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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

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1answer
25 views

Solution set for $ 0 < | x - c | < \delta$. Is $|x - c| < \delta$ equivalent to $0 \leq | x - c | < \delta$?

Let $x \in \mathbb{R}$. Let $c$ be a real number constant, and $\delta > 0$ and also a real number. Consider the following: $$ |x - c| < \delta \label{1}\tag{1}$$ $$0 \leq | x - c | < \...
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2answers
44 views

Proving an inequality involving absolute values

How can I prove the inequality $\left|x\right|+\left|y\right|+\left|z\right|\le\left|x+y-z\right|+\left|y+z-x\right|+\left|z+x-y\right|$ for all $x, y, z$ being real number. Can I prove this by ...
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1answer
33 views

Help $|nx|\le |n|\cdot|x|$, for $x\in K$ and $n \in \mathbb Z$?

How to prove: $|nx|\le |n|\cdot|x|$, for $x\in K$ and $n \in \mathbb Z$ ? The absolute value here is a nonnegative function from a field $K$ to $\mathbb R$ and in the definition there's a point; $|...
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1answer
47 views

Complex numbers problem with absolute value property

If a,b are complex numbers, k its an integer, $k \neq 0$ and $|a+k| + |b-k| + |a+b-k|=1$ then proof that $a,b$ are real numbers I've tried $a+k=x$ and $b-k=y$ Then I used absolute value ...
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1answer
64 views

What is the value of $|\sin(\cos\theta + i \sin\theta)| $ in complex analysis? [closed]

How would I compute the value to/simplify the following expression? $$\left|\sin\left( \cos\theta + i \sin\theta \vphantom{M^M} \right) \right| $$ Can I use the fact that $\cos\theta + i \sin\...
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4answers
61 views

Doubt on how to express $\sqrt{x}^2$ on derivative function… which is the right answer?

I was requested to find the derivative of $f(x)=\frac{x}{\sqrt{1-x^2}}$. When I was working on this, I found an expression of the form $\sqrt{1-x^2}^2$, which I translate to $|1-x^2|$. My calculus ...
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1answer
24 views

Inequality within complex

I step with a inequality and would like to know if it is truth... $||(a_1-a_2)^2+i(b_1-b_2)^2||\leq ||a_1^2+ib_1^2||+||a_2^2+ib_2^2||,\quad \forall a_1,a_2\in\mathbb{R}$. I tried to prove it but ...
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3answers
37 views

Some questions about proof for limit of sequence $a_n := 1 + \frac{1}{n}$ is 1

I'm trying to understand a proof for the limit of the sequence $a_n := 1 + \frac{1}{n}$ and $n \in \mathbb{N}$ which should be 1. So, the proof which I have here starts with: $$\left( \lim_{n \to \...
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2answers
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Is $y=|x^3| $a parabola?

I'm just curious. It seems to have the same shape and a similar form as parabolas such as $ x^2$ and $x^4 $. The odd exponent would normally give negative outputs for negative inputs, but the absolute ...
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2answers
88 views

Solve for $x\in\mathbb{R}$: $x^{2} + 2|x-3| - 10 \leq 0$

I went about taking cases for $x^{2} + 2|x-3|- 10\leq 0$. Taking $x-3$ and $-(x-3)$ as cases. Is it the correct approach? Taking cases I realise I get $x^{2} + 2x - 16 \leq 0$ and $x^{2}-2x-4 \leq 0$...
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0answers
44 views

How to solve an ODE with an absolute value?

I am a bit confused about how to solve an ODE of this type $y'= \mid x \mid - \mid y \mid $ , $y(0)=0$ Do I need to divide the ODE into four phases : $x>0$ & $y>0$ $x>0$ & $y<...
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2answers
26 views

What is $P\left(|\bar{x}-\mu|\leq 1.5\frac{0.2}{\sqrt{16}}\right)$ equal to?

I'm having a question regarding normalizing the distribution in 2-tailed test. I have $\mu=3.3$, $\sigma = 0.2$ and $n=16$. I need to determine $$ P\left(|\bar{x}-\mu|\leq 1.5\frac{0.2}{\sqrt{16}}\...
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1answer
20 views

How do you prove a natural logarithm with complex numbers equal to a natural logarithm with an absolute value?

The author stated that if $ln$ $z=a+ib$ then $ln |z|=a$ Can someone show me a proof of this? I have been looking and can not find one to see if its true What I do see is that if $z=a+ib$ then the $a$ ...
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2answers
166 views

If $h$ is twice differentiable, then $|h|$ is twice differentiable except on a countable set

Let $h:\mathbb R\to\mathbb R$ be differentiable. It can be shown that $$N:=\left\{a\in\mathbb R:h(a)=0\text{ and }h'(a)\ne0\right\}$$ is countable and $|h|$ is differentiable on $\mathbb R\setminus N$ ...
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0answers
28 views

Integral estimate for Hankle´s Contour

I have to proof the following estimate $\vert \int\limits_{H_k}z^{s-1}(e^{z}-1)^{-1}dz\vert \leq k^{\sigma}$ Where $H_k$ ist the Hankel Contour with radius $\rho_k = (2k+1) \pi$ From another ...
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2answers
71 views

|x²-2x| + |x-4| > |x²-3x+4| , How do I solve for all real x?

