For questions about or involving the absolute value function.

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5
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1answer
30 views

Does $|x^*|=|x|$ in a star ring with an absolute value?

Let $R$ be a star ring with an absolute value. Is it true that $|x^*|=|x|$ for all $x\in R$? Here a star ring is a ring with a function $*:R\to R$ called conjugation such that $(x+y)^*=x^*+y^*$ ...
1
vote
1answer
26 views

$|x^{-1}-y^{-1}|=|x-y|/|x||y|$ in a normed ring

I hit a slight snag when trying to prove that the inverse function $x\mapsto x^{-1}$ on the unit group is continuous in a ring with an absolute value, so I'd like some confirmation that the theorem is ...
1
vote
3answers
26 views

Absolute Value Inequality Including Itself

Given a real number $a$. Will it be correct to use the following inequality in the proof: $$-a\le|a|\le a$$ Although "less" and "greater" parts never actually happen, the whole equation will always ...
0
votes
2answers
78 views

How to solve inequalities with more than one absolute value expression

There are two parts to this question. 1. I'm seeing the correct method to solve these types of inequalities as something to do with "transition points". I don't quite understand this method. How do we ...
1
vote
1answer
41 views

Absolute value of product is less than product of absolute values: $|(1+a_1)(1+a_2)\dots (1+a_n)-1|\leq (1+|a_1|)(1+|a_2|)\dots (1+|a_n|)-1$

For a sequence $a_n\in\mathbb{C}$ I want to show that $$|(1+a_1)(1+a_2)\dots (1+a_n)-1|\leq (1+|a_1|)(1+|a_2|)\dots (1+|a_n|)-1$$ I think I should show this by induction on $n$. For the base case I'm ...
0
votes
1answer
35 views

Prove that $0 \leq \frac{x+|x|}{2} \leq |x|$

$\frac{x+|x|}{2}$ is superior or equal to $0$ but inferior or equal to $|x|$ where the $x$ is a real number. I must prove this by the method of proof by cases. I have no idea one how to begin ...
1
vote
3answers
101 views

Question about an inequality on a proof

I'm stuck on a proof. There's a step that says: $$ \left| \Im\left(\frac{1-e^{i(N+1)x}}{1-e^{ix}}\right)\right| \leq \left| (\frac{1-e^{i(N+1)x}}{1-e^{ix}}) \right|,\quad \text{with } N \in ...
1
vote
2answers
23 views

Module properties

I've got stuck on this problem Let $a$, $b$, $c> 0$ and $x$ a real number such that $$|ax - b| \leq c,$$ $$|bx - c| \leq a$$ and $$ |cx - a| \leq b.$$ Prove that $0 \leq x \leq 2$. What ...
-1
votes
1answer
29 views

Anita Tabacco include zero to N? [closed]

I have two problems actually, first one: Proffesor always grumbled that Anita Tabacco include zero to natural numbers, and proffesor dont include zero to the natural numbers when explain somethink or ...
0
votes
3answers
75 views

Some confusion with absolute value

Today at a math lecture, I solved the equation $|x+1|+|x-1|+|x|=4$ by using elementary arithmetic. But my professor did it a little bit differently: I didn`t pay attention to the teacher's ...
1
vote
1answer
65 views

Solve absolute value inequality

I have to show the inequality $$ \left|\frac{1}{2 + a}\right| < 1. $$ How do I do this? I know that a fraction is less than 1 when the denominator is greater than the numerator, but I cannot ...
0
votes
2answers
65 views

Finding the max absolute value of this analytic function

On the line segment from $z=R, R>0,$ to $z=R+i2\pi$, I want to find the maximum of: $\lvert e^{3z}/(1+e^z) \rvert$. If $z=x+iy$, this is equal to: $$ \lvert e^{3R}e^{i3y} / (1+e^Re^{iy}) \rvert ...
0
votes
2answers
34 views

Absolute Value derivative

What would the following derivative be? $$ \frac{\partial}{\partial x_k} |x_i - x_j| =?$$ Where $$ \frac{\partial x_i}{\partial x_k} = \delta_{ik}$$ For context, what I'm actually trying to do is ...
0
votes
3answers
106 views

The set of all real numbers $x$ such that $\sqrt{x^2}=-x$

The questions goes as The set of all real numbers $x$ such that $\sqrt{x^2} = -x$ consists of a. Zero only b. Nonpositive real numbers only c. positive real numbers only d. all real ...
0
votes
1answer
20 views

why it is not continuous for a absolute value division?

the question is : is y=|x-1|/(x-1)continuous on (-infi, +infi): I am wondering why this equation is not continuous when x = 1 I think when x=1, y will be 1
0
votes
2answers
45 views

Amplitude for sum of sinusoids

I am trying to plot the following function: $$ \max_t \left| \frac{1}{1-r^2} \big( \sin(r \omega_n t) - r \sin(\omega_n t)\big) \right| $$ By inspection, I have determined that the amplitude of the ...
-1
votes
1answer
36 views

Equation with logarithms and absolute values

I have this equation and I want to solve it for $x<0$. $$\frac{\ln|x|}{|x|}=\frac{\ln|x|}{x}$$ According to WolframAlpha, the solution is $x=-1$ but I don't know how to get that. My approach: ...
0
votes
1answer
24 views

How are absolute value of a field $F$ and norm of vector space $F/F$ related?

