For questions about or involving the absolute value function.

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2
votes
3answers
32 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
91 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
29 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
69 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
34 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
31 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
26 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
26 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
35 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
3answers
143 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
116 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
1answer
44 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
19 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
56 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
36 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
votes
0answers
42 views

Derivative of $g(x) = \frac{1}{2}(||x - x_0||^2 - \Delta^2) $ with $\Delta = x - x_0$

How do you take the derivative of the following equation? Note: Delta = x - x0. This is my attempt to get the solution at shown at the bottom: Solution:
1
vote
2answers
67 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 ...
3
votes
3answers
113 views

Absolute inequality derivation

I have been trying to prove an inequality that I am not even sure if it is even true or not. However I am experiencing great difficulties with this proof. I have an intuition that it is true and have ...
1
vote
2answers
42 views

Why if $B = \{x : |x+1| ≤ 3 \}$ then $B$ equals $[ -4, \infty )$?

I really don't understand why $B$ is from $-4$ to infinity because $x+1 ≤ 3$ $x ≤ 2$ and $-3 ≤ x+1$ $-4 ≤ x$ Shouldn't it be $B = [-4, 2]$?
6
votes
3answers
647 views

Arranging problem: 4 couples, 8 seats in a row… Am I making this too simple?

I am in a prob and stats course... haven't taken one in awhile and would like some help on these two problems. I think I am probably making these a little two simple. Four married couples have ...
1
vote
3answers
42 views

Rotate the graph of a function?

How do I rotate a graph of a function around a point, and show it in the related equation? An example could be $f(x)=\lvert x\rvert$ (absolute Value) and $f(x)=x^2$
1
vote
1answer
30 views

Examine the continuity of function $f(x)=\frac{2x^2-4x}{|x+1|+|x-3|-2}$

Using the definition of absolute value for $$|x+1|=\begin{cases} x+1, & x\ge -1\\ -x-1, & x>-1 \end{cases}$$ and $$|x-3|=\begin{cases} x-3, & x\ge 3\\ -x+3, & x>3 \end{cases}$$ ...
3
votes
1answer
73 views

Is there such a norm on any totally disconnected local field?

Let's set this definition of local field: Let $\mathbb{K}$ be a field and a topological space (non-discrete and totally disconnected). Then $\mathbb{K}$ is called a local field if both ...
0
votes
1answer
20 views

How do I see that $|Re^{it}((Re^{it})^2+4)| \ge R(R^2 - 4)$ for $R \in \mathbb R$ sufficiently large?

How do I see that $|Re^{it}((Re^{it})^2+4)| \ge R(R^2 - 4)$ for $R \in \mathbb R$ sufficiently large ? I've been trying to use triangle inequality and other ways, but I've not come to a conclusion. ...
1
vote
2answers
53 views

Proof of absolute inequality

I am new to proofs and would like some help understanding how to prove the following abs inequality. $$| -x-y | \leq |x| + |y|.$$ I think I should take out the negative in the left absolute value ...
1
vote
4answers
62 views

How to solve inequality for : $|7x - 9| \ge x +3$

How to solve inequality for : $|7x - 9| \ge x + 3$ There is a $x$ on both side that's make me confused...
1
vote
4answers
36 views

Integers $a$ for which the equation $\big\lvert 6\lvert x\rvert -8\big\rvert = a+x$ has the most solutions

$$\big\lvert 6\lvert x\rvert -8\big\rvert = a+x$$ I know this should be done graphically, looking at each case and seeing for which $a$ will it intersect the $x$-axis the most times, but I can't seem ...
3
votes
1answer
33 views

Give an example of a continuous function $f: (-1, 1) \rightarrow \mathbb{R}$ which attains a maximum at 0, but is not differentiable at 0

I need a little help with this exercise: Give an example of a continuous function $f: (-1, 1) \rightarrow \mathbb{R}$ which attains a maximum at 0, but is not differentiable at 0 I thought of the ...
3
votes
3answers
41 views

