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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

2
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2answers
27 views

Number of solutions of an equality with absolute value operator

Consider: $$ \left|\left|\left|x-1\right|-2\right|-4\right|=4 $$ What is the number of solutions for this equation? This one was particularly easy to me. If first observed that if this inequality ...
3
votes
4answers
53 views

Suppose $|z|\ge 2$, Prove $|z^8+135|\ge121$.

Suppose $|z|\ge 2$, Prove $|z^8+135|\ge121$. My work: $|z^8+135|=\sqrt{(z^8+135)(\bar{z}^8+135)}=\sqrt{|z|^{16}+135(z^8+\bar{z}^8)+135^2}\ge\sqrt{2^{16}+135(z^8+\bar{z}^8)+135}$ For the last term, ...
0
votes
1answer
14 views

Continuity of product in absolute value

Let $F$ be a field, and $\| \cdot\|$ be an absolute value (or norm). I want to prove that with this norm, multiplication is continuous map on $F$ in the sense: for $x_0,y_0\in F$, and any $\...
0
votes
3answers
36 views

How can $|n_1 - n_2| + |n_3 - n_4|$ be equal two different formulas

I have a formula to be calculated such as; $|n_1 - n_2| + |n_3 - n_4|$. I want to calculate it with two tuples of values such as; $(n_1,n_2)$ and $(n_3,n_4)$ When I try this method, it is equivalent;...
1
vote
2answers
37 views

Prove using the triangular inequality that: $|a+b| \geq |a| - |b|$

How can I prove using the triangular inequality that: $$|a+b| \geq |a| - |b|$$ I already proved it by considering all 8 possible scenarios (like a>b and b=0 ... etc) However I couldn’t manage to find ...
1
vote
3answers
52 views

Is it valid to take $|- \infty | = \infty$?

Is it valid to take $|- \infty | = \infty$? or is the absolute value e.g. not defined for infinity? Particularly, if one wishes to argue that operator $f(x)=x$ is not bounded below on $\mathbb{R}_{-...
3
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2answers
43 views

Natural logarithm with absolute value: Can I cancel the absolute value?

I was calculating basic rational integrals and came up with this kind of problem. I have this expression: $$2\ln|x|$$ I can re-write it down like that: $$\ln{x^2}$$ and thus cancel the modulus. ...
0
votes
0answers
30 views

Rewriting a $\max\left\{0,\dots\right\}$ function in order to integrate the function more properly

Yesterday I asked a question about a certain integral. In the integral is the term: $$\max\left\{0,\left|\text{n}\cdot\sin\left(2\pi\cdot x\cdot t-\frac{\pi}{2}\right)\right|-2\cdot\text{z}\right\}\...
0
votes
2answers
60 views

Solve for $x: |x+1| - |x-1| = 2$ [closed]

$$|x+1|-|x-1|=2$$ Well, what would be a very easy-to-follow step-by-step solution to this problem? I really don’t understand anything about the steps needed to successfully solve equations (and ...
3
votes
1answer
59 views

Find a set of points in the given complex plane

Here's the Question: Find a set of points in the complex plane that satisfies: $$|z-i|+|z+i| = 1$$ Now from triangle inequality I found: $$|z+z+i-i|=|2z|\geq1 $$ Which refers that there's no ...
-2
votes
1answer
35 views

Is it possible to give a purely syntactic proof of : $|x| <N$ is equivalent to $ x>-N \text{ AND } x<N$?

