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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

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20 views

Uncoditional formula for unsigned area under a linesegment w.r.t. $y=0$

I'm interested in finding the unsigned area under a line segment with respect to $y=0$. The line segment is defined by start point $(s_x, s_y)$ and end point $(e_x, e_y)$ Without loss of generality, ...
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1answer
38 views

For which $a \in \mathbb{R}$ is $x^2 + 4|x-a| - a^2 \geq 0$ true for all $x \in \mathbb{R}$? [on hold]

As the title says, I have a problem answering the question: for which $a \in \mathbb{R}$ is $$ x^2 + 4|x-a| - a^2 \geq 0 $$ true for all $x \in \mathbb{R}$?
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0answers
14 views

abs value equation with 3 solution, find value of constant in equation [on hold]

If $||4x+5|-b|=6$ is an equation in $x$ with 3 distinct solutions, find the value (s) of rational number $b$
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1answer
23 views

Linear equation with absolute value.

Find $x$ if $|x-|3x+1||=4$ I got 4 values of $x$ out of which 2 are obsolete... Why so??
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2answers
38 views

How to plot absolute value graphs?

I have to plot graph of $$f(x)=|x|+2|x-1|+|x-4|$$ See I know graphs of individual $|x|,2|x-1|,|x-4|$ But how can I draw their sum. I have to find minimum value of the sum using graph.
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3answers
30 views

Inequality involving absolute values.

I want to ask is whether there is a method to solve following inequality more easily and compactly or it is the only method. $$|x-2|+|x-8|\le x-2$$ What I know is taking $x<2,8>x>2,x>...
2
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2answers
26 views

Log Functions Inside Absolute Value

Is the function below always positive for $0< x <1$? (I am determining if the function requires the modulus sign or not.) $$\frac{1}{2}\log\left|\frac{1+\log(x)}{1-\log(x)}\right|$$ My first ...
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1answer
44 views

When do we need to take absolute values and why? [closed]

When do we have to take the absolute value when dealing with radicals and why? Surely $\sqrt[2]{x^2}$ is $|x|$ but I would like to know the mathematical reason behind it (not just words), which might ...
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0answers
31 views

Rephrasing what this proof is asking

I am new to proofs and am still struggling to parse them. I am not looking for a proof to the following statement; just guidance as to where to start or what the shape of a proof for it looks like. ...
2
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0answers
29 views

Absolute Value of a Integral, separation

This is a homework exercise, and ''Thomas early trascendentals'' book vol.4 says in property that : Def: To integrate an absolute value function, we have to look for specific cases when ; $ v(t)\...
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3answers
59 views

How to solve $|a+b|+|a-b|=c$?

It is intuitive that $a=\pm \frac{c}{2}$, with $-\frac{c}{2}\leq b\leq \frac{c}{2}$ or vice-versa are solutions to the problem. Can I get to these solutions without dividing the expression in all the ...
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5answers
51 views

Sum of two absolute values equal to a whole number

The following is the equation: $|x+1|+|x+2|=3$ How can I solve this problem? Do I have to reformat it to $|x+1|=3-|x-2|$? I would like a simple answer that by no means uses set theory. The answer ...
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1answer
14 views

Redefinition of a Constant Leading to Nullification of Absolute Value

I am currently taking a calculus 3 class in college, and my teacher did an intriguing problem about a tsunami in class which took up about 4 full white boards. Anyways, at a certain point in the ...
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1answer
38 views

solving equation involving absolute values [closed]

Given the equation $|x+2| + |3x+6| = 8,$ how can I find the sum of all its roots?
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2answers
78 views

How can I solve this absolute value equation?

This is the equation: $|\sqrt{x-1} - 2| + |\sqrt{x-1} - 3| = 1$ Any help would be appreciated. Thanks!
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1answer
21 views

Find the solution of $1 \le \left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +x\right\rfloor \lt 2$

Find the solution of $$1 \le \left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +x\right\rfloor \lt 2$$ My try: The only thing i know is that $$\left\lfloor \left\lvert \frac{x+1}{x-3}\right\rvert +...
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1answer
20 views

Solve for $x$: $3+|x-9|<\frac{2|x-1|}{x}$

I moved one part of the inequality to the other to create the following: $\frac{2|x-1|}{x}-3-|x-9|>0$ Eventually, I get to a case where I have 2 inequalities after opening 1 of the absolute ...
2
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4answers
51 views

If $|x|>|x-y|$ prove that $xy>0$

So the title says it all. I think using the square formula, not sure if that is legit. Thank you for your attention.
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1answer
46 views

Taking the absolute value in inequalities.

If I've a expression: $-4<3$ and take the absolute value $|-4|<|3|\implies 4<3$ which is false. So I though that maybe the inequality sign would change. But $|-2|<|3| \implies 2>3$ ...
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0answers
33 views

Writing Absolute Value For Equations With Inequalities

Write the absolute value equations in the form $|x−b|=c$ (where b is a number and c can be either number or an expression) that have the following solution sets: All numbers such that x≥5. ...
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1answer
37 views

Sketch the region in the plane consisting of all points $(x,y)$ such that $|x-y|+|x|-|y| \leq 2$

Sketch the region in the plane consisting of all points $(x,y)$ such that $|x-y|+|x|-|y| \leq 2$ I could consider the eight parts the plane gets divided into by the $x$-axis, $y$-axis, $y=x$ and $y=-...
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2answers
25 views

Do the $|$ around $|\langle u,v\rangle|$ refer to absolute value in the inner product version of the Cauchy-Schwarz inequality?

The full inequality is: $|\langle u,v\rangle| \leq ||u|| ||v||$ I understand that $||$ around the vectors $u$ and $v$ signifies the taking of their norm, but what do the single | around $\langle ...
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1answer
23 views

Minimization of multiple absolute sums

I know that on fist glance this seems as already explained problem on many internet places, but I haven't found solution anywhere. Anyway, here is my function: $$F=\sum_{i=0}^N |S_i|$$ where $$S_i=\...
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2answers
28 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
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4answers
79 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, ...
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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 $\...
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3answers
42 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;...
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2answers
42 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 ...
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3answers
54 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}_{-...
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2answers
45 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. ...
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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\}\...
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1answer
60 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 ...
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1answer
64 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 ...
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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
104 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
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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| \...
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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 ...
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1answer
17 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 ...
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3answers
59 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
19 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
24 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
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1answer
40 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 ...
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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 $|x^2+y^2| \geq 2|xy|$. ...
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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 ...
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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}-...
2
votes
3answers
53 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 ...
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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 ...
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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
vote
0answers
31 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
85 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 ...