x<0$ or $y<0$ This looks as though it's true from the start. Take $x=-4, y=4$. $|-4+4|<|-4|+|4|$ $0<8$ is true. The question is asking for a ...
3
votes
1answer
313 views

Why do definitions of distinct conic sections produce a single equation?

I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
2
votes
1answer
322 views

When should I put an absolute value sign around a function?

In Griffiths' introduction to QM book, I see $\int f^* f\ dx$ often written as $\int | f |^2 dx $, whereas $\int f^* g\ dx$ is understandably just that. Should I always write the absolute value sign ...
1
vote
2answers
417 views

Joint pdf of X and Y with absolute value range

I have the following joint pdf: $f(x,y)=0.5$ where $0 \leq|x|\leq|y|$, $0 \leq|y|\leq1$, and $0$ otherwise The question is: are $X$ and $Y$ independent and uncorrelated? I know that if ...
2
votes
2answers
85 views

How do I find this limit?

How would i find the limit as $\lim\limits_{x\to3}\frac{4x(x-3)}{|x-3|}$? that is the absolute value of x-3 in the denominator. I thought my professor told my class that we were able to omit the ...
2
votes
1answer
131 views

$|x-y| + |y-z| = |x-z|$ then $x \le y \le z$

I'm doing this exercise from Robert Bartle's Introduction to Analysis, it's a if only if excersise and I've done the half part, but I can't figure this part of the ...
3
votes
2answers
308 views

Why exactly can you take the absolute value of one side of this inequality and assume it is still true?

Exercise: Show that if $(b_n) \to b$, then the sequence of absolute values $\left| b_n \right|$ converges to $\left| b \right|$. Solution (partial): By the triangle inequality, ...
3
votes
1answer
165 views

Inequality with absolute value

I am unsure if have solved the following inequality correctly: $ \dfrac{2x+3}{x+5} \leq \dfrac{x+1}{|x-1|} \tag{1}$ I've proceeded as follows. If $x>1$ then $|x-1|=(x-1)$ If $x<1$ then ...
3
votes
1answer
666 views

Proving Absolute Value Inequality

I had posted a portion of this earlier asking about how to interpret min(). I received some excellent answers, however, I have run into problems and feel stuck. I am posting the question in its ...
4
votes
1answer
712 views

integral from 0 to $2\pi$ of $|\cos x|\operatorname{d}x$ not integrating as I'd expect

I drew a rough sketch of $|\cos x|$ and would guess the correct answer to this integral is $4$ because I know the area under the curve of $\cos x$ from $0$ to $\pi/2$ is $1$, and there are $4$ such ...
3
votes
3answers
117 views

Solving $ \left| \frac{-2x-6}{4} \right| \le 5$ for $x$

Say I have a statement like: $$ \left| \frac{-2x-6}{4} \right| \le 5. $$ And I want to find the closed interval form of $x$. i.e. I want to know what the maximum and minimum $x$ can be. How do I ...
2
votes
2answers
2k views

Square root of simple binomial function

Let's say I have the following formula: $$\sqrt{a^2-2ab+b^2}=\sqrt{(a-b)^2}=\sqrt{(b-a)^2}$$ When do I know which one of the following I should use?: $$\sqrt{(a-b)^2}=a-b\qquad\text{ or }\qquad ...
6
votes
2answers
2k views

How to use triangle inequality to establish Reverse triangle inequality

I need to use $|a+b| \leq |a|+|b|$ to show that $||a|-|b|| \leq |a-b|$. I have tried to represent $||a|-|b||$ as $||a|+(-|b|)|$, and then get $||a|+(-|b|)| \leq |a|+|-|b||$, but that isn't leading ...
2
votes
1answer
450 views

Rules applying to nested absolute values

I'm trying to use some algebra get $||x-5|-10|<\epsilon$ into a more manageable form (I'd like it in terms of $0<|x+5|<\delta$) but I'm not sure where to begin. I don't really know the rules ...
2
votes
3answers
671 views

Trouble with absolute value in limit proof

As usual, I'm having trouble, not with the calculus, but the algebra. I'm using Calculus, 9th ed. by Larson and Edwards, which is somewhat known for racing through examples with little explanation of ...
10
votes
5answers
461 views

Is -5 bigger than -1?

In everyday language people often mix up "less than" and "smaller than" and in most situations it doesn't matter but when dealing with negative numbers this can lead to confusion. I am a mathematics ...