For questions about or involving the absolute value function.

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6
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6answers
126 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 ...
0
votes
3answers
50 views

Peculiarities about Finding Absolute Values

It is known that the formula $\sqrt{x^2}$ is equal to the value of $|x|$. In my spare time last night, I wondered about $\sqrt[3]{x^3}$. After some thought and some graphing, I came up with this: ...
2
votes
3answers
89 views

absolute value inequalities

When answer this kind of inequality $|2x^2-5x+2| < |x+1|$ I am testing the four combinations when both side are +, one is + and the other is - and the opposite and when they are both -. When I ...
0
votes
1answer
49 views

Algebraic representation of an absolute value.

I tried several ways, but i could not come up with any way to have an equation as such: |n| = ... without using the absolute value signs on the right side of the ...
1
vote
1answer
69 views
1
vote
0answers
108 views

Integration of the absolute value of an unknown function

I'm doing a vector arclength problem, and have gotten to the part where I have $\int | r'(t) | dt. $ Both $r(t)$ and $r'(t)$ are unknown functions, though I do know that $0 \leq r(t)$ for $a ≤ t ≤ ...
0
votes
1answer
388 views

limit of an absolute value function

how would I go about finding the limit of the following absolute value function as it goes to infinity $\displaystyle\left|\frac{1}{x} - \frac{1}{y}\right|$ Ive never dealt with multivariable ...
2
votes
1answer
83 views

what is the value of $a+b?$

Can anyone help me to solve this problem: $x$ and$y$ are real numbers which satisfy $x>y$ and $xy<0$. If $\left | x \right | + \left | y \right | + \left | 42y-x \right | + \left | 23x-y \right ...
3
votes
2answers
99 views

Inequality proof with absolute values

How do you prove the following: $$ \varepsilon > 0 \\ \left | y-b \right | < \varepsilon\\ \left | x-a \right | < \varepsilon \\ \Longrightarrow \left | xy - ab \right | < ...
0
votes
3answers
76 views

To what extent can I square both sides of an absolute equation?

I am working on some absolute equation problems like the following: $$\begin{align} & {|x-4|} \lt 1 \\ & 1 \le |x| \le 4 \\ & |x+3| = |2x+1| \end{align}$$ Now, for both of these ...
1
vote
1answer
42 views

Evaluating Absolute Value Expression Within Ranges

I am trying to evaluate an absolute value expression but I am struggling to know whether to place a (+) or a (-) on each expression when evaluating each interval. For example, is there a quick ...
1
vote
1answer
64 views

Is this absolute value notation or something else?

In this document, in Figure 1 (second to last page) there are several uses of $\| \;\;\|$: Is this another notation for absolute value, or is this a notation for something to do with ...
1
vote
0answers
66 views

Banach spaces over complete fields with their own absolute value

Let $F\hspace{.03 in}$ be a field, and let $E\hspace{.03 in}$ be an ordered subfield of $F$. Let $\;\; |\hspace{-0.03 in}\cdot\hspace{-0.03 in}| \: : \: F \: \to \: E \;\;$ be such that for all ...
1
vote
2answers
96 views

How to prove this max absolute value equation?

How to prove this equation? $$\max(|x_1-x_2|,|y_1-y_2|) = \frac{\left|x_1+y_1-x_2-y_2\right|+\left|x_1-y_1-(x_2-y_2)\right|}{2}$$
2
votes
3answers
86 views

Question about absolute value

In $\mathbb{R}$, I know that \begin{equation*} |x|= \begin{cases} x&\mbox{$x\geq0$}\\ -x&\mbox{$x<0$} \end{cases} \end{equation*} What's the $|\cdot|$ in $\mathbb{R}^d$? Is it ...
3
votes
5answers
103 views

How is it, that $\sqrt{x^2}$ is not $ x$, but $|x|$?

As far as I see, $\sqrt{x^2}$ is not $x$, but $|x|$, meaning the "absolute". I totally get this, because $x^2$ is positive, if $x$ is negative, so $\sqrt{y}$, whether $y = 10^2$ or $y = -10^2$: $y$ is ...
0
votes
2answers
47 views

Is this a correct way to express $\left|f(x)\right| \leq \left|x\right|^9$?

If $\left|f(x)\right| \leq \left|x\right|^9$, then, is it correct to say that $f(x) \leq x^9$ and $f(x) \geq -x^9$ ? If it is not, could someone explain why? Thank you.
2
votes
2answers
124 views

Under what conditions is the identity |a-c| = |a-b| + |b-c| true?

