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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

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Absolute Value Problem Using Only Variables

I recently encountered this problem, and it does not make sense to me. It looks like Given $|a(b-cx)|=d$ , find the value of $|x-\frac{b}{c}|$ This was on a multiple choice test awhile ago, and I ...
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4answers
52 views

For what range of a-values does $||x|-2|=a$ have only 2 possible answers? (without drawing a graph)

How can i solve "For what range of a-values does the equation $||x|-2|=a$ have only 2 possible answers?" Without using graphs by using a graph I know the answer is a>2 or {0} as in the picture :
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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
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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
50 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
65 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
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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
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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
70 views

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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1answer
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Understanding the absolute value of a matrix.

I believe that the absolute value of a matrix is defined as $$ |A|=\sqrt{A^{\dagger}A} \ . $$ But the square root of a matrix is not unique wikipedia gives a list of examples to illustrate this. To ...
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2answers
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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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6answers
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Help me understand $y=f(x)$ vs. $y=f(|x|)$ intuitively

I am writing this because I want to fill the hole in my understanding of elementary functions in Euclidean plane. In class, we discussed parallel examples, such as $y = f(x)$ vs. $y = |f(x)|$. ...
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1answer
111 views

Inequation with absolute values

How would one proceed in solving this difficult inequation with multiple absolute values? is there a way one should proceed ? $$\frac{x}{||x|-2|} \le \frac{x-1}{|x-3|}$$ thanks
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0answers
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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
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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2answers
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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
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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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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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0answers
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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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2answers
2k views

Intersection of a point and absolute value function contained within a circle

I'm attempting some crazy ideas while programming a game and ran into the following math problem that has been bugging me for a few days: Given a unit circle and a random point $P$ within the circle, ...
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5answers
784 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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0answers
55 views

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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9answers
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How is the triangle inequality used to derive this inequality?

I am trying to prove that any non-constant complex polynomial tends to infinity as z goes to infinity. Someone asked this question on this website here. In that question, the following hint is given: $...
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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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7answers
203 views

Why does $\lvert x^2 \rvert < 16$ imply $\lvert x \rvert < 4$?

Suppose I have something like $\lvert x^2 \rvert < 16$. The properties of absolute value state that $\lvert x^2 \rvert = \lvert x \rvert^2$. While this makes sense, I'm having trouble understanding ...
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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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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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1answer
21 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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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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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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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
3k views

Modulus of Two Complex Numbers, Squared

I have a very silly question to ask! I have $|z_{1} + z_{2}|^2 = |z_{1}|^2+|z_{2}|^2+2|z_{1}||z_{2}|\cos{\theta}$, where $z_{1}$ and $z_{2}$ are complex numbers. For the life of me I cannot ...
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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
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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
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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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3answers
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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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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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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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2answers
711 views

How to set up equation to find the unknown values if a limit exist.

For what values of the constants $a$ and $b$ does the following limit exist? $$\lim_{x\to0}\frac{|x+3|(\sqrt{ax+b}-2)}x$$ for this question, $$f(x) = \frac{-(x+3)(\sqrt{ax+b}-2)}x,x<-3$$ $$f(x) =...
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8answers
42k views

The Median Minimizes the Sum of Absolute Deviations (The $ {L}_{1} $ Norm)

Suppose we have a set $S$ of real numbers. Show that $$\sum_{s\in S}|s-x| $$ is minimal if $x$ is equal to the median. This is a sample exam question of one of the exams that I need to take and ...
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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 ...
7
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1answer
134 views

bound for $a,b,c$ in $|ax^2+bx+c| \leq 1\;\forall x\in \left[0,1\right]$

Consider $a,b,c\in\mathbb{R}$ such that $|ax^2+bx+c|\leq 1\;\forall x\in \left[0,1\right]$. Prove that $|a|\leq 8\;\;,|b| \leq 8$ and $|c| \leq 1$. My Attempt: Set $x = 0$ in $|ax^2+bx+c|\leq 1$ to ...
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1answer
40 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?