Questions tagged [absolute-value]

For questions about or involving the absolute value function also known as modulus function.

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45 views

How can I handle$~\exp(\ln|x|)~$to solve 1st order linear DE?

RHS and LHS are same. $$\exp\left(\ln\left(x\right)\right)=\exp\left(\ln\left(x\right)\right)\tag{1}$$ Taking log. $$\ln\left(\exp\left(\ln\left(x\right)\right)\right)=\ln\left(\exp\left(\ln\left(x\...
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0answers
49 views

Solving $|3x-2y-11|+2\sqrt{31-8x+5y}=0$

$$|3x-2y-11|+2\sqrt{31-8x+5y}=0$$ I was confused at first as there are clearly 2 vars but only 1 equation, but the textbook does have an answer. What I tried: Setting some bound by letting the ...
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23 views

Absolute value equation with parameter p

I know how to calculate absolute functions without parameter: 𝑥 ⋅ |𝑥 + 6𝑝| = 36 The question is how parameter p changes the number of solutions. I tried to somehow calculate the discriminant for ...
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2answers
55 views

Area of the region bounded by $f(x)$ and $g(x)$? ($f(x)=|2-x|$ and $g(x)=2-|2-x|$)

$f(x)=|2-x|$ $g(x)=2-|2-x|$ Find the area of the region bounded by $f(x)$ and $g(x)$. I was told that you actually need two definite integrals to solve this problem, since it involved absolute value, ...
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20 views

Property involving Maximum and Absolute Value function

I am trying to show $$ |max\{a,b\} - max\{c,d\}| \leq max\{|a-c|,|b-d|\}. $$ There may be a slicker way to prove this but I considered the following cases. The trivial cases are when $a \geq b$ and $c ...
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9 views

Prove that $x∈(-∞; -1)∪[1;∞)$ is true for $|\frac{x-1}{x+1}| = \frac{x-1}{x+1}$

$x∈(-∞; -1)∪[1;∞)$ for $|\frac{x-1}{x+1}| = \frac{x-1}{x+1}$ I know that $x+1 \not= 0$ so $x \not= -1$ but am lost on what to do next
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Linear system of equations with absolute value

Assume I have a simple linear system of an equation $y=Ax$ , $y \in \mathbb{R}^{m}$, $x \in \mathbb{R}^{n}$ with full rank matrix $A\in \mathbb{R}^{m \times n}$. Is there any available technique to ...
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32 views

Integral of Exponent of L1 Norm of Linear Function

Let $A \in \mathbb{R}^{m \times n}$ be a matrix with $m \geq n$ of full rank. I'm trying to solve for the integral \begin{equation} f(\vec{x})=\int_{\mathbb{R}^n} \exp(- \lVert A \vec{x} \rVert_1) \, ...
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1answer
70 views

Is there a known equation for max or min of a set?

I know that $\max(a,b) = \frac{a + b + |a-b|}2$ What I want is to formulate a similar equation generalizing larger sets: $max(a,b,c)$, $\max(a,b,c,d)$ etc. I tried doing the algebra for $\max(a,max(b,...
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How can $\sqrt{x^2}=|x|$ [duplicate]

$$\sqrt{x^2}=|x|$$ So $\sqrt{x^2}$ is always positive And I know that $\sqrt{x^2}$ can be positive or negative For example if x = 4 or x=-4 $\sqrt{-4^2}$ = $\sqrt{16}$= -4 or 4 $-4$ is negative. Where ...
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1answer
26 views

Exponentiating expression containing ln(abs(x))

I am trying to figure out when we write +/- after exponentiating expressions containing natural log. So, say that we have integrated (1/x) with respect to x. Then we have ln(abs(x)) + C. That is, ln(x)...
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93 views

Minimizing $ |t|^2 + a(|t + u| - |u|) $ over $ t \in \mathbb{C} $

I'm not so experienced with Complex Analysis and I somewhere stumbled upon the following function $ f: \mathbb{C}^2 \rightarrow \mathbb{R} $ that I need to minimize with respect to the Complex ...
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1answer
24 views