In the document accessible at the following link [1]: https://i.stack.imgur.com/P2AvU.jpg, I've tried to explain through purely logical means ( mainly DeMorgan's law) why a $<$ absolute value ...
0
votes
1answer
30 views

Max and Min Inequality

I have to show that for $a,b,c,d \in \mathbb{R}$ $$|a\vee b - c\vee d| \leq |a-c|\vee |b-d|$$ I know this can be showed using cases, but I need help with a proof that doesn't involve cases. I found ...
5
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3answers
103 views

Prove that $\neg \exists x$ such that $ \lvert x-1\rvert+\lvert x+1 \rvert <1$ [duplicate]

In Spivak Calculus, chapter 1 question 11 vi. asks the reader to find all numbers $x$ for which $\lvert x-1\rvert+\lvert x+1 \rvert <1$. Intuitively speaking, it is quite obvious that there is no ...
2
votes
2answers
48 views

Need help with a proof about absolute value

Prove $||x| - |y|| \le |x-y|$ Here's my attempt of the proof: Since $x-y = |x-y|$ or $x-y = -|x-y|$, then $-|x-y| \le x-y \le |x-y|$. Also, $|x| = |-x|$ and $|y| =|-y|$, so we have that $|x|-|y| \...
1
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1answer
11 views

Difference of any two elements of a sequence being less than some number?

Considering a finite sequence of real numbers $x_0, x_1, ..., x_n$, which has no particular order, could I write a single statement that says the difference between any two elements of the sequence is ...
-1
votes
1answer
16 views

Unsure How to Proceed with Proof Related to Algebraic Limit Theorem

I'm independently studying Stephen Abbott's Understanding Analysis and am trying to follow the proof for $(b_n) \rightarrow b$ implies $(\frac{1}{b_n}) \rightarrow (\frac{1}{b})$ Specifically, I'm ...
1
vote
3answers
58 views

Is $|z^2 + 1| > |z|^2 - 1$?

I am often confused while using complex number formula involving comparisons. It is known that $|z^2 - a^2| > |z|^2 - a^2$. But is $|z^2 + 1| > |z|^2 - 1$? Where $z$ is a complex number. Also,...
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0answers
18 views

How Absolute changes power rules?

I have a question about power rules, we have f(m,n) : $\ a^{(m-n)}= \frac{a^m}{a^n}$ which is separable. What about: $\ a^{|m-n|}= ?$ Is it separable? I want f(m,n)=f(n,m) Thanks.
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0answers
22 views

Splitting expected value of absolute value

The following expected value is part of a limit I'm trying to evaluation (for the Lyapunov CLT). My question is essentially is this a valid approach, I have the expected value of the absolute value of ...
2
votes
1answer
37 views

1st order differential linear equation, question on absolute value

I'm trying to find the general solution to this equation: $$x \frac{dy}{dx}+3(y+x^2)=\frac{\sin(x)}{x} $$ Standard form puts it like this: $$\frac{dy}{dx}+\frac{3}{x}y=\frac{\sin(x)-3x^3}{x^2} $$ To ...
1
vote
3answers
33 views

Prove or disprove $|a| \geq 2|b| \implies |a+b| \geq |b|$

I conjectured that for all real $a, b$, that $|a| \geq 2|b| \implies |a+b| \geq |b|$. I'm using this to try to prove a limit where the denominator is $x^2+xy+y^2$ since I know $|x^2+y^2| \geq 2|xy|$. ...
1
vote
2answers
48 views

Maximum value of absolute sum in a polynomial

I have a great problem from Qvant magazine I can’t solve. Please help! Suppose that for any $-1\leq x\leq 1$, $|ax^2+bx+c|\leq 1$. Find the maximum possible value of $|a|+|b|+|c|$. My attempts: It ...
0
votes
3answers
71 views

Why $f(x)=|\sin{x}-1|=1-\sin{x}$? What if $\sin{x}-1$ is zero?

$g(x)=|\sin{x}-1|+|3-\cos{x}-\sin{x}|+2\sin{x}$ Answer: Above equality is simplified to $$1-\sin{x}+3-\cos{x}-\sin{x}+2\sin{x}=4-\cos{x}$$ $$-1 \le\sin{x}\le1$$ So , I know that $f(x)=|\sin{x}-...
2
votes
3answers
51 views

Inequalities, but working around an absolute value

So what I want to prove is $$ |xy+xz+yz- 2(x+y+z) + 3| \leq |x^2+y^2+z^2-2(x+y+z)+3| $$ for $x,y,z\in \mathbb{R}$, and I'm aware that the RHS is just $|(x-1)^2+(y-1)^2+(z-1)^2|$. Now I'm able to ...
0
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2answers
25 views