How do I solve this for all real x? |x²-2x| + |x-4| > |x²-3x+4| Looking at the question it is clear that it states |a| + |b| > |a-b|. How to proceed?
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5answers
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Solve the inequality : $ ||x|-1|<|1-x| $

Solve the following inequality: $$||x|-1|<|1-x|$$ My Attempt: I tried expanding this inequality by considering $8$ cases, but I am having trouble finding the range of the each of the solutions I ...
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5answers
782 views

Why isn't the definition of absolute value applied when squaring a radical containing a variable?

I recently learned about the following definition of absolute value: $|a| = \sqrt{a^2}$ Then I came across a solution to a problem that had the following step: $5 \geq \sqrt{5 - x}$ In order ...
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2answers
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Dealing with limits that contain absolute values

I've been asked to find the directional derivative of $f(x,y) = \|(x,y)\|$ at the point (0,0) in the direction of $v=(a,b)$: $$\lim_{t\to0} \frac{\|(0,0) + t(a,b)\| - 0}{t} = \lim_{t\to0} \frac{|t|\...
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2answers
58 views

Simplifying $\cos(2\arcsin(x))$ using only pythagorean trigonometric identity

I know that one can simplify $\cos(2\arcsin(x))$ using $\cos(a+b)=\cos(a)\cdot\cos(b)-\sin(a)\cdot\sin(b)$: \begin{alignat}{1} \cos(2\arcsin(x))&=\cos^2(\arcsin(x))-\sin^2(\arcsin(x)) \\&=1-2\...
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3answers
42 views

Absolute Value Rational Inequalities Help Please

I have been read plenty of questions on questions like this, but I still dont quite get it. For example, this question: $$ \left| \frac{2x+1}{x-3} \right| \ge 2 $$ How would I go about solving this?...
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1answer
20 views

Real Analysis introductory absolute value proof

let $x \in R$ Prove that $\vert x\vert \leq 2$ implies $\vert x^2 -4 \vert \leq 4 \vert x-2 \vert$ Here is my work: $\vert x-2 \vert \vert x + 2 \vert \leq 4 \vert x-2 \vert$ By the first ...
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1answer
42 views

Triangle Inequality and absolute value

I'm curious if the triangle inequality (and reverse triangle inequality) still hold if we only take the absolute value of one term. For example: $$||a| - b| \le |a - b|$$ If $b \ge 0$, then $|b|$ is ...
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0answers
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Integrating $\left|f(x)\right|$ by pulling out $\mathrm{sgn}(f(x))$ from the integral

I tried doing the following integral: $\int_{0}^{\pi/4}\sqrt{1-\sin2x}\mathrm dx$. Firstly I completed the square by rewriting $1$ as $\sin^2x+\cos^2x$ to get the integral revised to this form: $$I=\...
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1answer
28 views

Graphing $f(x)=\left|x^2-2x \right|-x$.

I'm dealing with absolute value inequalities. I've realized it would be so much easier to confirm the solution sets I'm obtaining with a curve of the function. Currently I'm doing $\left|x^2-2x\...
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2answers
45 views

Proving an inequality involving absolute value; how do I justify using a conjunction (and) instead of a disjunction (or)?

I'm putting together the following the proof, and I have a question about one of the final steps. Definition of absolute value: $\forall x \in \mathbb{R}, (x \geq 0 \Rightarrow |x| = x) \...
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0answers
36 views

Uncoditional formula for unsigned area under a linesegment w.r.t. $y=0$

I'm interested in finding the unsigned area under a line segment with respect to $y=0$. The line segment is defined by start point $(s_x, s_y)$ and end point $(e_x, e_y)$ Without loss of generality, ...
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1answer
26 views

Linear equation with absolute value. [closed]

Find $x$ if $|x-|3x+1||=4$ I got 4 values of $x$ out of which 2 are obsolete... Why so??
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2answers
38 views

How to plot absolute value graphs?

I have to plot graph of $$f(x)=|x|+2|x-1|+|x-4|$$ See I know graphs of individual $|x|,2|x-1|,|x-4|$ But how can I draw their sum. I have to find minimum value of the sum using graph.
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3answers
32 views

Inequality involving absolute values.

I want to ask is whether there is a method to solve following inequality more easily and compactly or it is the only method. $$|x-2|+|x-8|\le x-2$$ What I know is taking $x<2,8>x>2,x>...
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2answers
29 views

Log Functions Inside Absolute Value

Is the function below always positive for $0< x <1$? (I am determining if the function requires the modulus sign or not.) $$\frac{1}{2}\log\left|\frac{1+\log(x)}{1-\log(x)}\right|$$ My first ...
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0answers
32 views

Rephrasing what this proof is asking

I am new to proofs and am still struggling to parse them. I am not looking for a proof to the following statement; just guidance as to where to start or what the shape of a proof for it looks like. ...
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0answers
54 views

Absolute Value of a Integral, separation

This is a homework exercise, and ''Thomas early trascendentals'' book vol.4 says in property that : Def: To integrate an absolute value function, we have to look for specific cases when ; $ v(t)\...
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3answers
64 views

How to solve $|a+b|+|a-b|=c$?