Application of Lagrange theorem with absolute value

I have to solve an exercise using the Lagrange theorem but I have doubt. From the theorem I know that, let a continuous function in $[a,b]$ and differentiable in $(a,b)$ then: $$\exists c\in(a,b): ,\,...
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2answers
32 views

Removal of absolute signs

I'm a high school student and I'm currently studying differential equations. I encountered this one question: An object is dropped from a cliff. The object leaves with zero speed, and t seconds later ...
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Question about the proof that if x is a real number then there exists a postive integer n such that -n < x < n. [closed]

One of my homework questions of real analysis went like this: Prove that if $x$ is a real number then there exists a postive integer $n$ such that $-n < x < n$. My answer was this: First notice ...
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quadratic equation with parameter and absolute value

$p+((2p-3)x+3p-1)$$|x|$$=2$ for which P does this equation have 1 real solution. so, at first, I opened brackets and I get $(2p-3)|x|*x+(3p-1)|x|+p-2=0$ next I try to analyze $x$ $\geq$ $0$ and $x&...
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37 views

Absolute value inequality with two unknows: x and y

I have a problem that kept me all day and I can't see the solution. I'm in 10th Class, so if you can solve with easy steps, it will be very useful for me. So the exercise is: $|x+1|+|y-2x|≤4$ https:...
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1answer
45 views

Valuation ideal on non-archimedean field is a principal ideal?

First off, some notation/definitions. (EDITED) Given $k$ a field with an non-archimedean absolute value: Valuation ring:= $\mathcal{O}$:=$\overline{B}(0,1)$ (As in closed ball, not closure) Valuation ...
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1answer
63 views

Absolute Value in Integrals

I'm trying to solve this integral problem but I'm a bit confused as to how to evaluate the absolute value portion of it. For context, $a$ is a positive constant. $$ \int_{-a}^{a}\left(\sqrt{a^{2}-t^{2}...
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2answers
56 views

$x$ and $y$ are real numbers such that $y = |x - 2| - |2x - 12| + |x - 8|$. What is the least possible value of $y$?

I solved this question by assuming cases for various roots of this equation and I found that lowest value of $y$ is $-2$ for $x<2$ but this is a long process and took some time. I was wondering if ...
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1answer
32 views

Find alternate solutions to absolute value functions

For, the problem $|x + 1| = |2x - 1|$, I found one solution analytically: $x + 1 = 2x - 1$ $\to x = 2$ Since, these are absolute functions, they should intersect once more at $x = 0$. This solution I ...
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78 views

What's the trick to finding the minimum/maximum values of problems like $f(x) = |x + 1| - |x - 2| + |2x + 1|$

I often come across to problems like "find the minimum value of $f(x) = |x + 1| - |x - 2|$" or "find the maximum value of $f(x) = |x - 1| + |2x + 5| - |6x + 1|$" or find the ...
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43 views

subadditivity of complex numbers on the real line

I am trying to prove that an integral given by \begin{equation} \int^{\infty}_{-\infty} |f|^2 dx \end{equation} diverges where $f$ is complex. I can show easily that \begin{equation} |\int^{\infty}_{-...
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2answers
45 views

Absolute value inequality: $ |x_n y_n-x_n y|≤|x_n ||y_n-y|$

I am trying the prove the following theorem: If $(x_n)$,$(y_n)$ are sequences with $x_n→x$,$y_n→y$ and $λ∈R$. Then: $x_n y_n→xy$ A section fo my proof: $$|x_n y_n-xy| =|x_n y_n-x_n y+x_n y-xy|$$ $$ ...
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1answer
74 views

Derivative of absolute value using epsilon delta

I want to show that $$\lim_{h\rightarrow 0}\frac{|x+h|-|x|}{h}=\frac{x}{|x|},$$ for $x\ne 0$. Take $x\ne 0$ and let $\epsilon>0$. Then $$\left|\frac{|x+h|-|x|}{h}-\frac{x}{|x|}\right|\leq\frac{|x+h|...
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Absolute difference representation with integrals

I saw an interesting representation of the absolute difference using integrals and the indicator function and was wondering if someone could point me to a textbook where it shows the derivation a bit ...
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68 views

How do I compute this integral on the set A?