Absolute value squared - Meaning

I'm wondering if it's correct to say that the absolute value squared means that the values above $1$ are magnified, whereas the values below $1$ are damped (I'm considering only positive values ...
1
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1answer
19 views

$|(1 + j2\pi fT)^2| = 1 + (2\pi fT)^2$

I'm currently trying to understand why $|(1 + j2\pi fT)^2| = 1 + (2\pi fT)^2$ holds. So far I have: $|(1 + j2\pi fT)^2| = |-4\pi^2 f^2T^2 + j4\pi fT + 1|$. But why does $4\pi fT$ disappear? I know ...
0
votes
0answers
30 views

Is there a symbol that takes the absolute value of each component of a matrix or vector in the linear algebra?

Is there a symbol that takes the absolute value of each component of a matrix or vector in the linear algebra? For example, Abs$(-1,-2)=(│-1│,│-2│)= (1,2)$
1
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3answers
83 views

If |x + y| > |x - y|, then how to arrive at $xy > 0$?

When $\lvert x + y\rvert > \lvert x - y\rvert$, I am aware that we can square both sides to find that $xy > 0$. $x^2 + 2xy + y^2 > x^2 - 2xy + y^2$ $4xy > 0$ $xy > 0$ However, I'm ...
0
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0answers
17 views

Multiplicative Inverses with Equal Absolute Value

If algebraic systems may contain unequal elements of the same sign that 1) function as multiplicative inverses and 2) have the same absolute value, are there examples of such systems and ...
-2
votes
2answers
31 views

how to find a and b when there is absolute value inequalities

I was given a number line which cords were $x>5$ and $x<-7$, and I was given this equation $|x-a|>b$. I have no idea how to find the value of $a$ and $b$. I tried different ways of solving ...
0
votes
3answers
38 views

how to solve absolute value for a and b where x is some value

I am beginner at this stage with absolute value, suppose i have |x+a|=|x-b| and where as one of the x solution is 6 and the other is 0. Now how am i supposed to find a and b.
0
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3answers
26 views

Difference between the equation inequalities and absolute value inequalities

Using symbol lab I put this in $|x+4|\le |2x+10|$ and the answer I get is $x \le -6$ or $x\ge -14/3$, but when I manually worked out it was $\;x \ge -6\;$ or $\;x\ge -14/3$. My working out is in the ...
5
votes
5answers
209 views

limit and absolute absolute value problem

$$\lim_{x \to -2} \frac{2-|x|}{2+x}$$ If I calculate the left and right-hand limit I get different results. Left hand side: $$\lim_{x \to -2^-}\frac{2+x}{2+x}=1$$ Right hand side: $$\lim_{x \to -2^+...
0
votes
1answer
32 views

How do I solve $|-2x^2+1+e^x+\sin x| = |2x^2-1|+e^x+|\sin x|$ where x belongs to [0,2π]?

How do I solve $|-2x^2+1+e^x+\sin x| = |2x^2-1|+e^x+|\sin x|,$ where $x$ belongs to [0,2π]? My book solves it in this way: since RHS is positive, it concludes that $1- 2x^2 \ge 0$ and $ \sin x \ge 0$....
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0answers
25 views

On the Newton Polygon for $p-$adic Power series

I'm studyng a Book about $p-$adic numbers, and I have troubles with a "degenerate" case of a Newton polygon. Let $f(X)=\sum a_{i}X^{i}\in\mathbb{Q}_{p}[\![X]\!]$, we define the Newton poligon of $f$ ...
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1answer
35 views

What does $\|u\|^2_2$ mean?

