For questions about or involving the absolute value function.

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2
votes
2answers
110 views

Strategy to solve absolute value inequality

I was wondering if there is any strategy to solve absolute value On both sides inequalities, for example, $$| x^2 -3x + 2 | < | x + 2|$$ Thanks, Eli
0
votes
1answer
55 views

Complex number in polar coordinates

I have to get $\Im$, $\Re$, the absolut value as well as the argument $\phi$ of the complex number $$z = \left(-\frac{1}{\sqrt2}+\sqrt\frac{3}{2}i\right)^8$$ I do this by transforming $z' = ...
0
votes
2answers
44 views

Why is the following simplification possible?

I have seen the following simplification: $$\left|\frac{1}{(-1-\frac{1}{n})^4 - 1}\right| = \frac{1}{\left|-1-\frac{1}{n}\right|^4 - 1}$$ I really don't have a clue why this is possible... I am ...
1
vote
0answers
42 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
0
votes
1answer
27 views

Calculate the area that the following graphs form

I have been trying and trying to solve the following problem (I even used wolframalpha as an extra help, but no success, and I have like 100 calculations in my notebook): The Task: Calculate the ...
10
votes
6answers
1k views

How to calculate with absolute value.

Calculate:$$\frac{ \left| x \right| }{2}= \frac{1}{x^2+1}$$ How do I write the whole process so it will be correct? I need some suggestions. Thank you!
0
votes
2answers
61 views

What is the Laurent series of the complex absolute value?

What is the Laurent series of the function $f(z) = |z|$? It seems to be ill defined at $z=0$. Are there any other expansion techniques applicable for this function at $z=0$?
1
vote
1answer
436 views

How to express 2 absolute values as a piecewise function??

I understand how to solve 1 absolute value as a piecewise function. $f(x)=|x-1|$ $$ f(x)= \begin{cases} x-1& \text{if }x\ge1\\ 1-x&\text{if }x<1 \end{cases} $$ But when a function ...
1
vote
1answer
65 views

Change of variables - integrals

\begin{equation} \text{Let $\hspace{3mm}$ }f(t) = 2\int_{b}^{\infty} \sqrt{\frac{1}{2\pi t}}e^{-x^2/2t}dx. \end{equation} I found that this integral can be written with change of variables can be ...
0
votes
1answer
60 views

Are there any $x,y,z$ in $\mathbb{R}$ for which the following equations hold?

Are there any $x,y,z \in \mathbb{R}$ for which the following equations hold? $$|x+1| \leq 2\\ |y+1| \leq 3\\ |y-z| \leq 1$$ With the given we know that $x$ is between $[-3,1]$, $y$ is between ...
1
vote
1answer
44 views

Proving a claim $|c_n e^{in\theta}| = |c_n|$

I'm studying about Fourier series from a book called "Fourier series and its applications" by Folland and on page 40, the author makes the claim that: $$|c_n e^{in\theta}| = |c_n|,$$ where $n$ is an ...
1
vote
2answers
1k views

Prove that the absolute value of a product is the product of the absolute values of factors.

Theorem. $|a||b|=|ab|$ Proof. Applying the definition of absolute value, the left hand side of the equation could be either $a\times(-b)$ or $(-a)\times(b)$ or $a\times b$ or $(-a)\times(-b)$. For ...
0
votes
4answers
375 views

Find $z$ such that $|z+1|+ |z-1|=4$

I have this problem: Find all points of the complex plane wich satisfy: $$|z+1| + |z-1| = 4 $$ I know this is an ellipse with foci 1 and -1, and i know the answer is : $$3 x^2+4 y^2 \leq 12$$ but ...
4
votes
3answers
319 views

Proving continuity of a absolute value function

How can i prove the function $f: x \mapsto x|x|$ is continuous over $\mathbb{R}$ using epsilon-delta definition. I've tried: Given a certain $\epsilon$ we want to prove that there exists a $\delta$ ...
0
votes
2answers
500 views

Minimizing the sum of absolute values with a linear solver

I need a linear program to minimize the sum of several absolute values, but the inclusion of an absolute value means the linear solver won't work. I know there are ways around using an absolute value, ...
0
votes
1answer
119 views

Problem with absolute value

Say that $|\sqrt{x}-1| < \epsilon$. I am having a problem with handling this inequality. I want to exclude x. I. $|\sqrt{x}-1| < \epsilon$ $|\sqrt{x}| - |1| \leq |\sqrt{x}-1| < \epsilon$ ...
1
vote
2answers
84 views

Proof by contradiction: $c<a<d \wedge c<b<d \to |a-b|<d-c$

Let be $a,b,c,d \in \mathbb{R}$, I must proof "$c<a<d \wedge c<b<d \to |a-b|<d-c$". Proof by contradiction: I have $|a-b|\geq d-c$, therefore $a-b \leq c-d \vee a-b \geq d-c$ (or $a-c ...
2
votes
1answer
78 views

Solving inequation with two absoulte values

I need to solve the following inequation: $$ |x| \cdot |x-1|-1>-x\\ $$ I cant get the correct result. I tried to solve it like this: $$ |x| \cdot |x-1|-1>-x $$ I know that I can write $|x ...
2
votes
2answers
75 views

$|x|=\max\{-x,x\}=\max\{-x,x,0\}$?!