Plotting $|x - y| \leq 1/4$

I'm trying to plot $|x - y| \leq 1/4$ I've reduced this to $|x - y| \leq 1/4$, and then deduced via plugging in points that there should be two lines, one with y-intercept 0.25 and another with y ...
-2
votes
1answer
54 views

How to integrate a function with a nested absolute value: $|x^2 - 2|x||$? [closed]

I need help with this problem, $$\int_0^4|x^2 - 2|x||dx$$ what should I do with $2|x|$ ?
2
votes
1answer
45 views

Is this a correct way to use triangle inequality

If I have: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq f(x^*)$$ Can I proceed to say: $$|g_1(x) - g_2(x) - (g_1(a) - g_2(a))| \leq |g_1(x) - g_2(x)| - |(g_1(a) - g_2(a))|$$ $$ \implies |g_1(x) - ...
2
votes
1answer
17 views

Proof that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$

I have included a bolded comment in a step in the part of Gouvea's proof of Ostrowski's theorem where he shows that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$ (the ...
5
votes
2answers
66 views

Can every (convex) polygon be described by a single inequality (involving absolute values)?

For example, $$ |x| + |2x + y| + |x + 2y| + |y| + |x+y| < 4 $$ describes an octagon. I'm wondering whether an equation of this form always exist for any convex polygon, and if so, whether there ...
0
votes
1answer
27 views

What would the multifunctional inverse of $F(x)=|x|$ be?

What would the multifunctional inverse of $F(x)=|x|$ be, assuming $x$ is on the complex plane. Also, how would this usually be represented? Note that this won't be a 'true' function. (But assume a ...
-1
votes
1answer
40 views

Doubt with Intervals and Inequalities

This doubt has been bothering me for ages. I would be truly grateful for any help. Problem 1: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals Solution: $x\in(2,4)\cup(4,6)$ ...
1
vote
6answers
125 views

Proof of $ |a-b| = |b-a| $ [closed]

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 ...
-1
votes
3answers
48 views

find the derivative of the integral

Prove that the following integral $F(x)$ is differentiable for every $x \in \mathbb{R}$ and calculate its derivative. $$F(x) = \int\limits_0^1 e^{|x-y|} \mathrm{d}y$$ I don't know how to get rid of ...
2
votes
4answers
49 views

Solve $|1 + x| < 1$

I'm trying to solve $|1 + x| < 1$. The answer should be $ -2 < x < 0$ which wolframalpha.com agrees with. My approach is to devide the equation to: $1+x < 1$ and $1-x < 1$ and then ...
1
vote
1answer
24 views

Has this question on Absolute Value's been asked wrong?

I was going over some basics on Khan's Academy in preparation for a test. To my surprise I got this wrong: Has this been worded wrong? Surely the person farthest from sea level is Howard? This ...
0
votes
4answers
29 views

Soft absolute value

I'm looking for a "soft absolute value" function that is numerically stable. What I mean by that is that the function should have $\mp x$ asymptotes at $\mp\infty$ and behave smoothly in $[-1,1]$. ...
2
votes
2answers
49 views

Solve differential equation $y' = |1.1 - y| + 1$

How can the following differential equation be solved analytically? \begin{equation*} y' = |1.1 - y| + 1, \\ y(0) = 1. \end{equation*} I guess one must rewrite the differential equation piecewise ...
6
votes
5answers
887 views

If three complex numbers $z_k$ have modulus $1$, then $|z_1+z_2+z_3| = |\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$

Our teacher gave us a hard question (according to her, it is pretty hard for our level): Given that $|z_1| = |z_2|= |z_3|=1,z \in\mathbb{C}$, prove that $|z_1+z_2+z_3| = ...
1
vote
4answers
115 views

Can some explain very quickly what $ |5 x + 20| = 5 $ actually means?

I don’t mean to bother the community with something so easy, but for the life of me, I can’t remember how to do these. I even forgot what they were called, and I’m referring to the use of the “$ | $” ...
3
votes
1answer
140 views

mean value theorem sin(b) - sin(a)

It's too much hassle to post it here as latex, to so here's the screenshot. I don't understand why |cos(c)| = 1 Why 1? Why not $\frac {\sqrt{3}}{2}$? Why absolute value assumes the max value a ...