For questions about or involving the absolute value function.

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0
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2answers
26 views

Metric spaces, manipulating the absolute value function.

I have the following problem involving the set $Y$ of infinite sequences that absolutely converge such that, $$\sum_{i=0}^\infty x_i^2 \lt\infty$$ where $x_i$ is the $i$-th term in the infinite ...
14
votes
5answers
2k views

When I was teaching absolute function properties, I suddenly made this question …

I was teaching absolute function properties in a K-12 class. I made this question in my mind. Suppose $f(x)$ is a one-to-one function, and its definition is $f(x)=max\left \{ x,3x\right ...
3
votes
1answer
56 views

Rewriting $|x-10|+|y-5|\leq 7$ so that absolute values disappear - Algebra

Equation 1: $|x-10|+|y-5|\leq 7$ I want to rewrite this equation into equations that do not have the absolute value. $|A|\leq b$ can be written as $A \leq b$ $A \geq -b$ I want to apply the ...
1
vote
1answer
35 views

Complex Conjugation problem using the identity $|x|^2=xx^*$

Show that $$|c|^2= \frac{4k^2}{k^2 +\gamma^2}$$ given (1)$$a+b=c$$ and (2)$$ik(a-b)=-\gamma c$$ This was given in a lecture without proof, so there's probably a very simple way of proving the ...
2
votes
6answers
59 views

Why does $|x_1| = |x_2| \implies x_1 = \pm x_2$

I was doing a 'prove this is not surjective' practice problem and the step leading from my hypothesis, as listed, to the conclusion was not defined. I don't recall being exposed to a situation where ...
0
votes
1answer
52 views

Integral of reciprocal absolute value function

I'm having issues with the integral $$\int_{-1}^1 \frac{1}{|x|}dx$$ Solving it conventionally gives me values such as $\ln 0$ and $\ln(-1)$ which are indeterminate on the real plane. Is there a way to ...
0
votes
0answers
35 views

Epsilon delta limit to show that [closed]

show that $$\left|\frac{28}{3x+1}-4\right| = \left|\frac{12}{3x+1}\right| \cdot |x-2| $$ using $\epsilon$-$\delta$ definition of a limit. I have no idea where to start since the question is not ...
-7
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2answers
47 views

If $a$ is not equal to $0$, then $|a| = -a$ is never true? [closed]

Pleaae explain why the following statement is wrong If $a \neq 0$, then $|a| \neq -a$ . What two concepts are being confusing?
0
votes
1answer
11 views

Normal Distribution $r-1$ th moment with absolute value

I was stuck for this problem whole night and I tried numerical solution using MATLAB and the following result seems hold for x follow normal N(0,1) and for any positive number (not integer only) ...
-1
votes
2answers
52 views

$|f(x)g(x)| = |(f(x)||g(x)|$ [duplicate]

I was wondering if $|f(x)g(x)| = |f(x)| |(g(x)|$ is true all the time as in the case of real numbers. I was not convinced enough that that was true. But I can't think of any counterexample. Thank ...
4
votes
1answer
40 views

Taking out absolute value on the solution to integral equation

I have this equation:$$y=2+\int_2^x (t-ty(t))dt$$ After solving it I got the answer $-\ln|1-y|=\frac {x^2} 2-2$ although the book has the same answer without the absolute value in the logarithm, why ...
3
votes
1answer
49 views

Difficulty in finding the Range of x

$x^2 - | x-2 | + 6 > 0 $ , where x belongs to $R$ I am not sure about my own approach to this ques. I solved it as: $x^2 + 6 > | x-2 |$ , thereafter i got 2 cases Case 1: $-(x^2 + 6) ...
0
votes
0answers
28 views

What is the expected value of the absolute value of a Wiener Process?

I am trying to show that the with a Wiener Process $w(t)$, then $\mathbb{E}[|w(t_1)w(t_2)|] = (\frac{2a}{\pi}) \sqrt{t_1 \cdot t_2} (\cos \theta + \theta \sin \theta)$, given $\sin \theta = ...
0
votes
1answer
23 views

2nd derivative of a functions absolute value

So on wolfram alpha I am told that if $y=y\left ( x \right )$ then $\frac{d^{2}}{dx^{2}} \left | y \right |= \frac{y}{\left | y \right |}y^{''}+2\delta \left ( y \right )y^{'2} $ See it at this link ...
1
vote
0answers
23 views

Least Square Approximation Using Legendre Polynomials

Obtain a fourth degree least squares polynomial for $f(x) = 1/|x|$ over $[-1,1]$ by means of Legendre Polynomials I got stuck when trying the integral over the given interval. Is there another way ...
3
votes
3answers
301 views

Number of real roots

Find number of real roots of the equation $$3^{|x|}-|2-|x||=1$$ My try:I have tried to remove the modulas by assuming x in some intervals and moved the linear part to RHS and drawn the rough graph ...
1
vote
2answers
24 views