As the title suggests, I need to find out under what conditions the identity |a-c| = |a-b| + |b-c| is true. I really have no clue as to where to start it. I know that I must know under what ...
6
votes
3answers
512 views

How prove this inequality: $\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|$? [duplicate]

Let $x_{1},x_{2},\cdots,x_{n}$ be real numbers. Show that $$\sum_{i,j=1}^{n}|x_{i}+x_{j}|\ge n\sum_{i=1}^{n}|x_{i}|.$$ I think this problem may be solved using nice methods, but I can't find ...
0
votes
1answer
74 views

Prove: If $\lim_{n\rightarrow\infty}|a_n| = 0$, then $\lim_{n\rightarrow\infty}a_n = 0$

Using the squeeze theorem, prove the following: If $\lim_{n\rightarrow\infty}|a_n| = 0$, then $\lim_{n\rightarrow\infty}a_n = 0$. Let $f(x) = |a_n|$. Let $g(x) = a_n$ such that $\forall x : ...
0
votes
3answers
443 views

I need help finding the x intercept of an absolute value equation.

$y= |2x-3| + 2x +6$ Find the $x$ intercept. (P.S.: In my Algebra teacher's answer document it says that there is no $x$ intercept for this equation. I'm confused as to why that is. I keep ...
1
vote
0answers
65 views

Basic question about $p$-adic expansions

I was recently introduced to the $p$-adic numbers, and have been asked to show that for $n > 0$, the $p$-adic expansion of $\frac{1}{1 - p^n}$ is $\sum_{i=0}^\infty p^{in}$. Could someone tell me ...
2
votes
4answers
68 views

Need help on proofs

I'm not familiar with proofs and I dunno if I am doing them right. Here's one from Spivak's Calculus: 14. (a) Prove that |a| = |-a| (The trick is not to become confused by too many cases.) So I did: ...
1
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3answers
72 views

Value and simplify

I want to find the value and simplify square root 36 ? Square root of 36 is 6 But I would know how to find the value and simplify it .
2
votes
1answer
28 views

Finding x-intercept of absolute values

What is the x-intercept of $|x-12|+8$? I don't know how to solve it with the absolute values and whether there will be two answers.
0
votes
1answer
2k views

Chain rule and the derivative of absolute value functions

Is it possible to use the chain rule to calculate the derivative of $|x^4|$ and $|x|^4$ in $x=0$? Does the derivative to these functions exist in $x=0$?
1
vote
3answers
129 views

Prove that: $\left | 2x-y-4 \right |\geq 4\sqrt{2}+4$

Let $x,y\in \mathbb{R}$ know that $4x^2-9y^2=36$ Prove that: $$\left | 2x-y-4 \right |\geq 4\sqrt{2}+4$$
3
votes
3answers
140 views

Finding the derivative of $|x|^4$ using the chain rule.

I am presented with the following task: Can you use the chain rule to find the derivatives of $|x|^4$ and $|x^4|$ in $x = 0$? Do the derivatives exist in $x = 0$? I solved the task in a rather ...
0
votes
0answers
79 views

Finding the number of integral values of a for which the inequality $3- |{x-a}|>x^2$ is satisfied by at least one negative value of $x$.

Finding the number of integral values of a for which the inequality$$ 3- |{x-a}|>x^2$$ is satisfied by at least one negative value of $x$.Here's a duplicate post ->{{Determine all the values of ...
1
vote
1answer
62 views

Help with showing that $|x+y|=|x|+|y| \longleftrightarrow xy\ge0 $

I need to prove that $|x+y|=|x|+|y| \longleftrightarrow xy\ge0 $. I proved before that for any $x,y\in R$ holds $|x+y|\le|x+y|$ and I thought maybe it could help me with my arguments to show what I ...
0
votes
0answers
26 views

Questions about $|f(1+a+bi)|<|f(1+a)|$

Let $a,b >0$ and $|*|$ denote the absolute value. Let $f(z)$ be a realvalued analytic function defined for $Re(z)>1.$ For any $a,b$ we have $|f(1+a+bi)|<|f(1+a)|$. Some questions : $1)$ If ...
1
vote
1answer
211 views

Finding domain of a rational function

Find the domain and graph: $$f(t)=\frac{-t}{|t|}$$ My book says to define it piecewise. My questions: $\mathbf{1)}$ Do all rational functions have to be defined piecewise, or just this ...
0
votes
1answer
65 views

A simpler proof for an equality regarding sums of absolute values

Question Let $I$ be a set of indices, and for all $i \in I$, assume $$0 \le {a_i},{b_i},{c_i},{u_i},{v_i} \le 1 \;,$$ with $u_i+v_i=1$. Assume that $L$ is an upper bound on both ${\sum\nolimits_i ...
3
votes
1answer
101 views

Indicating when $|x + y + z| = |x| + |y| + |z|$ holds

This is a problem from Spivak's Calculus $3^{rd}$ ed., Chapter I, Problem $12$(vii) Indicate when $|x + y + z| = |x| + |y| + |z|$ holds, and prove your statement. My attempt: Clearly, $|x + y + ...
0
votes
2answers
222 views