Given a vector $u = (x, y, z)$ what is $\|u\|_2^2$ ?
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2answers
48 views

Prove by cases that $|x|≤R \iff -R≤x≤R$

Prove by cases that: $$|x|≤R \iff -R≤x≤R$$ $R$ is defined as $R≥0$. I consider the two relevant cases to be $x≥0$ and $x<0$. However, for the latter, if $x<0$ then $|x|=-x$. This yields $-x≤...
0
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1answer
36 views

Using the order axioms of $\mathbb{R}$ to prove the semi-definite positivity property for the absolute value?

How can I use the order axioms of $\mathbb{R}$ to prove the semi-definite positivity property for the absolute value: For all $x \in \mathbb{R}, |x|\geq0$ and $|x|=0$ if and only if $x=0$?
0
votes
1answer
36 views

How do you express a fraction with a modulus as the numerator and denominator as a piece-wise function? [closed]

Could somebody please show me how to write it as a piece-wise function? $$g(x)=\frac{|x^3+x-2|}{|x|}$$
0
votes
1answer
21 views

rational function with absolute values

How can i write a rational function with absolute values as a piecewise function, for example $$f(x)= \frac{|x+1|}{|x+2|}$$
0
votes
1answer
23 views

Product of real solutions of the equation [closed]

Product of real solutions $x$ of the equation: $x^2+4|x|-4=0$ ? $4(2\sqrt2-3)$ : Is it the correct answer?
0
votes
0answers
39 views

Why is $\mathbb{C}$ over $\mathbb{R}$ considered ramified?

For a number field $K/\mathbb{Q}$, we say that a finite place of $Q$ is ramified if there exists a valuation $v_{p_i}$ in $K$ lying over $v_p$ such that it is ramified in the sense of the associated ...
3
votes
1answer
59 views

Solving $\lvert z\rvert^2+iz+i=0$

I want to solve: $\lvert z\rvert^2+iz+i=0$ Let $z=x+iy$ $$\implies(x^2+y^2)+i(x+iy)+i=0 \\ \iff x^2+y^2-y+ix+i=0 \\ $$ Comparing left and right side:$$\iff (x^2+y^2-y)+i(x+1)=0+0i$$ $$\implies x+1=...
3
votes
2answers
54 views

$||x-1|-|x+2||=p$ find p for which the equation has one solution

Consider the equation $||x-1|-|x+2||=p$ Find the value of $p$ for which the above equation has one solution.
0
votes
2answers
48 views

Inequality related to a bijection $x\mapsto |x|^{-2}x$

Let $x,x'\in \mathbb{R}^d$ with usual norm. \begin{equation} \frac{|x-x'|}{(1+|x|)(1+|x'|)} \leq \left|\frac{x}{|x|^2}-\frac{x'}{|x'|^2}\right| \end{equation} I have read this inequality, however, ...
1
vote
2answers
66 views

How to derive that $|y-z| - |x-z| \le |x-y|$

So I am reading a derivation and I came to a point where they reach this point: $$ \text{Something} = |y-z| - |x-z|.$$ Then they continue, and say, that from triangle inequality $|y-z| - |x-z| \le |x-...
1
vote
2answers
36 views

For which values of $a$ we will get two different roots?

In given the following system of equations: $$ |x-1| > 2x+2 $$ $$ x^2 + ax + a -1 = 0 $$ For which values of $a$ we will get two different roots?
4
votes
4answers
73 views

Solve the equation |x-1|=x-1

Solve the equation:$|x-1|=x-1$ My solution: Case 1 :$ x\ge1$, Hence $x-1=x-1$, therefore infinite solution Case 2 :$ x<1$, Hence $1-x=x-1$,$x=1$, hence no solution But the solution i saw ...
0
votes
1answer
21 views

Modulus (absolute value) function

Find all values of a where 'a' belongs to all real numbers, for the equation a³ + a²|a + x| +|a²x + 1| = 1 has not less than four different solutions which ...
0
votes
1answer
33 views

taking the absolute value of complex numbers to an arbitrary power [closed]

I need $|\frac{i^{n}}{n}|$ and I have seen the problem simplified to $\frac{|i^{n}|}{n}$ and I am confused by this as isn't $\frac{1}{n}$ the coefficient of i so we could just square it and take the ...