For questions about or involving the absolute value function.

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3
votes
1answer
54 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
32 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
58 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 ...
13
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 ...
0
votes
0answers
35 views

Epsilon delta limit to show that [on hold]

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
votes
2answers
46 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?
-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
48 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
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) ...
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 ...
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 ...
3
votes
3answers
300 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 $ ...
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
1answer
91 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
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 ...
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.
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 ...
2
votes
2answers
86 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 ...
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, ...
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.)\ ...
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 ...
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
37 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 $x\le2.$ ...
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 ...
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.
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" ...
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 ...
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 ...
0
votes
1answer
22 views

How are the following inequalities concluded based on this first one?

$$I-\frac{\epsilon}{3} \leq s(f,T) \leq \underline{I} \leq \overline{I}\leq S(f,T) \leq I+ \frac{\epsilon}{3}$$ from this, the following is concluded, but how? $$1.\ \ \ 0 \leq |I-\underline{I}|\leq ...
1
vote
0answers
17 views

Identifying a real parameter in an equation

I'm not really sure how to go about this problem, as I've never encountered anything similar before. I'm supposed to find all the values $m$ for which the following equation has $3$ distinct real ...
2
votes
2answers
25 views

$ 2\log ^2_{4}(|x+1|)+\log_4(|x^2-1|)+\log_{\frac{1}{4}}(|x-1|)=0$

Find the sum of solutions to: $$ 2\log^2_{4}(|x+1|)+\log_4(|x^2-1|)+\log_{\frac{1}{4}}(|x-1|)=0 $$ I'm not sure about what to do with the absolute values, how can I get rid of them? Should I solve ...
1
vote
0answers
60 views

How $|x|<a\implies a>0$

The title is not exactly what I'm asking, so sorry for that. I was doing a problem in my mathematics text book. It is given that $|x|<a$, I thought if $a=2$ then we can put $x=1$ but what if ...
5
votes
1answer
38 views

Is every non-archimedean absolute value on a number field equivalent to a $|\cdot|_{\mathfrak{p}}$?

Let $K$ be an algebraic number field, i.e. a finite field extension of $\Bbb{Q}$. I would like to prove that every non-archimedean absolute value on $K$ is equivalent to $$ |x|_{\mathfrak{p}} := ...
1
vote
0answers
20 views

Question on valuation axioms - Relating to $\mathbb{R}$

In an earlier thread, I asked if there was a standard generalization of the absolute value of $\mathbb{C}$ that could be placed on a field, but might not take values in $R_{\geq 0}$. What somebody ...
1
vote
2answers
70 views

Solutions for $|x^2-5x+2|=4$

Problem: Find all values of $x$ such that $|x^2-5x+2|=4$ The only way I can see to solve this would be to square both sides of the equation so as to eliminate the modulus sign. However, that ...