Absolute value of polynomial

I don't seem to grasp why $$|1+6ωi-9ω^2| = 1+9ω^2, ω\gt 0$$ Where does the $6ω$ go? I'm thinking of $|6ωi|=6ω$.
2
votes
2answers
111 views

An Inequality Involving $\min(x, y)$

The following problem is from Spivak's Calculus (4th ed., pg. 18): Prove that if: $|x-x_0| < \min(\frac{\epsilon}{2(|y_0|+1)}, 1)$ and $|y - y_0| < \frac{\epsilon}{2(|x_0|+1)}$ then $|xy-x_0 ...
1
vote
1answer
255 views

Maximum of an absolute value complex function

I'm working my way through Marsden's Basic Complex Analysis book and I can't solve this problem. It's problem 23 of section 1.2 if that helps. Let $a$ be a complex number, find the maximum of ...
3
votes
2answers
187 views

Complex number with z to the power of 4

I have to find all $z\in C$ for which BOTH of the following is true: 1) $|z|=1$ 2) $|z^4+1| = 1$ I understand that the 1) is a unit circle, but I can't find out what would be the 2). Calculating ...
0
votes
1answer
73 views

Matrix integral of absolute exponential item

If $A=(a_{ij})$ is an $n\times n$ symmetric positive matrix, is it possible to calculate the following matrix integral? $$\int_{0}^{\infty}\left | e^{-A(t+1))}-e^{-At)} \right |\mathrm dt,$$ where ...
1
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1answer
2k views

Normal distribution with absolute value

I am new to the normal distribution topic. While I have understood and solved various different kind of questions, the normal distribution questions with absolute value, are the ones I have no idea ...
0
votes
2answers
101 views

How to solve $|2x +1|< 1/4$?

How do you solve $$|2x +1|< \frac{1}4$$
1
vote
3answers
159 views

Solve the equation : $x^2 − 6 |x − 2| − 28 = 0$

The following is an absolute value quadratic equation that I want to solve: $$x^2 − 6 |x − 2| − 28 = 0$$ Here is what I did : $x^2 − 6 |x − 2| − 28 = 0$ $x^2 − 6 |x − 2| − 28 = 0$ ...
0
votes
2answers
82 views

Prove $|x+1|\leq 4$ implies that $-4\leq x\leq 2$.

How do I prove that if $x$ is a real number, then $\lvert x+1 \rvert\leq 3$ implies that $-4\leq x\leq 2$. EDIT: $\lvert x+1 \rvert\leq 4$ should be $\lvert x+1 \rvert\leq 3$
2
votes
4answers
136 views

Prove that $||x|-|y|| \leq |x-y|$ [duplicate]

$||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ In Principles of MA(Rudin), the author said one sees easily that $||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ (p.88, Rudin) from the triangle ...
0
votes
1answer
99 views

Determine all the values of the parameter $a$ for which the inequality $3-|x-a|>x^2$ is satisfied by at least one negative $x$.

I wanted to know, how can I determine all the values of the parameter $a$ for which the inequality $3 - |x-a| > x^2$ is satisfied by at least one negative $x$. I tried for $x<a, |x-a|=-(x-a)$ ...
1
vote
3answers
121 views

Inequalities - Absolute Value

$$|2x-1| \leq |x-3|$$ Answer is $$-2 \leq x \leq \frac43$$ My Question is HOW?
1
vote
2answers
56 views

Finding $x$ from inequality: $\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}$

Find $x$ in $\mathbb{Z}$ satisfying this inequality: $$\left | \frac{3^n + 2}{3^n + 1} - 1 \right | \le \frac{1}{28}.$$ I tried something, but I don't think it's correct. $$-\frac{1}{28} ...
-1
votes
1answer
478 views

Describe the set of points on the complex plane…

Describe the set of points on the complex plane for which $|z-2| + |z+2|=4$... So, I know you can solve this instantly, just by using definition, but I want to do it the long way.. So, $$|x- i*y ...
2
votes
5answers
182 views

Prove the triangle inequality [duplicate]

I want to porve the triangle inequality: $x, y \in \mathbb{R} \text { Then } |x+y| \leq |x| + |y|$ I figured out that probably the cases: $x\geq0$ and $y \geq 0$ $x<0$ and $y < 0$ $x\geq0$ ...
1
vote
2answers
768 views

Proof the maximum function $\max(x,y) = \frac {x +y +|x-y|} {2}$ [duplicate]

I want to prove the maximum function max: $\mathbb{R} \rightarrow \mathbb{R}$, which is defined by $$\max(x,y) = \begin{cases}x, \text { if } x \geq y , \\ y, \text { if } x < y \end{cases}$$ ...