For questions about or involving the absolute value function.

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0answers
51 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 ...
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1answer
33 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 ...
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1answer
30 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 ...
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1answer
28 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
61 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
49 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
41 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 ...
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1answer
17 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
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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
44 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
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1answer
59 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
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4answers
86 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...
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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 ...
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2answers
70 views

Proving uniform continuity of absolute value

Prove that the function $f(x) = |x-a| - |x-b|$ is uniformly continuous on $\mathbb{R}$.
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1answer
29 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?
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1answer
46 views

Solve the inequality…

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! $$2\sqrt{(x-1)(x+2)}\ge|x+1|-2$$
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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
50 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
42 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
40 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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1answer
23 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 ...
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1answer
47 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
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2answers
100 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 ...
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0answers
39 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
94 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: ...
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1answer
63 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
34 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} }{ ...
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1answer
81 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
37 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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4answers
120 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 ...
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1answer
25 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
56 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
27 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
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1answer
54 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
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7answers
129 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 ...
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1answer
47 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.
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0answers
41 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 - $$ ...
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1answer
21 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
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1answer
33 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 ...
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2answers
21 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 ...
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1answer
34 views

How to take the laplace of $e^{-|t|}$

I seem to be having some trouble trying to compute the laplace transform of this function. I looked on Wolfram and it said the answer was simply $$\dfrac{1}{s+1}$$ but I highly doubt that is the ...
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1answer
48 views

Finding the solutions of nonlinear system with absolute value

I need to show that the initial value problem: $\dot{x}=|x|^{1/2}$ $x(0)=0$ has 4 different solutions through the point (0,0). The problem also says that I have to sketch the solutions in the ...
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2answers
59 views

Proving absolute value inequality by contradiction

Prove that for $|x|, |y|, |z| \geq 2$ the following holds: $|x^2 + y| + |y^2 + z| + |z^2 + x| \geq |x| + |y| + |z|$ So I thought about a simple proof by contradiction but am not sure whether it's a ...
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1answer
44 views

Explanation for |x|=-x if x<0

Can someone explain $|x|=-x$ if $x<0$ . I've proven various theorems in my real analysis text for homework, but I cannot see how $|x|=-x$ if $x<0$ makes sense.
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3answers
103 views

Proof for absolute value inequality of three variables: $|x-z| \leq |x-y|+|y-z|$ [duplicate]

$|x-z| \leq |x-y|+|y-z|$ We know that both LHS and RHS are non negatives. So, I thought of proving this by comparing the squares of both sides but can't advance beyond that step. Any help would be ...
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1answer
43 views

How do I expand absolute values?

If we have this expression: $$f = uu-\left( u + \frac{\partial u}{\partial x} \delta x \right) \left( u + \frac{\partial u}{\partial x} \delta x \right)$$ we can expand it to this: $$f = u^2-\left( ...
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1answer
176 views

Logarithmic inequalities

Full disclosure: This is a homework problem, but my question is regarding a concept that came about during solving the problem, not the actual solution to the problem. Problem: Rewrite as geometric ...
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2answers
45 views

How to integrate absolute function

I have this absolute e-function, but I don't know how to calculate the integration $$ \int_{-2}^{2} e^{\frac{1}{2}j\omega |x|}dx $$ Any idea?
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0answers
34 views

Inequality with false solutions. Why? [duplicate]

When you have a question like $|x| = 3x – 2$, why do false solutions occur? if $x>0$, $x = 3x- 2$ $-2x = -2$ $x = 1$ If $x<0$, $x = -3x + 2$ $4x = 2$ $x = 1/2$ The $1/2$ solution is ...