For questions about or involving the absolute value function.

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0
votes
2answers
57 views

Graphs for mod functions

Can someone please teach me how to obtain graphs for the following types of functions: $2+3|x-1|$ $|x-1|+|x|+|x+1|$ $|x-1|-|x|-|x+1|$ $|x-1|^2$ Thanks.
0
votes
1answer
20 views

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$

Does $|(aj+b)^{-1}| = (|aj+b|)^{-1}$, where $aj+b$ is a complex number, and $|f(x)|$ is the modulus function. In the past I've been calculating $|(aj+b)^{-1}|$ by multiplying the numerator and ...
4
votes
3answers
95 views

Is it always true? $\left|A-B\right| \le \left|A\right| + \left|B\right|$

Is it always right to claim that: $$\left|A - B\right| \le \left|A\right| + \left|B\right|$$ where $A, B \in \mathbb{R}$ ?
2
votes
0answers
34 views

Solving $n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt$

I have to solve $$ n \int_{\mathbb{R}}{\left|\frac{1}{n}\sum_{j=1}^n{e^{(itY_j)}}-e^{-\frac{1}{2}t^2}\right|^2}\psi(t)dt $$ where $\psi(t)=(2\pi)^{-\frac{1}{2}}e^{-\frac{1}{2}t^2}$ is the density ...
0
votes
1answer
52 views

$\left | -(x+2)^2+6(x+2) \right |>13$

I did $-(x+2)^2+6(x+2)>13$ and $-(x+2)^2+6(x+2)< -13$. The first inequality had complex solutions and therefore can be disregarded but the second one has two real solutions, $x \approx -3.7$ and ...
0
votes
3answers
85 views

Absolute value quadratic inequalities not the usual?

$ | -x^2 + 6x | \gt 13 $,for example. I would start off solving $ -x^2 + 6x = \pm 13 $ and either get 4 solutions, 3 solutions or two simply do the the nature of the graph. Without knowing if the two ...
3
votes
0answers
89 views

Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
1
vote
4answers
799 views

How to solve equations involving modulus function of the type $|x+1| - |1-x|=2 $ and $ |x-1|=|x|+a$?

I am able to solve equation of the type $ |5x+1|=|11-2x|$. I square both the side and my equation becomes $ (5x+1)^2=(11-2x)^2 $ further simplification gives me $ (5x+1)=\pm (11-2x)$. I get have ...
1
vote
1answer
60 views

Don't understand inequality in order to prove Algebraic Limit Theorem

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I'm stuck on Theorem 2.3.3 on page 45, i.e., the Algebraic Limit Theorem. In particular, letting $\lim a_n = a$ and $\lim ...
2
votes
1answer
33 views

If $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ are complex numbers, then $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$

Let $w_1=a_1+ib_1$ and $w_2=a_2+ib_2$ be two complex numbers. Ahlfors says that $|e^{w_1}-e^{w_2}|\geq e^{a_1}-e^{a_2}$. I don't understand why that is. Any help would be greatly appreciated.
2
votes
4answers
68 views

Evaluate the integral $\int_{-1}^{1}\left\vert\, x^{3} - x\,\right\vert\,{\rm d}x$

I'm trying to solve: $$\int_{-1}^{1}\left\vert\, x^{3} - x\,\right\vert\,{\rm d}x$$ I tried to solve this integral as follows: solving $x^{3} - x = 0$ which implies $x = 0$ , $x = -1$ or $x = 1$. ...
0
votes
0answers
22 views

A system of absolute value equalities

Background: I'm trying to show that the transformation $T:\Bbb R^n\to\Bbb R^n$ defined by $T(x_1,\dots,x_n) := (|x_2-x_1|,|x_3-x_2|,\dots,|x_1-x_n|)$ is (or is not, this is out of curiosity only) ...
2
votes
2answers
126 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
1
vote
2answers
40 views

Where did I go wrong with this inequality involving absolute value function?

Question: Find all $x \in \mathbb R$ such that the inequality $4<|x+2|+ |x-1|<5$ is satisfied. This is my attempt at solving the problem: Case (i): If $x+2 \geq 0 $ and $ x-1\geq0$, then ...
0
votes
2answers
80 views

$|x| - |y| \leq |x-y|?$

Is there a clever way to show that $$|x| - |y| \leq |x-y|$$ I believe I can think of a way to solve this using cases, but the book I'm working out of said that "A very short proof is possible if you ...
0
votes
3answers
37 views

I do not quite understand this difference in limits

According it my study material: $\lim_{x\to 0^-}\frac {x}{|x|}= -1$ and $\lim_{x\to0^-} \frac {1}{|x|}= \infty$ Why does $\lim_{x\to0^-} \frac {1}{|x|}\ne -\infty$ as 1 still devided by a negative ...
1
vote
3answers
107 views

Proving a limit exists - solving for epsilon with absolute values

I have the equation that I want to prove the limit goes to 1: $$\lim_{n \to \infty} \frac {(n+8)(n+1)}{n(n-10)} = 1$$ Using definition of limit, I get this equation: $$ \left | \frac ...
2
votes
3answers
68 views

For which $a$ does $|x+1|+|2-x|=a^2 -1$ have exactly two solutions?

