For questions about or involving the absolute value function.

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1answer
80 views

Linear Programming : Alternative to summation of absolutes in constraints

I am solving a placement problem, i.e. map $integers\ i\ from\ 0\ to\ 6$ to $(x_i,y_i)\ st\ 1 \le x_i,y_i\le 3$ such that : $ \sum\limits_{i=0}^6 \sum\limits_{j=0}^6 Cost(i,j)*(|x_i - x_j | + | y_i ...
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1answer
32 views

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $?

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $ ? WTS $x \ge |a| \rightarrow x \ge a \land x \ge -a $     Since $|a| > -a$ then we have $x \ge -a$ ...
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1answer
27 views

Absolutet Value Inequality with cases number line

I was wondering if anyone knows how to solve $|ax+b|<cx+d$ type questions by using cases and the number line to finish. I am personally struggling with the number line, I have half-finished a ...
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3answers
262 views

Limit of the absolute value of a function

Say I have a real valued function $f(x)$, is it true that if $\lim_{x \rightarrow c} |f(x)| = 0$ then $\lim_{x \rightarrow c} f(x) = 0$?, where $c$ can be a real number or $\pm \infty$.
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4answers
357 views

Inequality for absolute values

How do you show either of the equivalent inequalities: $$2(|a|+|b|+|c|)\leq |a+b+c|+|a+b-c|+|a-b+c|+|a-b-c|$$ or $$|x+y|+|x+z|+|y+z|\leq |x|+|y|+|z|+|x+y+z|$$ Hold for complex numbers or in $n$ ...
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0answers
16 views

Cumulative distribution function of a model similar to the multinominal distribution

I would like to solve a problem similar to the multinominal distribution (http://en.wikipedia.org/wiki/Multinomial_distribution): For k independent trials each of which leads to a success for ...
3
votes
2answers
31 views

$5-3|x-6|\leq 3x -7$

I have this inequation: $$5-3|x-6|\leq 3x -7$$ i solved this this way: i said, for $x\geq6$ is the modulus positive, so I made 2 cases in which the modulus gives + or - : 1) for $x\geq6$ ...
1
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1answer
63 views

Quadratic inequality with absolute values

I've decided to study calculus on my own, and I've started working on "A First Course in Calculus" by Serge Lang, 5th edition. Now I'm just reading the chapter on preliminaries, and there is a section ...
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2answers
40 views

Need help with this absolute value equation

I need to solve the following equation involving absolute value: $$|x-1| = 1-x$$ Looking at the term $x-1$, I thought I'd divide the interval into parts: $x < 1$ and $x \geq 1$. Now, when ...
2
votes
2answers
330 views

Why can't absolute values be expressed with negative numbers. [closed]

The answer to this question seems obvious. 'An absolute value expresses the quantity of ones between any number and 0'. But does that mean it must be positive? I took a shot at answering my ...
0
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1answer
36 views

Finding best fitted value for power function. please help!

I need to find: 1. the best fitted value for $a$ in the power function 2. the best fitted value for $b$ in the power function Data given: I know that $b=bi$ and $a=e^{bo}$ --> my question is how ...
0
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3answers
27 views

What's the best method to graph the following function by hand…

Here in my exercise I have to study the function and draw its graph. Can you please tell me what's the best method to do this, because I don't think that's reasonable to use the input output method, ...
2
votes
2answers
101 views

Basic question about solving modulus equation

It common in the literature to solve the modulus equation like $|x+5|+|x-1|=8$ by dividing into cases when $x<-5$, $-5\leq x<1$ and $x\geq1$. My question is whether dividing into cases is ...
0
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3answers
125 views

Absolute values don't work

I don't understand, how absolute valued could possibly be considered well defined. As shown here, $|a| = |-a| , ||a|| = |-|a||$ So lets take $a=-2, |a| = -2 = |-a|,$ but $|-a| = |2| = 2$ But it ...
1
vote
2answers
534 views

Suppose f(x) is an odd function. Prove that g(x) = |f(x)| is an even function.

