For questions about or involving the absolute value function.

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3
votes
4answers
48 views

What can be said about the relationship between the complex numbers $\lvert z\rvert^n$ and $\lvert z^n\rvert$?

I've been playing around with this for a while without much progress. More precisely, I suppose, I'd like to know if one always less than or equal to the other? The fact that one never sees this in ...
6
votes
1answer
63 views

Basic question $|x^2| < 9$

I have a rather basic question. Let's assume that $|x^2| < 9$, where $x\in \mathbb{R}$. Then everyone knows that $x \in$ (-3,3). However, I have trouble arriving at the answer based on basic ...
0
votes
0answers
22 views

Is it always possible to define an absolute value in an ordered field?

I am trying to show (not sure if possible) that i can generalize all basic arithmetic operations between limits of sequences of real numbers to any ordered field, so i need to build a generalized ...
5
votes
5answers
11k views

Converting absolute value program into linear program

I have the generic optimization problem: $$ \max c^T|x|$$ $$ \text{s.t. } Ax \le b $$ $x$ is unrestricted How do I convert it into a linear programming problem? Online I read something about ...
1
vote
1answer
15 views

Trouble with an inequality between magnitudes of complex numbers

We are supposed to show that $$|ab^* + a^*b| \leq 2|ab|$$ where a and ba re complex numbers and a* and b* are their respective conjugates (so $a = x_1+iy_1$, $a^* = x_1-iy_1$, $b = x_2+iy_2$, $b^* = ...
5
votes
2answers
388 views

Why do definitions of distinct conic sections produce a single equation?

I understand how to get from the definitions of a hyperbola — as the set of all points on a plane such that the absolute value of the difference between the distances to two foci at $(-c,0)$ and ...
3
votes
3answers
48 views

Find the area of the region described by $|5x|+|6y| \le 30 $

Find the area of the region described by $|5x|+|6y| \le 30 $ (where $|z|$ denotes the absolute value of $z$). My effort Imagining a number line and interpreting the problem as the request to ...
1
vote
0answers
21 views

Derivative of the maximum of a function on a interval

My question is as follows: Given a function $f: [-h,\infty) \rightarrow \mathcal{R}$ and the maximum function given by \begin{equation} \max_{s\in [-h,0]} |f(t+s)| \end{equation} for $t\geq0$. Then ...
0
votes
2answers
30 views

Inequality with absolute value.

Show that $\forall a,b\in \mathbb{R}$: $$ \left| \frac{a}{1+a^2} - \frac{b}{1+b^2} \right| \leq |a-b| $$ Being honest, I do not know where to start (apart from common denominator form) and would ...
2
votes
1answer
108 views

Is this a valid proof for $\left | a+b \right | \leq \left |a \right| + \left | b \right | $?

$\left | a+b \right | \leq \left |a \right| + \left | b \right | \Rightarrow$ $\sqrt{{(a+b)}^2} \leq \sqrt{{a}^2} + \sqrt{{b}^2}$ ${(\sqrt{{(a+b)}^2})}^2 \leq ({\sqrt{{a}^2} + \sqrt{{b}^2}})^2 ...
1
vote
4answers
92 views

Prove that $\left|(|x|-|y|)\right|\leq|x-y|$

Prove that $\left|(|x|-|y|)\right|\leq|x-y|$ Proof: $$\begin{align} \left|(|x|-|y|)\right| &\leq|x-y| \\ {\left|\sqrt{x^2}-\sqrt{y^2}\right|}&\leq \sqrt{(x-y)^2} ...
0
votes
2answers
34 views

If $\left| x \right| \ge \left| y \right|$ then Can we say $\left| {x + y} \right| \ge \left| x \right| - \left| y \right|$?

Let $x,y\in \mathbb{R}$ and $\left| x \right| \ge \left| y \right|$. Can we say $\left| {x + y} \right| \ge \left| x \right| - \left| y \right|$?
1
vote
2answers
453 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, ...
1
vote
2answers
44 views

Solve the equation $\Bbb|x-\sqrt{x}|=\lfloor x\rfloor-\sqrt{x}$ [closed]

$$\Bbb|x-\sqrt{x}|=\lfloor x\rfloor-\sqrt{x}$$ Do equation for positive numbers that are incorrect answer?I did not find them. $$With great wishes for you$$
1
vote
1answer
40 views

Absolute value of product is less than product of absolute values: $|(1+a_1)(1+a_2)\dots (1+a_n)-1|\leq (1+|a_1|)(1+|a_2|)\dots (1+|a_n|)-1$

For a sequence $a_n\in\mathbb{C}$ I want to show that $$|(1+a_1)(1+a_2)\dots (1+a_n)-1|\leq (1+|a_1|)(1+|a_2|)\dots (1+|a_n|)-1$$ I think I should show this by induction on $n$. For the base case I'm ...
0
votes
1answer
24 views

solve and skecth $\log{|z|}=-2\arg(z)$

Ive asked this question a week ago, but nobody managed to answer but it is doing my heading from then. I know usually You demand some initial work done on the question but I just dont know how to ...
0
votes
1answer
23 views

Double integral over the set with an absolute value of $y$

I need to calculate an integral over the set: $$D \colon 0\leq x\leq \pi\text{ and }|y|\leq x$$ from the set (definite integral) $D \int \cos(y)dA$ I don't understand what $|y| \leq x$ means. Can ...
5
votes
1answer
206 views

Does the triangle inequality for the absolute value hold for matrix trace?