So I think it's true that a field $F$ and its respective vector space $F/F$ are isomorphic, since they consist of the same elements, and the operations of $F$ (addition,mult.) and $F/F$ (vector ...
0
votes
0answers
17 views

Question related to $F(x) = |x-a_1| + |x-a_2|+ … + |x-a_N|?$

Suppose $a_1 < a_2 < \cdots < a_N $. So $F(x) = S_N - 2S_i + (2i-N)x$ if $a_i < x < a_{i+1}$ with $S_i = a_1 + ... + a_i$. Assume $ a_i < u < a_{i+1} < v $, we have: $F(u) = ...
-1
votes
1answer
23 views

How do you calculate the absolute value of trigonometric functions? [closed]

How do you calculate: $|\tan t|= \sqrt{3}$ The answer to this is required to be in the interval $[0,2\pi]$
3
votes
3answers
110 views

Why does the integral $\int\frac{1}{x+i}dx$ not require the absolute value in the logarithm?

Going through some old calculus exams, I find a solution to an integral via partial fraction decomposition. The solution manual does not perform full decomposition to avoid complex numbers, however; I ...
0
votes
1answer
43 views

How to prove: if $x\in [-3,4]$ then $5\leq |x-3|+|x+2| \leq 7$

So far I have that if $-3 \leq x \leq 4$ then we have that: $-6 \leq x -3 \leq 1$ and $-1 \leq x+2 \leq 6$ So $|x-3| \leq 6$ and $|x+2| \leq 6$ but I'm not sure how to continue the proof.
0
votes
1answer
80 views

Complex Analysis Questions - $|z + 2| + |z - 2| = \sqrt{10}$

I'm just starting in a Complex Analysis course, and I am stuck on a couple questions. The questions are as follows: If $z = a + bi$ is a point on the curve $|z + 2| + |z - 2| = \sqrt{10},$ find ...
-2
votes
4answers
40 views

Solving $|x+2| = 4 + |x-7|$ [closed]

How would one solve the following equation over $\mathbb R$? $$|x+2| = 4 + |x-7|$$
0
votes
2answers
26 views

question about absolute value inequalities

We know that $|a|<b$ implies $-b<a<b$. Would that still hold if $-|a|<b$? That is, would that imply $-b<-a<b$? Thanks
1
vote
4answers
114 views

Compute $\int_{a}^{b}\left|x\right|\mathrm{d}x$

The reason I ask this is because $$\int_{x=a}^{x=b}\left|x\right|\mathrm{d}x$$ gives exactly the same result as ...
0
votes
0answers
29 views

Absolute Value Inequality Proof - Hint needed

I am having difficulty on a problem. If someone could explain where I should start or what I can use to help solve the proof it would help. Prove that if $|(x+2)| \lt 1$ then $|(x^2 +2)| \gt 3$
0
votes
0answers
20 views

Functional Derivative of Complex Absolutes

It is said, that the lowest order complex amplitude equation shows potential dynamics with the functional \begin{align} V[A] = \int\limits_{a}^{b} \text{d}X\left[- \vert A \vert^2 + \frac{1}{2} \vert ...
2
votes
0answers
35 views

absolute values and integals

I have the following integral $$\int_{- \infty}^\infty e^{-|x|} dx$$ and the following two questions (1) Since the preimages $x$ determine the the images $e^{-|x|}$ for nonnegative and negative ...
2
votes
2answers
54 views

Absolute value inequality $3 > |x + 4| \geq 1$

I've just started with absolute value equations and I have a real hard time understanding how to solve this. I got the following question, and I can't make heads or tails out of it. Assume that $x, ...
2
votes
2answers
45 views

How to solve complex equation with same variable on two sides

What is the analytic solution of X for the equation below? $$conjugate(X)= \frac{-2\times A}{B\times X} $$ A, B and X are complex numbers. Would the magnitude of X be given by this? $$ |X| =\sqrt ...
1
vote
3answers
76 views

If {$b_n$} converges to b, then prove that {|$b_n$|} converges to |b|.

Is my proof good or does it need more work? Let $\epsilon$ > 0, we want N s.t. $\forall$ n $\geq$ N $\subset$ ||$b_n$| - |b|| < $\epsilon$. If |$b_n$| $\longrightarrow$ |b|, then $\exists$ N $\in$ ...
1
vote
2answers
43 views

Prove that |a -b| $\geq$ ||a| - |b|| $\forall$ a,b $\in$ $\mathbb{R}$

Here is my proof can anyone check if it needs more detail or if it's good? Since a = a - b + b, then by the triangle inequality, |a| $\geq$ ||a| - |b|| + |b| so that |a| - |b| $\geq$ ||a| - |b|| ...
3
votes
1answer
44 views