If it is not a problem, I would really appreciate if someone could explain to me how to solve and graph the following equation: For which real numbers $a$ does the equation $|x+1| +|2-x|=a^2 -1$ ...
2
votes
1answer
73 views

Does “Expected Absolute Deviation” or “Expected Absolute Deviation Range” exist in stats and have another name?

So everyone is familiar with Variance and Standard Deviation from high school, but it seems no one has any familiarity with a philosophical justification for such weird, seemingly arbitrary measures. ...
1
vote
4answers
90 views

How to solve Absolute Value Inequality: |x-1| ≥ 3-x

I am learning the topic of solving absolute value inequality question. I had mostly understood the steps in order to solve for an inequality. However, I'm still clueless of a step to solve the ...
1
vote
2answers
50 views

Spivak Absolute Value Problem (Prologue 9-v)

I'm working on the following problem Express the following with at least one less pair of absolute value signs $$|(| \sqrt2 + \sqrt3| - |\sqrt5 - \sqrt7|)|$$ Now I can see that the ...
0
votes
4answers
69 views

solving the system

solve the system : $$ y+|x-2|=3 $$, $$ |x+y|= m $$ graphicly when $m$ equals $6$. I can easily (realtively) skecth the first graph , however, how the bloody hell do you sketch $|x+y|= 6$??
5
votes
1answer
71 views

Maximum value problem

A function $\hspace{0.1cm}$$f:[0,1]\to[-1,1]$$\hspace{0.1cm}$ satisfying$\hspace{0.1cm}$ $|f(x)|\leq x$$\hspace{0.1cm}$ $\forall x\in[0,1]$. Then find the maximum value of: ...
1
vote
2answers
51 views

Restore the signum of abs(sinc(x))

Is it possible, by any means, to restore the signum of sinc(x) after being transformed to its absolute value, abs(sinc(x))? How it got to abs() is irrelevant, I only want to know if the reverse is ...
2
votes
2answers
43 views

Concerning Rules of Exponents & Absolute Value

I understand that one of the accepted definitions of the absolute value function is $\left| x \right| = \sqrt{x^2}$. However, I do not understand why if I substitute $-5$ in for $x$ that I can't do ...
0
votes
0answers
14 views

Difference in magnitude between two cross-correlations by two different way of calculations.

I think there are two ways of calculating cross-correlations for two difference random variables, X and Y. I am assuming discrete functions. 1) Multiplication $$ \sum_{m=-\infty}^\infty x[m]y[m+n] ...
0
votes
2answers
112 views

Prove: If $a\in\mathbb Z$ and $|a| > 1$, then $1/a \notin \mathbb Z$.

Prove: If $a$ is an integer and $|a| > 1$, then $1/a$ is not an integer. Hi, I need help proving this either by contradiction or contrapositive. I'm not sure where to begin
2
votes
4answers
249 views

inequality method of solution

Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$ I've tried first determining when the absolute value ...
1
vote
1answer
48 views

Limit of functions absolute value

$$ \begin{align} &\lim\limits_{x\to0} \frac{|3x-1|-|3x+1|}x\\ =&\lim\limits_{x\to0} \frac{(3x-1)^2-(3x+1)^2}{x(|3x-1|+|3x+1|)}\\ =&\lim\limits_{x\to0} \frac{-12x}{x(|3x-1|+|3x+1|)} = ...
2
votes
3answers
214 views

Derivative and integral of the abs function

I would like to ask about how to find the derivative of the absolute value function for example : $\dfrac{d}{dx}|x-3|$ My try:$$ f(x)=|x-3|\\ f(x) = \begin{cases} x-3, & \text{if }x \geq3 \\ ...
3
votes
3answers
113 views

Please help with absolute value $|x^2 - 3x| = 28$

Just a question about solving an absolute value equation: $$|x^2 - 3x| = 28$$ Do I just solve this as if the absolute value brackets weren't even there? $$x^2 - 3x - 28 = 0$$ $$(x+4)(x-7) = 0$$ ...
1
vote
1answer
48 views

Do modulus and absolute value operations use the same sign?