Suppose f(x) is an odd function. Prove that g(x) = |f(x)| is an even function. I understand that an odd function is where f(-x) = -f(x), and an even function is where f(-x) = f(x), but am struggling ...
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0answers
74 views

Absolute values nested multiple times

Is there any algorithm to quickly determine "zero points" (i.e. points with undefined derivation) of absolute values functions which are nested multiple times? I do know, that any part of this ...
3
votes
2answers
55 views

no. of positive integral solutions of ||x - 1| - 2| + x = 3

What are the no. of positive integral solutions of ||x - 1| - 2| + x = 3 ? My effort ||x - 1| - 2| = 3 - x |x - 1| - 2 = 3 - x OR |x - 1| - 2 = x - 3 |x - 1| = 5 - x OR |x - 1| = x ...
0
votes
1answer
36 views

Absolute value being an odd function

Correct me if I am wrong, but I learned that for a function to be symmetrical to the origin, it can be rotated 180 degrees and still appear the same. How is ${x^2 - y^2 = 0}$ an odd function if when ...
0
votes
1answer
2k views

Integral of The Absolute Value of x

$$ \int_{1}^4|x|dx $$ I know how to take the integral of a more complex function (like f(x)= |x+2|) but I don't understand what to do if it's just the absolute value of x. If the lowest number is 1 ...
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5answers
70 views

Definition of absolute value

If ${f(x) = \sqrt{x^2}}$, then f(x) can also be expressed as: C. ${|x|}$ D. $ \pm x$ I thought the answer was D, but it's C. Couldn't it be both?
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1answer
110 views

Absolute extrema of a multivariable function bounded by an ellipse

I have a function $f(x,y) = 2x + x^2 + y^2$ bounded by the ellipse $x^2 + 4y^2 \leq 24$ I know how to determine the extrema within the ellipse by getting the partial derivatives and setting them to ...
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4answers
45 views

Find the minimum value of this expression with absolute values

The expression is $$|x-3| + |x-1| + |x| + |x+2| + |x+4|$$ I know that the minimum values for this expression is when x = 0 but is there any algebraic way to find this out? I did it on the ...
0
votes
1answer
23 views

Find the value of parameter $m$ such that the equation has real solutions…

For which values of real parameter "m" the equation:$$\sqrt3*|\tan x+\cot x|=4m$$ has real solutions? My only thought is that $m\gt 0$ because the right part of the equation is an absolute value which ...
2
votes
1answer
37 views

Find how many solutions has the following equation…

Determine how many real solutions has the following equation: $$x^2(|x|-6)=-15$$ I noticed that $|x|-6$ should be negative because $x^2$ is always a positive value. Thus, $x\in(-6;6)$. I made a ...
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2answers
138 views

Prove that $|x+y| \leq |x|+|y|$ [duplicate]

How to Prove the triangle inequality which says for all x (no matter how big or small) and for all y (no matter its size) in the set of irrational+rational numbers, this holds: $|x+y| \leq |x|+|y|$
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1answer
35 views

Solve the equation given below…

I have such an exercise: $$\color{teal}{{|x|\over{x}}\sin^2x-\cos|x|\cos x=1} $$ What I did is so: If $x\ge 0$ then we have: $$\sin^2x-\cos^2x=1$$ $$\sin^2x=1$$ So: $$\sin x=1$$ or $$\sin ...
2
votes
1answer
61 views

Proving something about $|f(x)|$ when the lim of $f(x)/x^2$ is known

I've been trying to crack this issue for 2 days and I got pretty much nothing Given that $f$ is a continuous function and the following limits exists and are finite: $$ (1) ...
3
votes
4answers
110 views

Solve $z^2 - iz = |z - i|$

I have the equation: $z^2 - iz = |z - i|$ The solutions are $i$, $-\sqrt3/2 + i/2$, $\sqrt3/2 + i/2$ Can someone please walk me through or give me a hint...
0
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1answer
16 views

Trying to differentte $\ln(|2+f(x)|)=2+e^{x*x}$

I am trying to solve this differential $\ln(|2+f(x)|)=2+e^{x*x}$ so far I did this much; $$ \ln(|2+f(x)|)=2+e^{x*x}\\ |2+f(x)|=e^{2+e^{x*x}}\\ \text{now I have two situations/solutions, because of ...
0
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2answers
217 views

Proving uniform continuity of absolute value

Prove that the function $f(x) = |x-a| - |x-b|$ is uniformly continuous on $\mathbb{R}$.
0
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1answer
41 views

Modulus Inequality

Solve the inequality $$2|x-3| > |3x+1|$$ Is sketching the only way I can solve ALL modulus equations and inequalities? Does an algebraic technique work for all modulus equations and inequalities?
2
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1answer
56 views

Solve $2\sqrt{(x-1)(x+2)}\ge|x+1|-2$

Can you please show me how can I solve this inequality. I would like to see how it can be done without the graph of the functions. Thank you! $$2\sqrt{(x-1)(x+2)}\ge|x+1|-2$$
0
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1answer
48 views

