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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

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3answers
72 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}-...
1
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3answers
139 views

How to prove this inequality $x,y\in\Bbb R$, $|x|<1,|y|<1$ show that $\bigg|\frac{x-y}{1-xy}\bigg| < 1$ (and similar ones)

I have to show that the inequality below is true, i tried some thing but got stuck, i tried to eliminate the absolute value $-1<\frac{x-y}{1-xy}<1$ and then solve for $x$ and $y$ with no luck......
82
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4answers
125k views

Reverse Triangle Inequality Proof

I've seen the full proof of the Triangle Inequality \begin{equation*} |x+y|\le|x|+|y|. \end{equation*} However, I haven't seen the proof of the reverse triangle inequality: \begin{equation*} ||x|-|...
2
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3answers
64 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 ...
2
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3answers
1k views

Proving a property of modulus function by exhaustion: $\forall\ x \in\mathbb{ R} : |xy| = |x||y| $

I would like some clarification of a quick proof of the properties of the modulus function to make sure I'm doing the right thing. $$\forall\ x \in\mathbb{R} : |xy| = |x||y| $$ If I let $ x,y \in\...
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2answers
35 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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2answers
781 views

When will equality holds in reverse triangle inequality?

Prove the reverse triangle inequality :$|z\pm w|\ge||z|-|w||$ for all $z, w \in \mathbb C$, with equality holds if and only if either $z$ or $w$ is a real multiple of another. I have proved the ...
3
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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=...
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1answer
20 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 ...
1
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0answers
36 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$(-1,-2)=(│-1│,│-2│)= (1,2)$
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3answers
87 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 ...
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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 ...
0
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3answers
39 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.
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1answer
24 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?
-1
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2answers
329 views

Sketching a set of complex numbers and deducing the value of $|z +1 - i|$ for such numbers

The point $P$ represents the complex number $z$. a) Given that $\arg(\frac{z-2i}{z+2}) = \frac{\pi}{2}$ , sketch the locus of $P$. Ok so I've sketched this and this is what it looks like : b) ...
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2answers
34 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 ...
5
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5answers
213 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^+...
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3answers
31 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 ...
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3answers
2k views

limit of absolute value

$$ \lim_{x \to 0} \frac{\lvert2x-1\rvert - \lvert2x+1\rvert}{x} $$ Defining the function piecewise reveals the limit is in fact, continuous about 0 However when I go to solve it in a normal ...
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1answer
34 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
29 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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2answers
3k views

Absolute value and credit card balance

I'm embarrassed to ask this question, but my child has the following homework question: "Use absolute value to describe the relationship between a negative credit card balance and the amount owed." ...
0
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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
49 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
48 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 $x=0$?
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1answer
45 views
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1answer
24 views

rational function with absolute values

How can i write a rational function with absolute values as a piecewise function, for example $$f(x)= \frac{|x+1|}{|x+2|}$$
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0answers
45 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
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2answers
61 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.
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2answers
49 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
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2answers
67 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-...
4
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4answers
77 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 ...
1
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2answers
37 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?
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1answer
36 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 ...
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3answers
59 views
1
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6answers
73 views

How to do $\lvert x - 1 \rvert < \lvert x-3 \rvert$

Solve for $x$ such that $$\lvert x - 1 \rvert < \lvert x-3 \rvert$$ I understand that the question is essentially saying what are the possible values for $x$, centered at $1$ with a distance of $x-...
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3answers
84 views

What happens to an absolute value graph if $|x|$ has a coefficient

I skipped Algebra I in school, and we have Mathematics midterms next week. While going through our review packet, I noticed graphing absolute values, something I had never seen before. I've figured ...
2
votes
2answers
184 views

Absolute value of numbers

The absolute value of the sum of -5 and twice a number is 19. Find the number. I have a problem with this question because i do not fully understand absolute value and this question is a little trucky ...
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2answers
32 views

Absolute value with factors in it [closed]

$$ \lvert1+x(1-x)\rvert< \lvert1-x\rvert$$ How do I solve for $x$? I'm having a hard time finding the intervals.
2
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1answer
36 views

When is the polynomial $P(|x+y|)$ total differentiable?

If $P \in \mathbb{R}[x]$ is a polynomial, under which sufficient condition is the function: $$f: \mathbb{R}^2 \to \mathbb{R}: f(x,y) = P(|x+y|)$$ total differentiable? So for a function to be total ...
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1answer
30 views

Find $x+y$ for the given conditions

Here's a question that was asked in the International Kangaroo Math Contest 2017. Since I get a little confused when I solve equations having the absolute function, so I couldn't get the required ...
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4answers
493 views

Any way to solve $|x-8| = |2-x|-6$ algebraically?

Everything I've tried has given me $x = 2$ (which is obviously incorrect, since $-6 \neq 6$). The actual answer is $x \geq 8$ which I obtained by observing a graph. Would love assistance!
0
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0answers
43 views

When do absolute values $| \cdot|$ retain inequalities? What about for norms?

Can I substitute stuff inside $| \cdot|$ and retain inequalities? Example: want to make $$ \left|\frac{m}{n} - 1 \right| < \epsilon, \quad \epsilon > 0; \quad n,m \in \mathbb{N}$$ Could I say:...
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2answers
61 views

Why can I state $ \left| 3-y \right|=e^{ -4t }$ iff $3-y=e^{ -4t } $?

I have this differential equation problem. $$ \frac { dy }{ dt } =\quad 12-4y,\quad y\left( 0 \right) \quad =\quad 0 $$ Walking through my steps. $$ \frac { dy }{ dt } =\quad -4\left( y-3 \right) ...
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5answers
93 views

Method to solve $|x| + |2-x| \leq x+ 1$

Even if it seems really easy, I'm struggling to solve $$|x| +|2-x |\leq x+1.$$ The book says that $ x \in [1,3] $. I first rewrote as $x+(2-x)\leq x+1$ with $x\geq 0$ and $-x-(2-x)\leq x+1$ with $x&...
2
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0answers
25 views

Max function with absolute values for arbitrary number of arguments

The maximum between two numbers $x$ and $y$ can be easily written as $$ max(x,y) = \frac12\left(x+y +|x-y|\right). $$ We can obviously generalize this to any number of arguments as $$ max(x_1,\dots,...
1
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3answers
82 views

How to you prove $|a|=|-a|$ is true

Let $a \in R$ prove that: $|a|=|-a|$ I am new to proofs so this is my attempt: Case 1: $|a|=|-a|$ $$(a)=-(-a)$$ $$a=a$$ Case 2: $|-a|=|-a|$ $$-(-a)=-(-a)$$ $$a=a$$ Case 3: $|a|=|a|$ $$a=a$$...
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votes
3answers
83 views

$\log_{0.1}(x^4) - 4 \geq 0$ - Solution verification

My question: are these steps ok? To be more precise, is the step where I take fourth root ok? $$\log_{0.1}(x^4) - 4 \geq 0$$ $$\iff$$ $$\log_{0.1}(x^4) \geq 4$$ $$\iff$$ $$\log_{0.1}(x^4) \geq \log_{0....
0
votes
1answer
32 views

Gaussian distribution with absolute value

I am doing my homework about continuous random variable and Im struggling with this problem : Given a Gaussian random variable $T(85,10)$, find $c$ satisfying $\mathbb{P}[|T| < c] = 0.9$. Could ...
1
vote
1answer
49 views

Graphing absolute value inside absolute value equation

How would I graph an equation with an absolute value inside an absolute value? For example, $\left|-\left|x\right|+1\right|+\left|y-2\right|=3$ I tried graphing this on Desmos and it gave me some ...