For questions about or involving the absolute value function.

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7
votes
4answers
388 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 $n$ ...
1
vote
0answers
17 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
vote
1answer
91 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 ...
0
votes
2answers
47 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
623 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
votes
1answer
39 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
votes
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
183 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
votes
3answers
139 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
947 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 ...
1
vote
0answers
89 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
59 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
43 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
5k 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 ...
1
vote
5answers
73 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?
1
vote
1answer
210 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 ...
1
vote
4answers
71 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
30 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
44 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 ...
1
vote
2answers
216 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: $|x+y| \leq |x|+|y|$
1
vote
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
158 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
votes
1answer
17 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 ...
-1
votes
2answers
327 views

Proving uniform continuity of absolute value

Prove that the function $f(x) = |x-a| - |x-b|$ is uniformly continuous on $\mathbb{R}$.
0
votes
1answer
70 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
votes
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
votes
1answer
50 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.)
1
vote
1answer
427 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
votes
1answer
132 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 + ...
0
votes
0answers
71 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 ...
-1
votes
1answer
28 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
117 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
205 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
5answers
114 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
224 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 ...
1
vote
2answers
41 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
votes
1answer
175 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, ...
1
vote
1answer
95 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?
1
vote
4answers
224 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
29 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 ...
-1
votes
4answers
103 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?
2
votes
1answer
60 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?
2
votes
6answers
140 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
0
votes
1answer
49 views

How do I prove that $|x+y| \ge \big||x|-|y|\big|$?

I don't know where to start with proving this. Any help will be greatly appreciated.
2
votes
0answers
114 views

Fourier Transform of inverse powers of the absolute value

I don't think this question has been asked previously, so here goes. I need to evaluate the following integrals - $$ ...
0
votes
1answer
31 views

finding an absolute value inequality

The question asks, "find an absolute value inequality whose solution's are x>2 and x<-12". I have no idea where to start and was wondering if anyone could help
1
vote
1answer
58 views

Rearranging absolute values (limit proof)

My textbook ends a proof with the following: $|x-9| \over \sqrt(x) + 3$ < $\epsilon$ can be rearranged to conclude: |$x-9 \over \sqrt(x) -3$ - 6| < $\epsilon$ However, I don't understand ...
0
votes
2answers
24 views

Determine a value $t_0$ such that $|u(x,t)| = |-4e^{-2t/5}\cos2x\;| < 0.0001$

My problem is the following: Determine a value $t_0$ such that $|u(x,t)| = |-4e^{-2t/5}\cos2x\;| < 0.0001,$ for $t > t_0$, with $0<x<\pi$. How to approach this problem? According to my ...