Solve the following inequality…

Can you please verify if I've done this exercise correctly, and if you have a better solution, please, show it to me. Thank you! (The exercise is in the left top corner.)
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1answer
277 views

Triangle inequality frobenius norm

I'm trying to show that the frobenius norm is a norm. however it appears as if triangle inequality isnt met. $$||A+B||_F = \sqrt{\sum_{i,j=1}^n |a_{ij}+b_{ij}|^2} \leq \sqrt{\sum_{i,j=1}^n ...
0
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1answer
75 views

Equivalent form not using absolute values

Looking at the solution of Trench´s Introduction to Real Analysis exercises, I am struggling with this: Write the following expression in equivalent form not involving absolute values: $a + b + 2c + ...
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0answers
53 views

Simple proof explanation - Possibly triangle inequality involved

I'd like some help with understanding the following statements...I saw it on the internet while searching for a proof, and I'd like to understand why its true: let $A$ be a diagonally dominant matrix ...
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votes
1answer
26 views

Help determining if an equation is a function of x

Graph: ${y\over|y|}={x\over|x|}$ ${\lfloor x \rfloor \lfloor y \rfloor = 1}$ Determine if each graph represents a function of x and explain your answer. I've never seen anything like the before ...
0
votes
1answer
80 views

Periodic absolute value function

Define $$h(x)=|x|$$ on the interval $[-1,1]$ and extend the defintion of $h$ to all of $\mathbb{R}$ by requiring that $h(x+2)=h(x)$. Now define the function: $$h_n (x)=\frac{1}{2^n} h(2^n x)$$ ...
2
votes
2answers
157 views

Solving inequalities with absolute values

This is the question: $$ \left| \frac{x+2}{3(x-1)} \right| \leq \frac{2}{3} $$ And this is my working out, first I squared both the numerator and denominator, then solved it as if it was a normal ...
0
votes
0answers
65 views

Looking for a counter example: limit of absolute value of $f(x)$

Consider the following: $$\lim_{x\rightarrow a}f(x)=L\Rightarrow \lim_{x\rightarrow a}|f(x)|=|L|$$ I proved it using the "second triangle inequality", but I tried to think why is the reversed ...
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5answers
98 views

Why is $f(x)=|x|$ not differentiable?

Consider the function $f(x)=|x|$, I know that $f$ is not differentiable at $x=0$, but still, when you try to differentiate $f(x)=\sqrt{x^2}$ (which is exactly the same), you get: ...
0
votes
1answer
125 views

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 ...
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2answers
38 views

I need help on this differential equaion problem?

Let equation $(1)$ be $\overrightarrow{F}= m \cdot \overrightarrow{a}$ and equation $(2)$ be $\overrightarrow{F}= \frac{-G \cdot M \cdot m}{ | \overrightarrow{r^2} |} \frac{\overrightarrow{r} }{ ...
0
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1answer
126 views

Derivatives of Norms and Absolute Values (distributions)

For example we have for $x \in \mathbb{R}$, $$\frac{\partial}{\partial x}\left| x\right| = 2\Theta(x) -1 $$ and thus $$\frac{\partial^2}{\partial x^2}\left| x\right| = 2\delta(x) $$ We also have, ...
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1answer
71 views

How to take derivative of sums of absolute values

Take the derivative of $f(m) = \sum_i | x_i - m |$. I've been told that derivative of each term is +1 or -1. How do you show that?
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vote
4answers
179 views

Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$

Let $a,b,c$ real numbers. Prove the inequality $|a-b| \leq |a-c| + |c-b|$. Prove that equality holds if and only if $a \leq c \leq b$ or $b \leq c \leq a$. I've tried starting with just $a \leq ...
2
votes
1answer
28 views

Real parameter equation

I'm having a problem with this question: For which values of the real parameter a the equation: $$||x|-1|=a$$ has exactly 4 solutions? The solution is this: $$0 < a < 1$$ What I tried was ...
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4answers
81 views

A function where absolute maximum is also absolute minimum?

What is an example of a real-valued function where an absolute maximum is also an absolute minimum?
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0answers
32 views

Formula to convert value to absolute value

This is probably a 'dumb' question (it's a while since I studied maths) but is there a way to convert a value to an absolute value using only the +,-,x,/ symbols? I'm pretty certain that the only way ...
2
votes
1answer
56 views

$|x|^{|x|}$ is continuous in $\mathbb{R}$

I'm trying to show this now my self, but still no go. There isn't really a concrete attempt that I can show.. Help?