Let $x \in \mathbb{R}$, $|x|=\max\{-x,x\}$, is correct also $|x|=\max\{-x,0,x\}$? Thanks in advance!
0
votes
3answers
94 views

Absolut value of cubic polynomial roots lower than 1

Assume we have a cubic polynomial $ x^3 +bx^2+xc+d=0 $, with b,c,d real numbers. Let $x_1, x_2, x_3 $ be the roots, either real or complex. What is the relation of the coefficients b,c and d in ...
1
vote
2answers
54 views

I want to check that $\left|\left|a+b\right|-\left|a\right|-\left|b\right|\right|\leq2\left|b\right|\forall a,b\in\mathbb{R} $.

I want to check that $\left|\left|a+b\right|-\left|a\right|-\left|b\right|\right|\leq2\left|b\right|\forall a,b\in\mathbb{R} $. It 's equivalent to ...
-4
votes
3answers
92 views

How to solve this: $|3-x|\ge2$ [closed]

How to solve $|3-x|\ge2$ ? I know that if $|x| < y$, then $-y < x < y$. But in this case what to do? Thanks. Here, $|x|$ is the absolute value of $x$.
4
votes
4answers
131 views

Let $x$ be in the set of real numbers $\mathbb{R}$ and let $f(x)=|2x-1|-3|2x+4|+7$ be a function, write $f(x)$ without the absolute value.

Let $x$ be in the set of real numbers $\mathbb{R}$ and let $f(x)=|2x-1|-3|2x+4|+7$ be a function, write $f(x)$ without the absolute value. I thought of it this way: $$f(x)=\begin{cases}2x-1-3(2x+4)+7 ...
0
votes
1answer
37 views

Absolute Value Inequality - Precision

So I was writing a computer program, which is supposed to check whether $x$, an approximation of $\sqrt{a}$, is close enough to $\sqrt{x}$. Since these definitions aren't very precise, I defined ...
14
votes
4answers
644 views

How find this inequality $\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$

let $a,b,c,d,e\in R$,and such $$a^2+b^2+c^2+d^2+e^2=1$$ find this value $$A=\max{\left(\min{\left(|a-b|,|b-c|,|c-d|,|d-e|,|e-a|\right)}\right)}$$ I use computer have this $$A=\dfrac{2}{\sqrt{10}}$$ ...
3
votes
2answers
221 views

How does the triangle inequality work for $|x-y|$?

I know that $|x+y|\leq |x|+|y|$... But is it similar for $|x-y|$? That is, is $|x-y|\leq |x|+|y|$? I ask because of the following: $x-y=x+(-y)$, so $|x+(-y)|\leq |x|+|-y|=|x|+|y|$ Is it possible ...
-2
votes
1answer
292 views

Example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points [duplicate]

So I had an exam today and one of the questions were: Give an example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points. At first I had no idea how to do it ...
3
votes
4answers
167 views

Is $\sqrt{x^2}=|x|$ or $=x$? Isn't $(x^2)^\frac12=x?$ [duplicate]

$|x|=\sqrt{x^2}$ as Wolfram|Alpha shows. But, as $(x^2)^\frac12=x$, I can't understand where am I wrong interpreting Square-root.
1
vote
3answers
111 views

Finding the limit: $\lim_{x\to3^+} \frac{\sqrt{x-3}}{|x-3|}$

Can anyone tell me how to properly solve this limit? $\displaystyle \lim_{x\to3^+} \frac{\sqrt{x-3}}{|x-3|}$ I know the answer is positive infinity, and I would know how to do the problem if $x$ was ...
1
vote
1answer
54 views

When is $|f(x)|$ equivalent to $f(|x|)$

Specifically for functions of a complex variable. Are there any rules of thumb?
1
vote
0answers
711 views

Properly Solving Absolute Value Inequality and Quadratic Inequality Problems

How do I solve the following absolute value inequality and inequality problems properly? 1) $\newcommand\abs[1]{|#1|}\abs{2x+9}>x$ Solving this problem algebraically, I get When $x > 0, x ...
1
vote
1answer
57 views

What is the Fourier transform of an M like function

Given the function $$ f(x)= \begin{cases} \vert x \vert& \text{, for }\;\vert x\vert\le M \\ 0 & \text{, otherwise} \end{cases} $$ for some constant $M$. What would be the form for the ...
0
votes
2answers
36 views