It is intuitive that $a=\pm \frac{c}{2}$, with $-\frac{c}{2}\leq b\leq \frac{c}{2}$ or vice-versa are solutions to the problem. Can I get to these solutions without dividing the expression in all the ...
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5answers
56 views

Sum of two absolute values equal to a whole number

The following is the equation: $|x+1|+|x+2|=3$ How can I solve this problem? Do I have to reformat it to $|x+1|=3-|x-2|$? I would like a simple answer that by no means uses set theory. The answer ...
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1answer
14 views

Redefinition of a Constant Leading to Nullification of Absolute Value

I am currently taking a calculus 3 class in college, and my teacher did an intriguing problem about a tsunami in class which took up about 4 full white boards. Anyways, at a certain point in the ...
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1answer
39 views

solving equation involving absolute values [closed]

Given the equation $|x+2| + |3x+6| = 8,$ how can I find the sum of all its roots?
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2answers
79 views

How can I solve this absolute value equation?

This is the equation: $|\sqrt{x-1} - 2| + |\sqrt{x-1} - 3| = 1$ Any help would be appreciated. Thanks!
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1answer
22 views

Find the solution of $1 \le \left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +x\right\rfloor \lt 2$

Find the solution of $$1 \le \left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +x\right\rfloor \lt 2$$ My try: The only thing i know is that $$\left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +...
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1answer
20 views

Solve for $x$: $3+|x-9|<\frac{2|x-1|}{x}$

I moved one part of the inequality to the other to create the following: $\frac{2|x-1|}{x}-3-|x-9|>0$ Eventually, I get to a case where I have 2 inequalities after opening 1 of the absolute ...
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4answers
54 views

If $|x|>|x-y|$ prove that $xy>0$

So the title says it all. I think using the square formula, not sure if that is legit. Thank you for your attention.
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1answer
48 views

Taking the absolute value in inequalities.

If I've a expression: $-4<3$ and take the absolute value $|-4|<|3|\implies 4<3$ which is false. So I though that maybe the inequality sign would change. But $|-2|<|3| \implies 2>3$ ...
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1answer
38 views

Sketch the region in the plane consisting of all points $(x,y)$ such that $|x-y|+|x|-|y| \leq 2$

Sketch the region in the plane consisting of all points $(x,y)$ such that $|x-y|+|x|-|y| \leq 2$ I could consider the eight parts the plane gets divided into by the $x$-axis, $y$-axis, $y=x$ and $y=-...
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2answers
29 views

Do the $|$ around $|\langle u,v\rangle|$ refer to absolute value in the inner product version of the Cauchy-Schwarz inequality?

The full inequality is: $|\langle u,v\rangle| \leq ||u|| ||v||$ I understand that $||$ around the vectors $u$ and $v$ signifies the taking of their norm, but what do the single | around $\langle ...
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1answer
24 views

Minimization of multiple absolute sums

I know that on fist glance this seems as already explained problem on many internet places, but I haven't found solution anywhere. Anyway, here is my function: $$F=\sum_{i=0}^N |S_i|$$ where $$S_i=\...
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2answers
55 views

Number of solutions of an equality with absolute value operator

Consider: $$ \left|\left|\left|x-1\right|-2\right|-4\right|=4 $$ What is the number of solutions for this equation? This one was particularly easy to me. If first observed that if this inequality ...
3
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4answers
82 views

Suppose $|z|\ge 2$, Prove $|z^8+135|\ge121$.

Suppose $|z|\ge 2$, Prove $|z^8+135|\ge121$. My work: $|z^8+135|=\sqrt{(z^8+135)(\bar{z}^8+135)}=\sqrt{|z|^{16}+135(z^8+\bar{z}^8)+135^2}\ge\sqrt{2^{16}+135(z^8+\bar{z}^8)+135}$ For the last term, ...
0
votes
1answer
18 views

Continuity of product in absolute value

Let $F$ be a field, and $\| \cdot\|$ be an absolute value (or norm). I want to prove that with this norm, multiplication is continuous map on $F$ in the sense: for $x_0,y_0\in F$, and any $\...
1
vote
3answers
44 views

How can $|n_1 - n_2| + |n_3 - n_4|$ be equal two different formulas

I have a formula to be calculated such as; $|n_1 - n_2| + |n_3 - n_4|$. I want to calculate it with two tuples of values such as; $(n_1,n_2)$ and $(n_3,n_4)$ When I try this method, it is equivalent;...
1
vote
2answers
48 views

Prove using the triangular inequality that: $|a+b| \geq |a| - |b|$

How can I prove using the triangular inequality that: $$|a+b| \geq |a| - |b|$$ I already proved it by considering all 8 possible scenarios (like a>b and b=0 ... etc) However I couldn’t manage to find ...