Show that $|a+b|=|a|+|b|\Leftrightarrow ab\geq0$

Assume $a, b \in \mathbb{R}$. Show that $$|a+b|=|a|+|b|\Leftrightarrow ab\geq0$$ The triangle inequality says that for $a,b\in\mathbb{R}$, $|a+b|\leq |a|+|b|$. So I belive I can say ...
2
votes
0answers
37 views

Around an inequality

I have a very general question, hopefully not too general. Assume that we have real numbers $a_{ij}, b_{ij}$ $(1 \leq i, \: j \leq n)$ such that $-1 \leq a_{ij}, b_{ij} \leq 1$ for all $i,j,$ for ...
0
votes
1answer
71 views

Need help using the triangle inequality

Use the triangle inequality and the reverse triangle inequality to find an upper bound for the set of all numbers of the form $\left\lvert\frac{x^2-3}{x-2}\right\rvert$ as $x$ ranges over the interval ...
2
votes
2answers
40 views

Solving the absolute value equation $2-3|x-1| = -4|x-1|+7$

$$2-3|x-1| = -4|x-1|+7$$ This is an example from my text book, and I do not understand how they got the answers. Solution: (this is the solution in my textbook) Isolate the absolute value of ...
0
votes
1answer
33 views

If we bound x on an interval, how can we bound |x|?

We are given $a<x<b$, where $a$ and $b$ are constants. This means that $x$ belongs to the interval from $x=a$ to $x= b$ excluding $a$ and $b$. What is the interval that $|x|$ (the absolute ...
0
votes
0answers
31 views

Minimizing the absolute value of the sum, |sum(cx)| ,by linear program

Now I need to use linear program to minimize the absolute value of the sum.The mathematical model of linear program can be expressed as Min |sum(cx)| Subject to |Ax|<=b ...
0
votes
1answer
28 views

Absolute value in Hasse's theorem

The Hasse's theorem says that for an elliptic curve $E$ defined on $\mathbb{F}_p$ where $p$ is a prime number, we have: $|n-(p+1)| < 2\sqrt{p}$ with $n$ the order of $E$. I am wondering why the ...
2
votes
4answers
60 views

Solve $\vert x-2\vert+2\vert x-4\vert\leq \vert x+1\vert$

I was helping someone with abolute values and inequalities and found this question. What is the easiest way to solve this? The only thing I thought of is to add the L.H.S and graph it with the R.H.S ...
-2
votes
1answer
48 views

Solving an absolute value equality without using a graphical method [closed]

How to solve this equality without using a graphical method ? $|1-x|-|x+1|=2$
2
votes
1answer
75 views

Lower bound for norm of sum of vectors

It is common to use: $$\| \sum_{i=1}^n \bar X_i \| \le \sum_{i=1}^n \|\bar X_i \|. $$ Now for 2 dimensions, I know that: $$ \|\bar A + \bar B \|^2 = (\bar A + \bar B)\cdot(\bar A + \bar B)$$ $$ = ...
0
votes
1answer
34 views

Absolute change and percentage change.

This page says- absolute change = new quantity – old quantity. For ex- Example 1: On September 20, a gallon of gas at my usual gas station cost $1.83. On September 30, I noticed that the price had ...
1
vote
1answer
37 views

Absolute Value $|1-(1/x)| = |(1/x)-1|$

Can someone please explain to me how: $$|1-(1/x)| = |(1/x)-1|$$ Im working on a limit problem in my calculus book and I cant seem to understand how they reversed this and it equals the same thing. ...
0
votes
1answer
81 views

Need help with proof with absolute value and complex numbers. [duplicate]

Had some trouble trying to prove the following problem. Prove that if $|z| < 1$ and $|w| < 1$, then $$ \frac{|z-w|}{|1-\overline{z}w|} < 1 $$ Would appreciate some help.
0
votes
2answers
32 views

Polynomial equations with absolute values

How can I solve: $x^2 + 2|x| - 3 = 0$ ? My attempt: $|x| = \frac{3 - x^2}{2}$ $x = \pm \frac{3 - x^2}{2}$ $x^2 \pm 2x - 3 = 0$ The solutions to this 2nd degree polynomial is $x_1 = -3$ $x_2 ...
1
vote
2answers
40 views

absolute value binomial split into two absolute values

$$ |a-b| = |a|-|b| $$ I think I might missing something with absolute values. Can I split a binomial into two separate absolute values like above?
0
votes
1answer
74 views

How to prove triangle inequality in How to Prove It Sec. 3.5 Question 12c?

(a) Prove that for all real numbers $a$ and $b$, $$|a| \le b \text{ iff } -b \le a \le b.$$ (b) Prove that for any real number $x$, $$-|x| \le x \le |x|.$$ (Hint: Use part (a).) (c) Prove that ...
1
vote
4answers
94 views

Prove that $\left|(|x|-|y|)\right|\leq|x-y|$

Prove that $\left|(|x|-|y|)\right|\leq|x-y|$ Proof: $$\begin{align} \left|(|x|-|y|)\right| &\leq|x-y| \\ {\left|\sqrt{x^2}-\sqrt{y^2}\right|}&\leq \sqrt{(x-y)^2} ...