$\iint_V |y-x^{2}| \operatorname{d}x \operatorname{d}y$ with $V = [-1,1] \times [0,2]$

it's especially difficult because i don't understand how to integrate absolute value terms. I only know that if you function, say $x^{2}-1$, is below the $x$-axis i need to integrate $1-x^2$ between ...
1
vote
1answer
46 views

limit of an absolute sequence: ${b_n} = |{a_n} - 1|$

$$\eqalign{ & \mathop {\lim }\limits_{n \to \infty } {a_n} = 3 \cr & {b_n} = |{a_n} - 1| \cr} $$ Hence, $$\mathop {\lim }\limits_{n \to \infty } {b_n} = |3 - 1| = 2$$ Is it right to ...
0
votes
2answers
297 views

Absolute Value Properties

I'm attempting to prove that $|x|-|y| \le |x-y|$. I've come up with the following proof. The proof relies on these results obtained from previous exercises: $-|x| \le x \le |x|$ ${|x-y|=|y-x|}$ ...
0
votes
1answer
160 views

usage of absolute value within natural log in solution of differential equation

y=2^x sinx rewriting, |y|=2^x |sinx| my questions, before taking the natural log for both sides and rearrange why do we need to rewrite using absolute value? why this particular question need to have ...
0
votes
1answer
55 views

Help solving a problem with inequalities with absolute values

I have these statements presented: $|x - x_0| < \frac{\epsilon}{2(|y_0| + 1)}$ , $|x - x_0| < 1$ , $|y - y_0| < \frac{\epsilon}{2(|x_0| + 1)}$ And I must prove that: $|xy - x_0y_0| < ...
1
vote
4answers
85 views

Inequalities and absolute values

My book asks that if $$-5\leq x\leq 1$$ then find the boundaries of absolute value of $x$. Can you please help me in finding that?
0
votes
1answer
962 views

Absolute value of limit [closed]

Expain why the following is true: If $$\lim_ {x\to a}\ f(x) = k$$ then $$\lim_ {x\to a}\ |f|(x) = |k|$$
1
vote
2answers
63 views

Definite integral of an absolute function

What is the value of $$\int_{-2}^2 \left|x+1\right|\;dx ?$$ I know the answer is 5 but I don't know how to work it out.. (Original scan)
0
votes
1answer
73 views

Taking “Absolute Value Operator” as a common factor?

If I have an equation like this and Im trying to solve for X |x| + 4|x| = 40 Can I take the absolute Value (Modulus) as a common factor? ...
2
votes
2answers
2k views

Sum of absolute values and the absolute value of the sum of these values?

I'm working on a proof and I need some help with this: I determined that for some situations ($x$ or $y$ are negative but not both): $|x| + |y| > x + y$ How can I conclude using that statement ...
6
votes
3answers
165 views

How would I prove $|x + y| \le |x| + |y|$?

How would I write a detailed structured proof for: for all real numbers $x$ and $y$, $|x + y| \le |x| + |y|$ I'm planning on breaking it up into four cases, where both $x,y < 0$, $x \ge 0$ ...
0
votes
1answer
256 views

How do I evaluate left and right limits?

I have this assignment: $$\lim_{x \to 1} \frac{x^2 - 1}{|1 - x^3|}$$ I do not understand how I should do to separate this into two problems (one for $x \to 1^-$ and one for $x \to 1^+$ and then get ...
0
votes
3answers
584 views

Integrating absolute value function

I'm working on a problem drawing phase plane diagrams in my applied mathematics course. I'm supposed to draw the phase line diagram of $x''+\vert x\vert=0.$ In the process, I get to the differential ...
0
votes
1answer
82 views

Question about one of the first problems in Spivak's Calculus

It's about Chapter I, Problem 21 from Spivak's Calculus: Prove that if: $|x - x_0| < \frac{\epsilon}{2}$ and $|y - y_0| < \frac{\epsilon}{2}$ then $|(x + y) - (x_0 + y_0)| < \epsilon$ ...
3
votes
3answers
154 views

Absolute Value inequality help: $|x+1| \geq 3$

Find the solutions to the inequality: $$|x+1| \geq 3$$ I translate this as: which numbers are at least $3$ units from $1$? So, picturing a number line, I would place a filled in circle at the ...
0
votes
2answers
94 views

Different ways of defining Absolute Value

Calculus I presents this definition of absolute value: $$f(x)=y=|x|=\left\{\begin{array}{}\;\;\;x&\text{ if}\,\,\,x\geq 0\\-x&\text{ if}\,\,\,x<0\end{array}\right.$$ But you can also ...
0
votes
1answer
58 views

Algebra Absolute Value

Let $a,b,c$, and $d$ be real numbers with $$|a-b|=2, \hspace{.2in} |b-c|=3, \hspace{.2in} |c-d|=4$$ What is the sum of all possible values of $|a-d|$? I am completely clueless on how to begin! It's ...
6
votes
6answers
128 views

why minimum of these functions happen at a special place?

why minimum of these functions happen at a special place? how to use derivative to find the minimum of these functions? $$|x-1| + |x-2| + \dots + |x-9|$$ minimum is for $x = 5$ $$|x-1| + |x-2| + \dots ...