Finding first and second derivative of an function with an absolute value

Given the equation $f(x)= |x^2-9|$ where $-4\le x\le 5$, I must find the extremes, as well as the concavities. This I know how to do. The issue is I'm unfamiliar on how to find the first and second ...
0
votes
2answers
53 views

Solve $1<\left(\dfrac{3x^2-7x+8}{x^2+1}\right)\leq 2,\ \ x\in\mathbb{R}$

Solve $1<\left(\dfrac{3x^2-7x+8}{x^2+1}\right)\leq 2,\ \ x\in\mathbb{R}$ options $a.)\ 1<x<6\\ b.)\ 1 \leq x<6\\ c.)\ 1<x\leq 6\\ \color{green}{d.)\ 1\leq x \leq 6}$ I ...
2
votes
5answers
36 views

solve $\dfrac{x^2-|x|-12}{x-3}\geq 2x,\ \ x\in\mathbb{R}$.

solve $\dfrac{x^2-|x|-12}{x-3}\geq 2x,\ \ x\in\mathbb{R}$. options $a.)\ -101<x<25\\ b.)\ [-\infty,3]\\ c.)\ x\leq 3\\ \color{green}{d.)\ x<3}\\ $ I tried , Case $1$ ,for $ ...
2
votes
1answer
92 views

Why is the definition of the absolute value $|x+1|$ the way it is?

In my notebook it is given that for the above function, we would have: $f(x) = {-(x+1), x<-1; (x+1), x\geq-1}$ What I don't get is why did we take $-1$ instead of $0$ as is the case for the ...
2
votes
2answers
87 views

Distribution of minimum absolute value

Consider $K$ independent Laplace variables $X_k, k=1,\ldots,K$, with mean 0 and scale $\lambda$ (so that their PDF is $f(x)=\frac{1}{2\lambda}e^{-\frac{|x|}{\lambda}}$. Let $Y$ be the variable taking ...
0
votes
1answer
20 views

Solving a system of equations with an absolute value term

$x$ and $y$ are two integer numbers and $x \geq y$. The values of $x$ and $y$ are positive or negative integers. When the sum of these two numbers are multiplied by $y$ we obtain $P$ and when the ...
2
votes
2answers
30 views

Explaining why the absolute value of an odd function is even.

For the following: If $f(x)$ is an odd function, then $|f(x)|$ is _____. I said even, because I graphed an odd function and then the absolute value of it and ended up with an even function. The ...
1
vote
1answer
102 views

Quick question about absolute value

Hello I am just having a quick question in the textbook intro to real analysis, during one of the limit examples the author notes, if $$|x-c| \lt 1$$ then $$|x| \lt |c| +1$$ What rules are used to ...
2
votes
3answers
167 views

Find $\int_a^b \sin |x| \, \mathrm{d}x $

How to find the integral $$\int_a^b \sin |x| \, \mathrm{d}x \,?$$ I'm able to obtain definite integral of form $ \int_a^b \lvert\sin x \rvert \, \mathrm{d}x$ but not when the modulus operator is ...
0
votes
5answers
29 views

Roots of Unity: second largest value and absolute value

Consider the $n$th roots of unity $e^{2 \pi i k/n}$ for fixed integer $n \geq 2$ and $0 \leq k < n$. Now I am interested in the second largest value (in absolute value) of the values ...
-3
votes
2answers
92 views

Under what conditions is $|x+y|=|x|+|y|$ true? [duplicate]

What instance that this equation would be true? $|x+y|=|x|+|y|$ Given that $x$, $y$ are elements of real numbers.
3
votes
2answers
547 views

When does the equality hold in the triangle inequality? [duplicate]

Hi guys could you please help me on this question I'm confused. question: when does the equality hold in the triangle inequality: my attempt : $|x + y| \leq |x| + |y|$ this implies $(|x+y|)^2 = ...
3
votes
3answers
4k views

Equality holds in triangle inequality iff both numbers are positive, both are negative or one is zero

How do we show that equality holds in the triangle inequality $|a+b|=|a|+|b|$ iff both numbers are positive, both are negative or one is zero? I already showed that equality holds when one of the ...
0
votes
1answer
21 views

Jargon for maximum/minumum absolute value in a set

Given a group of numbers $-5,-3,1,2$, the maximum is 2, the minimum is -5. What is the mathematical jargon for the maximum and minimum in absolute terms (i.e. -5 and 1 respectively)? Basically, I ...
7
votes
2answers
57 views

Basic absolute value property

Hello all I am wondering if anyone has the correct proof that I should use for Spivak calculus ( chapter 1, question 12 ) that says $$|xy|=|x| \cdot |y|$$ from past times I know it is true , but I ...
3
votes
10answers
396 views

Prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$

I'm trying to prove that if $a,b \in \mathbb{R}$ and $|a-b|\lt 5$, then $|b|\lt|a|+5.$ I've first written down $-5\lt a-b \lt5$ and have tried to add different things from all sides of the ...
2
votes
6answers
149 views

Proof of $ |a-b| = |b-a| $

While working out some intriguing qualities of absolute values for my studies of calculus, I frequently used the formula below. I know that the formula below is clearly correct but how would I prove ...
4
votes
3answers
166 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 ...
2
votes
2answers
66 views

Proving that $x_n\to L$ implies $|x_n|\to |L|$, and what about the converse?