I have the following problem: Compute the integral $\int_{\partial A} \langle F,\nu \rangle dS$ where $\nu$ is the normal vector and $$F(x,y,z)=(2x-3y+z,\,\,\,x-y-z,\,\,\,-x+y+2z)$$ and $$A=\{(x,y,z):...
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1answer
46 views

Solving $|1 - \ln(1 - |2x| + x)| = |1 - |3x||$

I am asking you for help (that is, just verify my passages and perhaps to right my wrongs) about this exercise. $$|1 - \ln(1 - |2x| + x)| = |1 - |3x||$$ Now here is what I have done. First of all I ...
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21 views

Integral inequality with absolute value and switch of the integration variable

What is the rule/theorem/inequality that can be applied to this integral: $$ \int_a^b {d^k \over dt^k}x(t)e^{-j2\pi ft}\,df $$ so that I can write this inequality ? $$ \left\lvert\int_a^b {d^k \over ...
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1answer
30 views

Do we consider functions positive in indefinite integrals?

In integrals like $\int \sqrt{x^{2}} d x$ where we have to take the radicand out of the radical, we have to put first $\int|x| d x$ but here I'm stuck : is $|x|$ = $+x$ or $-x$? Photomath gives the ...
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2answers
49 views

Absolute Value equations in 2 variables ($|x-y|+|x+y|=\sqrt5$) [duplicate]

I'm struggling a lot figuring out how to sketch the following $$|x-y|+|x+y|=\sqrt5$$ This is what I know for$|x-y|$ $x-y \geq 0$ then $y \leq x$ so $|x-y|= x-y$ $x-y < 0$ then $y >x$ so $|x-y|= ...
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Why can't $|x-2|$ be less than zero?

Just confused with the concepts in Absolute Value. So we know that to solve absolute value equations such, $$|x-2| = 5 \tag1$$ In this case we have, $x-2 = 5$ and $x-2 = -5$. then solve for $x$. ...
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3answers
48 views

Solving $|2x+8|^2 -|9x+36|-9=0$

it looks easy but I messed up something with steps. $$|2x+8|^2 -|9x+36|-9=0$$ This is what I got $$|2(x+4)|^2 - | 9(x+4)| - 9 = 0$$ Then I used $u = (x+4)$ and here where it got complicated.
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4answers
119 views

Solve $|2x−3| < |x+5|$

I've tried solving $$|2x-3|=|x+5|$$ but the same method does not look applicable here. $$2x-3 = x+5, \text{if}\; x>3/2$$ $$2x-3= -(x+5), \text{if} \; x<3/2$$ This method was not working here.
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1answer
67 views

How to determine $\sup_{x \in \mathbb{R}} \left|\frac{\sin \left(x^{2}+1\right)}{x^{2}+1}\right|$

It is clear (especially using a graphic calculator) that the sup of the function is the value of the function for $x=0$, but I want to prove it analytically. First, because of Weierstrass, the sup of $...
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2answers
64 views

How to deal with this inequation with 3 absolute values? [closed]

Solve:$$|x-2|+|3x+2|-x-2|x+4|\le -x-4$$ I am having trouble even starting with this inequation. Do I find out first the zeroes of each absolute value and then I would get intervals for each absolute ...
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3answers
112 views

Why do we require the $p$-adic norm to satisfy multiplicativity?

I suppose my question is really "why do we require norms in general to satisfy multiplicativity?". I ask this because for the usual absolute value on $\mathbb R$, I never feel like ...
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1answer
62 views

Prove inequality using Lagrange's remainder

My problem is the following: Prove that $\left | e^{x}+e^{-x}-2-x^{2} \right |\leq \frac{1}{6}x^{4}$ when $\left | x \right|\leq1$. Using Maclaurin's expansion formula, I get that $$e^{x}=1+x+\frac{x^{...
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3answers
61 views

Solve the equation $|2x-1|+|3x+2|+|x|=3$

Solve the equation $$|2x-1|+|3x+2|+|x|=3$$ My question is regarding the case when $x\le-\dfrac23$. According to the answer of the problem, $x=-1$ is NOT a solution, but I simply don't see why. In ...
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1answer
53 views

How to derive easily : $|\sqrt{x+3} -2|=\frac {|x-1|}{\sqrt{x+3}+2}$? ( Context : a limit problem).

