For questions about or involving the absolute value function.

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0
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1answer
35 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
635 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
199 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
276 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
161 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.
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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
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1answer
54 views

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

Specifically for functions of a complex variable. Are there any rules of thumb?
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0answers
695 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
54 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 ...
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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
274 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
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1answer
142 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
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4answers
83 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
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1answer
888 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
68 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
162 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
235 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
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3answers
573 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
79 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
153 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
93 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
57 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 ...
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3answers
55 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: ...
3
votes
3answers
115 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
60 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 ...
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1answer
71 views
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0answers
142 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
503 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
87 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
131 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 | < ...
1
vote
3answers
158 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
52 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
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1answer
70 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 ...
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0answers
73 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 ...
2
votes
2answers
150 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
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3answers
90 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 ...
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5answers
133 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
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2answers
49 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.
0
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2answers
72 views

Solving an equation with absolute values: $ | 2x - 5| + | 2x - 3 | = m $

Given that the following equation does not have solutions in $\mathbb{R}$, find the value of $m$: $$| 2x - 5| + | 2x - 3 | = m $$ I try to resolve this equation on cases, when $| 2x - ...
2
votes
2answers
136 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
530 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 ...
3
votes
2answers
113 views

Help checking proof of reverse triangle inequality $|x| - |y| \le |x + y|$?

Let $x, y \in \mathbb{R}$. Prove $|x| - |y| \le |x + y|$. By the the triangle inequality $|x| + |y| \ge |x + y|$, hence $$ \begin{align} &|y| \ge |x+y| - |x| \\ &|x+y| \ge |x+y| - |y| \\ ...
0
votes
1answer
80 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
1k 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
78 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 ...