It is well-known that, $\left|m-n\right|\ge\left|\left|m\right|-\left|n\right|\right|$ for real numbers. But if one defines $\left|M\right|=\sqrt{M^2}$ for a symmetric matrix $M$, does one have ...
1
vote
1answer
41 views

Double integration over function with absolute values

I have having difficulty in how to solve the following double integral problem involving absolute values and the assumption that $\alpha > 1$: $\iint_{-\infty}^{+\infty} \frac{1}{1+|x|^\alpha} ...
0
votes
1answer
30 views

Does every non-archimedean absolute value on field take value in $\mathbb{Q}$

Let $K$ be a field, a non-archimedean absolute value is defined to be a map $K\to \mathbb{R}$ satisfying $|x|=0\Rightarrow x=0$, $|x|\cdot|y|=|xy|$ and $|x+y|\leq\max(|x|,|y|)$. Is there an example ...
17
votes
6answers
2k views

Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
6
votes
5answers
312 views

Absolute value and max/min function: why $a + b + |a - b|=2\max(a,b)$? [duplicate]

I am being told that $a + b + |a - b|$ is equal to $2\max(a,b)$. What is the reasoning behind this?
3
votes
2answers
58 views

Prove $|x|^2$ = $x^2$

My first attempt at this proof divided into 2 cases, one where $x^2$ is greater than or equal to 0, and another where $x^2$ is less than 0. For the first case, I said that the definition of absolute ...
5
votes
1answer
158 views

Verify integration of $ \int\frac{\sqrt{2-x-x^2}}{x^2}dx $

This is exercise 6.25.40 from Tom Apostol's Calculus I. I would like to ask someone to verify my solution, the result I got differs from the one provided in the book. Evaluate the following integral: ...
1
vote
5answers
65 views

$\left|x\right| < \left|\tan(x)\right|$ close to $0$

I was trying to prove this inequality $\left|x\right| < \left|\tan(x)\right|$ in a neighborhood of $0$. I tried splitting into the four cases opening the modulus but still wasnt able to solve it. I ...
0
votes
1answer
19 views

Why/how do I ignore these absolute values while using Variation of Parameters?

I'm given the initial value problem: $x' = \frac{3}{t}x + e^{3t}$, $x(1) = 2$ Using the variation of parameters formula, I end up with: $2e^{3 \ln|t|} + e^{3 \ln|t|} \int^{t}_{1}e^{-3\ln|s|} e^{3s} ...
4
votes
3answers
87 views

What is the fastest way to find the range of functions having modulus: $f(x) = |x+3| - |x+1| - |x-1| + |x-3|$

While solving problems I saw a question in which I was supposed to find the range of a function $$f(x) = |x+3| - |x+1| - |x-1| + |x-3|$$ I know the way in which I can take different cases of $x$ ...
2
votes
2answers
54 views

Absolute value inequality $3 > |x + 4| \geq 1$

I've just started with absolute value equations and I have a real hard time understanding how to solve this. I got the following question, and I can't make heads or tails out of it. Assume that $x, ...
2
votes
4answers
102 views

Techniques as to solving absolute value equation

Solve absolute value equation with absolute value variable one one side or even both side, without a number outsides of absolute value signs are typically easy. In my high school, I was taught to ...
0
votes
1answer
35 views

Complex numbers converge if their absolute values and arguments converge

Let the sequence $\{z_n\}_{n>0}$ and $w \not=0$ be such that $|z_n| \to |w|$ and $\operatorname{Arg}(z_n) \to \operatorname{Arg}(w)$. Show that $z_n \to w$. My proof: $z_n= |z_n|e^{i \arg(z_n)} ...
3
votes
2answers
45 views

Prove $||a| - |b||$ is less than or equal to $|a-b|$

I was given the hint to split it into two cases ($|a| - |b|$ being positive and negative) and then use the triangle inequality. However, since the triangle inequality says that $|a+b|$ is less than or ...
5
votes
1answer
54 views

Prove that $|x|^2$ = $x^2$.