Source : Lebossé & Hemery, Algèbre et Analyse ( Terminales CDT ), $1966$. $|\sqrt{x+3} -2|=\frac {|x-1|}{\sqrt{x+3}+2}$ This equality is asserted in the solution of a limit problem , namely : show ...
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3answers
107 views

Find the maximum value of a constant $\psi$ for which the inequality $\psi |x - y| \leq |f(x) - f(y)|$ holds

Let $n$ be a natural number not equal to $0$. Find the maximum value of the constant $\psi(n)$ for which for every polynomial $f(x)$, with $\deg f = n$, with the property that for every integer $x$, $...
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1answer
48 views

Help with ODE - Cannot get to computed solution - Possible problem with absolute values?

I'm struggling with the following ODE, which computed gives a different answer than the one I got doing it analytically. Maybe I have made a silly mistake or I do not get how to work with the absolute ...
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1answer
132 views

what does m<|x| imply?

I know $|x|\le M \ \implies -M\le x \le M$ but what does $m\le|x|$ imply? Can I just plainly interpret this as $m\le x$? and if I say $m\le x \le M$ does this still mean $-M \le x \le M$?
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2answers
73 views

Solving an absolute value inequality with fractions

I'm having a hard time figuring out this inequality: $\bigg|\dfrac{x-4}{x+5}\bigg| \le 4$ I use one of the absolute value properties, which results in: $-4 \leq \dfrac{x-4}{x+5} \leq 4$. From there, I ...
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1answer
40 views

How would I find the critical values of an absolute value function? $y=|x+1| + |x-1|$ over $[3, -2]$

Would I just have to graph this to find the critical points? Because I tried the method where you rewrite the expressions in the absolute brackets as $\sqrt{(x+1)^2}$ and $\sqrt{(x-1)^2}$ and take the ...
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1answer
65 views

Complex number inequality with absolute value

In the middle of problem i've got inequality: Find all $z\in\mathbb{C}$ such as: $$ ||z|-1|<|z+1|$$ So i start to elaborate on that for $z=x+iy$: $$ |\sqrt{x^2+y^2}-1|<\sqrt{(x+1)^2+y^2} $$ Now ...
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17 views

How to simplify the following error(absolute percentage error) formula?

I have three indicators to estimate: income, cost, profit, which are represented by I, C, and P respectively, and of course P = I-C. Therefore three errors(absolute percentage error): $error(I)=\frac{...
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1answer
42 views

Are the conditions for completeness over a valued field equivalent?

I am a beginner in learning perfectoid fields, and I have several questions about complete valued fields. Let $K$ be a nonarchimedian valued field(not necessarily discrete). Let $m_K$ be the maximal ...
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1answer
80 views

Finding the minimum of $|S_1-S_2|+|S_3-S_4|+|S_5-S_6|+|S_7-S_8|$ where $S$ is a sum of $x,y$, and $z$.

Consider $24$ positive numbers; $x_1, \dots, x_8,y_1, \dots, y_8,z_1, \dots, z_8$. Let $$M=|\text{sum of }(\text{one }x, \text{one }y, \text{one }z)-\text{sum of }(\text{different }x,\text{different }...
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1answer
81 views

prove that the equation has root when x>0

prove that $\left\lvert{x-a_1}\right\rvert$+$\ldots$+$\left\lvert{x-a_i}\right\rvert$=$\frac{n}{2}$has root when x>0 given that ${a_1}$+$\ldots$+${a_i}$=1 , $i=1\ldots n$, and $0\leq a_i\leq1$I ...

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