Do modulus and absolute value operations use the same sign? If so, do we always assume that a modulus is intended when the number is complex? If an expression says $|a+bi|$, this means I should ...
0
votes
2answers
88 views

Derivative of $f(x)=|x|$

Okay, so $\displaystyle \frac{d}{dx} |x| = \frac{|x|}{x}$. But I have trouble seeing why. Here's what I've tried: $$\frac{d}{dx}|x|=\begin{cases} \frac{d}{dx}x & \text{if }x > 0 \\ ...
0
votes
1answer
99 views

Infimum of absolute values versus absolute value of infimum

Let $A\subseteq\mathbb R$. Is there a nice proof of the inequality $\displaystyle\inf_{a\in A} |a|\le|\inf_{a\in A} a|$? The only proof I know is, though not very difficult, annoying because it ...
0
votes
1answer
27 views

How do I solve the following absolute value equation?

I'm having trouble solving this equation: $$|x+1| = |2x-2|$$ For $x+1 = 2x-2$ and $-(x+1) = -(2x-2)$ I received $x = 3$ and for $-(x+1) = 2x-2$ and $x+1 = -(2x-2)$ I received $x = 1/3$ I tried ...
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vote
3answers
54 views

How do you solve two equal absolute value expressions?

I'm having trouble understanding how the following is solved. $$|x+1| = |x-2|$$
3
votes
4answers
168 views

Show that $|z+1|\le|z+1|^2 +|z|$ for all $z \in \mathbb{C}$

Question: Show that $|z+1|\le|z+1|^2 +|z|$ for all $z \in \mathbb{C}$ So far I have, Suppose $1\le|z+1|$ $|z+1|\le|z+1|^2$ $|z+1|\le|z+1|^2+|z|$ Now I must show $|z+1|<1$ but this is where ...
0
votes
1answer
31 views

Filling in the derivative of the absolute value at zero

I have a function $f(x)$ such that $f(x_0)=0$ and I'm interested in the derivative $\frac{d |f(x)|}{dx}$ evaluated at the point $x_0$. I realize that this is usually undefined. However, if ...
0
votes
1answer
37 views

Modulus function (working out coordinates)

Lets say you have $y = -|3x - 1|$ when working out where it cuts the axis, particularly the x-coordinate you do the following when $y = 0, 3x - 1 = 0$ therefore $x = 1/3 $ the modulus and the ...
2
votes
1answer
95 views

Solving absolute inequality

I have the following inequality: $$|4 - k^2| > |10 + 13k|$$ So how to solve this ?
2
votes
5answers
130 views

Finding the minimum value of a sum [closed]

Let $x,y,z$ be real numbers . Find the real number $a$ so that $S$ has a minimum value , where $$S=|x-a|+|y-a|+|z-a| .$$
0
votes
2answers
69 views

Does the absolute value of +3 lose its positive direction yet have its positive value? [closed]

We have no sigh with the absolute value of +3, yet its value is positive.(Wikipedia) Does this mean that the absolute value doesn’t have its positive direction (+3 is located on positive direction ...
0
votes
1answer
26 views

$|2- (\sqrt{n^2+4n} - n)| ≥ \frac{1}{10}$

Any suggestions how to solve the following equation: $|2- \sqrt{n^2+4n} + n| ≥ \frac{1}{10}$ Thank you in advance.
-1
votes
2answers
68 views

Sketching a set of complex numbers and deducing the value of $|z +1 - i|$ for such numbers

The point $P$ represents the complex number $z$. a) Given that $\arg(\frac{z-2i}{z+2}) = \frac{\pi}{2}$ , sketch the locus of $P$. Ok so I've sketched this and this is what it looks like : b) ...
5
votes
5answers
239 views

Evaluating the following integral: $ \int \frac{x^2}{\sqrt{x^2 - 1}} \text{ d}x$

For this indefinite integral, I decided to use the substitution $x = \cosh u$ and I've ended up with a $| \sinh u |$ term in the denominator which I'm unsure about dealing with: $$\int ...
1
vote
2answers
43 views

Maximal distance between points on a line

Two points A and B are on different sides of a line. Find a point Y on the line such that the absolute value of the difference from Y to A and Y to B is maximal. My thoughts are as follows. Let's ...
0
votes
1answer
83 views

Integral of absolute value = absolute value of the integral

Let $(a,b) \in \mathbb{R}^2$ and $f \in C^0([a, b] , \mathbb{C})$ Find the condition on $f$ so that $$|\int_a^b f|=\int_a^b|f|$$ My try : The function $f: t \mapsto r(t)\exp(i\theta)$ where $r$ is a ...
1
vote
5answers
78 views

Calculate $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$

I know that $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$ does not exist, because the lateral limits are different and I also know that the absolute value on the denominator has ...
1
vote
4answers
182 views

Simple question about the range of possible values for a function

So we have $2 |3-x| + 5 = k$, where $k$ is a constant. Provided this equation has two real solutions for $x$, what is the range of possible values for $k$?
2
votes
6answers
111 views

Solving $|\frac{x+1}{x}|< 1$

I need some help/suggestions solving the following math problem. I don't know how to continue from step 2. Find x. 1.) $\displaystyle\left|\frac{x+1}{x}\right|< 1$ 2.) ...