Problem 3. Show that for a sequence $(x_n)$ the following are true: (i) $\lim x_n=0$ if and only if $\lim |x_n|=0$. (ii) $\lim x_n=L$ implies $\lim |x_n|=|L|$. Is the converse true? Prove or ...
3
votes
3answers
81 views

Why is $\max(x, x')$ equivalent to $\frac{1}{2}( x + x' + |x - x' |)$?

Why is it that $$\max(x, x') = \frac{1}{2}( x + x' + |x - x'|)$$ is true? Is it supposed to be obvious? Because it seems to come out of thin air for me. Anyway, I've verified this by plotting it in ...
1
vote
2answers
54 views

Quadratic Absolute Value Equation

Problem: Find all $x$ such that $|x^2+6x+6|=|x^2+4x+9|+|2x-3|$ I can't understand how to get started with this. I thought of squaring both sides of the equation to get rid of the modulus sign, ...
0
votes
4answers
105 views

I have discovered a way to calculate the absolute value (area,volume, etc) of a n-dimentional shape, using it's coordinates only, how do I publish it?

Firstly, I want to preface by saying that I am no experience with the maths community at all, however I did take Maths and Further Maths for my A-Levels. What I have discovered is a way of using ...
3
votes
3answers
39 views

solve $|x-6|>|x^2-5x+9|$

solve $|x-6|>|x^2-5x+9|,\ \ x\in \mathbb{R}$ I have done $4$ cases. $1.)\ x-6>x^2-5x+9\ \ ,\implies x\in \emptyset \\ 2.)\ x-6<x^2-5x+9\ \ ,\implies x\in \mathbb{R} \\ 3.)\ ...
1
vote
3answers
31 views

Trigonometry - log/ln and absolute sign in equations

Will this equation still hold if the absolute sign is being used at different places For example, This trigonometry identity; ...
0
votes
1answer
72 views

Is there a number whose absolute value is negative?

I've recently started to think about this, and I'm sure a couple of you out there have, too. In Algebra, we learned that $|x|\geq0$, no matter what number you plug in for $x$. For example: ...
3
votes
1answer
38 views

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $ I tried $x^2-|5x-3|-x<2$ , case $1$ , $x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\ $ ...
1
vote
0answers
32 views

Integral of $|\cos(ax))|\times e^{-x^2/b}$

I can compute the following integral very easily ($a$ and $b$ are real and positive): $$\int_{-\infty}^{\infty} \cos(ax)\times \frac{1}{\sqrt{\pi b}}\cdot e^{-\frac{x^2}{b}}\,dx = ...
6
votes
0answers
38 views

Lower bound on absolute value of determinant of sum of matrices

I needed to find a lower bound on $|\det(A+B)|$ where $|.|$ is the absolute value operator. Because I was unable to get such a bound so I was trying to guess a bound and prove it. But ...
1
vote
2answers
39 views

Quadratic Absolute Value Inequality

Problem: Find all $x$ such that $|x^2-3x+1|<1$ I can't understand how to get started with this. I've never tried to solve quadratic Inequalities before. At first I thought of working with the ...
5
votes
2answers
45 views

Doubt with Absolute Value Inequality

Problem: Find all values of $x$ for which $\dfrac{|x-2|}{x-2}>0$ My incorrect attempt: Using the definition the Modulus, $|x-2|=x-2$ for all $x\ge2$ and $|x-2|=-x+2$ for all $x\le2.$ ...
1
vote
4answers
117 views

Why is $\sqrt{x^2}= |x|$ rather than $\pm x$? [duplicate]

Shouldn't the square root of a number have both a negative and positive root? According to Barron's, $\displaystyle \sqrt{x^2} = |x|$. I don't understand how.
0
votes
3answers
56 views

Proving that $|a-b|≤|a|+|b|$ [closed]

Can someone prove this to me: $$|a-b|≤|a|+|b|$$ I am in 8th grade and I have this for my homework. Thanks for helping.
0
votes
2answers
47 views

Graph $y=|x+8|+|x-8|$

Graph $y=|x+8|+|x-8|$ I tried to simply this with $$y=(x+8)+(x-8) \implies y=2x,x>0\\ y=(-x+8)+(-x-8) \implies y=-2x,x<0$$ But this looks quite different from the original. I look ...
2
votes
1answer
34 views

How to solve equations containing multiple $|x|$s?

Suppose I have an equation which looks like: $$|x-2| + |2x+1| = 3$$ or, $$|x-1| + |x-3| - |5x-1| = 2$$ How should I solve such problems? What i do is generally a kind of "hit-and-trial" ...