This is what I did, but I'm not sure if it's a good enough proof: Since $|x|$ is equal to $x$ when $x$ is greater than or equal to 0, and is equal to $-x$ when $x$ is less than 0, I said that $|x|^2$ ...
0
votes
2answers
37 views

Absolute Value Rational Inequalities

Ok so I have the following two inequalities: \begin{equation} \left| \frac{x+6}{x-2}\right| \leq 4 \end{equation} and \begin{equation} \frac{x^2-1}{\left| x+2\right|} \leq 3(1-x) ...
7
votes
2answers
589 views

A ''strange'' integral from WolframAlpha

I want integrate: $$ \int \frac{1}{\sqrt{|x|}} \, dx $$ so I divide for two cases $$ x>0 \Rightarrow \int \frac{1}{\sqrt{x}} \, dx= 2\sqrt{x}+c $$ $$ x<0 \Rightarrow \int \frac{1}{\sqrt{-x}} \, ...
-2
votes
1answer
102 views

How to prove that $\lim_{(x, y) \to (0, 0)} \left(\frac{1}{|x|} + \frac{1}{|y|}\right) = 0$? [closed]

Prove that $$\lim_{(x, y) \to (0, 0)} \left(\frac{1}{|x|} + \frac{1}{|y|}\right) = 0$$ I couldn't prove this. Please suggest a solution.
1
vote
4answers
62 views

Prove $\lvert x\rvert$ = $\lvert-x\rvert$ for all real numbers $x$ [closed]

Been at this one for a long time. I'm trying to use the fact that $|x|$ = $x$ if $x$ is greater than or equal to 0, and $|x|$ = $-x$ if $x$ is less than 0. Then I want to split the proof into these 2 ...
0
votes
2answers
27 views

Piecewise from Rational Absolute Value Function

How would one separate a function like the following into piecewise? $$f(x)={\left|4-x\right|\over{\left|x-4\right|}}$$ I've been taught that with a rational function with an absolute value in the ...
4
votes
3answers
65 views

Proving $\max$ of $a, b$.

How do I prove that $$\max{\{a, b\}} = \frac{a + b + \left | a - b \right |}{2}$$ I have no idea how to even start the proof, any idea / intuition that can get me started is greatly appreciated. ...
0
votes
3answers
40 views

Square divided by absolute value

First time posting on Math SE, with kind of a basic algebra question. Question Does the relation: $$\dfrac{(ab)^2}{|ab|} = \left|ab\right|$$ with $a,b \in \mathbb{R_{\ne 0}}$ always hold? It seems ...
0
votes
2answers
984 views

Absolute value and credit card balance

I'm embarrassed to ask this question, but my child has the following homework question: "Use absolute value to describe the relationship between a negative credit card balance and the amount owed." ...
0
votes
0answers
20 views

Function with absolute value in denominator - limits

f(x)=(x-1)/(|2-x|-1) |2-x|= { |2-x|; x < +2} {-|2-x|; x >= +2} State domain, range and the equations of the asymptotes. D(f)= {x | x > 3 or x < 3} R(f)= {y | y > 1 or y <= -1} ...
2
votes
6answers
89 views

Solving the absolute value inequality $\big| \frac{x}{x + 4} \big| < 4$

I was given this question and asked to find $x$: $$\left| \frac{x}{x+4} \right|<4$$ I broke this into three pieces: $$ \left| \frac{x}{x+4} \right| = \left\{ \begin{array}{ll} ...
2
votes
2answers
59 views

Evaluating the Fourier coefficients of $abs(x)$

Let's get started: $$\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi} |x|e^{-inx} dx$$ since $|x|$ is an even function: $$= \frac{1}{\pi}\int_0^{\pi} xe^{-inx} dx$$ Integration by parts yields: ...
1
vote
1answer
40 views

Identifying a real parameter in an equation

I'm not really sure how to go about this problem, as I've never encountered anything similar before. I'm supposed to find all the values $m$ for which the following equation has $3$ distinct real ...
0
votes
0answers
35 views

Show that absolute value satisfies triangle inequality, how? [duplicate]

I wish to show that given $a,b,c \in \mathbb{R}$, the following holds: $|a - c| \leq |a-b| + |b-c|$ Using the definition $| x | = \max\{x, -x\}$ I can't seem to be able to show that $|a - c| \leq ...
0
votes
0answers
23 views

Continuity properties of an example function $f:\mathbb{R}^n\to\mathbb{R}$

Consider the function $f:\mathbb{R}^n\to\mathbb{R}$ defined as follows: $$ f(x)=\begin{cases} ||x||^2 & \text{if $||x||\le 1$,}\\ 1/||x||^2 & \text{if $||x||> 1$,} \end{cases} $$ where ...
0
votes
3answers
41 views

Nested absolute-value inequality

I try to solve a problem in two ways, but the results are not the same. Method 1. $$\lvert \lvert x \rvert + x \rvert \le 2$$ For $x < 0$, we have $\lvert x \rvert = -x$. Therefore: $$\lvert ...
0
votes
1answer
25 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 ...
1
vote
1answer
56 views

Prove those inequalities are true [duplicate]

I want to prove that those inequalities are true for $a, b ∈ R$: $$ |a + b| ≤ |a| + |b| $$ $$ ||a| − |b|| ≤ |a − b| $$ $$ |a − b| ≤ |a − c| + |c − b| $$ Now I can see that they are true, and I could ...
3
votes
2answers
113 views

Derivative of $x\cdot|x|$ on $x=0$?

$$f(x) = x |x|$$ Wolfram Alpha says is: $$f'(x) = \frac{2x^2}{|x|}$$ and thus $f'(0)$ is indeterminate, while an HP48 says that: $$f'(x) = |x| + x \operatorname{sgn} x,$$ which would yield $f'(0) ...