Questions tagged [absolute-value]

For questions about or involving the absolute value function.

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Proofs involving strict inequalities

let $a,b \in R$ Prove that if $3 \lt a \lt 5$ and $b= 2 + \sqrt{a-2}$ then, $3 \lt b \lt a$ My approach was simply to start with the first inequality and transform it into b and see what happens. ...
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1answer
20 views

Nonstandard inequality with parameter and absolute value

The given inequality is $|x^2-ax+1|<3(x^2+x+1)$. The question is: For which values of $a$, every $x$ is a solution? I am trying to solve it by making the graphics of the two sides of the ...
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1answer
34 views

Using triangle inequality to find $\lim_{(x,y)\to(0,0)}\frac{x^3-x^2y}{x^2+y^2+xy}$

This is an exercise from my textbook where the problem is to find the limit of the function $\frac{x^3-x^2y}{x^2+y^2+xy}$ when $(x,y) \to (0,0)$. So after changing to polar coordinates and ...
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1answer
81 views

An $\arcsin$ inequality

Show that if $0<|x|,|y|<1$, then $$\arcsin |x| +\arcsin |y| > \arcsin\left|\frac{x+y}{1+xy}\right|.$$ I found a proof (see below). Is there a different way (hopefully simpler) to show that ...
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0answers
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equivalent discrete absolute values

If an absolute value $|\cdot|$ on a field $K$ is discrete, then the value group $|K^*|$ is a discrete subgroup of $\mathbb{R}_{>0}$. Hence $|K^*|=\lambda^{\mathbb{Z}}=\{\lambda^n \mid n \in \mathbb{...
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1answer
44 views

Differentiability of $f(x)$ at $x=0$

If $f(x)=a_0\cos|x|+a_1\sin|x|+a_2|x|^3$ is differentiable at $x=0$, then (A) $a_1=0, a_2=0$ (B) $a_0=0, a_1=0$ (C) $a_1=0$ (D) $a_0, a_1, a_2$ can take any real value The ...
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1answer
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What does c mean in Calculus (Absolute Value Inequality)? [on hold]

I am graphing Absolute Value Inequalities and I came across this problem. |x-c| < 0.1. I'm not sure what c represents in this inequality. Thanks for your help!
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1answer
31 views

When given the difference between numbers, do we use the absolute value?

This question is more about wording than computation. The sum of the first number and the square of the second number is 18. The difference between the square of the second number and twice the ...
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0answers
13 views

$|\frac{\sin^3a-2}{(1-\frac{\cos b}{2})^n}|$ give the best upper and lower bounds

$$\left|\frac{\sin^3a-2}{\left(1-\frac{\cos b}{2}\right)^n}\right|$$ Give the best upper and lower bounds of the form $cA^n$, where $c$ and $A$ are fixed numbers. $$\left|\frac{\sin^3a-2}{\left(1-\...
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4answers
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Range of a function $|\sin x|+|\cos x|$

What is the range of function $Y=[|\sin x|+|\cos x|]$ where $[ ]$ denotes the greatest integer function. And what is range of function $Y=|\sin x|+|\cos x|$ For the second one, I have tried squaring ...
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6answers
108 views

Finding $a$ for which $x^2a-2x+1=3\lvert x\rvert$ has exactly $3$ distinct real solutions.

Find all real numbers a for which the equation $$x^2a-2x+1=3\lvert x\rvert$$ has exactly $3$ distinct real solutions in $x$. I have tried the question very much but a big doubt is how a quadratic ...
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4answers
97 views

When is $y=|P(x)|$ is differentiable?

Let $P:\mathbb{R}\to\mathbb{R}$ denote the polynomial function. When is $y=|P(x)|$ differentiable? I found out that $y=|P(x)|$ may be differentiable through-out $\mathbb{R}$ or it may not be. When it ...
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0answers
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Minimum Absolute Difference

Given 'n' numbers, how do we choose 'k' (k<=n) numbers such that the sum of the absolute difference between each element in 'n' and the closest element in 'k' is the minimum. Each of these 'n' ...
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3answers
88 views

Algebraic equation of first degree with absolute values [closed]

How to solve $$\left| a^2-2 a-b^2-4 b-x \right| + \left| a^2-2 a-b^2-4 b-3 x+2\right| +\\ \left| a^2-2 a+b^2+4 b+2 x \right| + a^2-2 a+b^2+4 b+18 \left| x-2 \right| +11 x=20 $$ over the reals if ...
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3answers
41 views

Minimum of a trigonometric function involving absolute value

Given $f(x) = | \sin( | x | ) | $, I am told to found the local and global minimum and maximum of $f$ (If they exist). Simply from sketching the graph of $f$ I get that the function maximizes ...
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1answer
58 views

Calculating the expecation of the supremum of absolute value of a Brownian motion

I got a Brownian motion $B(t)$ that starts in $0$ and want to calculate the expectated value of the supremum on the interval $[0,1]$ of the absolute value of it, i.e. $E \left (\sup \limits_{t \in [0,...
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2answers
47 views

Real analysis question $ a = b$ if and only if for every $\epsilon \gt 0, \left\lvert a-b \right\rvert \lt \epsilon$

I've encountered the proof that BWOC, $|a-b| \lt \epsilon$, we assume that $a \ne b.$ Without loss of generality, $a\gt b, a-b\gt 0$, let $\epsilon = a-b.$ Then $|a-b| \lt a-b$, which is a ...
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1answer
61 views

How to graph $x|2x-1|-3$?

I found it on an exam and it stumped me. Is it considered a function? Because I get the same outputs from $x = 0$; and $x = 1/2$
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2answers
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What is the proof that $|\ln (1-x)|\le 2|x|$ if $|x|\le \frac{1}{2}$?

I'm trying to prove that $$|\ln (1-x)|\le 2|x| \, \text{if} \, |x|\le \frac{1}{2}.$$ What I've got so far is $$\begin{align*}e^{x}&=1+x+\dfrac{x^{2}}{2!}+\cdots\\1+x&\le e^{x}\\1-x&\le e^{-...
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5answers
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Find all numbers x which satisfy $|x^2 + 2| = |x^2 − 11|$.

This question is taken from book: Exercises and Problems in Calculus, by John M. Erdman, available online, from chapter 1.1, question $4$. Request help, as not clear if my approach is correct. (4)...
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2answers
107 views

Maximisation of a piecewise affine function over an ellipsoid

Given vectors $\mathrm a, \bar{\mathrm x} \in \mathbb R^n$ and matrix $\mathrm P \in \mathbb S^n_{++}$, how to deal with the absolute value in the objective function of this optimization problem in $\...
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1answer
50 views

Find minimize value of $P=(a^2+b^2+c^2)(|a-b|+|b-c|+|c-a|)$

For $a,b,c$ are real number satisfied: $a^3+b^3+c^3=3abc+32$. Find minimize value of $P=(a^2+b^2+c^2)(|a-b|+|b-c|+|c-a|)$ This is my try: WLOG, I suppose that: $a\ge b\ge c$. Then we have $P=(a^2+b^...
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1answer
51 views

Absolute value of complex $\exp(z^2)$

(This is my first post on stackexchange. Please tell me, if I made any formatting errors and such.) This question is about how the absolute value function works with the complex exponential. We have ...
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2answers
25 views

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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1answer
29 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
48 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
35 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
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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
69 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
94 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
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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
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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
22 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
170 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
34 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
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|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
102 views

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
793 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
24 views

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
63 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
44 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
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
48 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
59 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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1answer
31 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
46 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) \...