Questions tagged [absolute-value]
For questions about or involving the absolute value function also known as modulus function.
2,867
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Trying to solve $|2x-15| = -x^2 - 5x -8$
My first instinct was to take the positive and negative of the right hand side, resulting in
$2x-15 = -x^2 - 5x - 8$, and $2x-15 = x^2 + 5x + 8$, which results in the first giving me two real answers ...
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Question about $p$-adic absolute value
Let $x,y \in \Bbb{Q}^×$, and for $\varepsilon＞0$, suppose $|x-y|_p＜\varepsilon$, where $| \ \ |_p$ is the $p$-adic absolute value.
My book says that if $\varepsilon$ is small enough, then $|y|_p＞(1/2)|...
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What is the proof for the $(\sqrt[n]{a})^n$ equaling $a$ or $|a|$ if $n$ is odd or even?
If I use $64$ and $-64$ as the radicands, and have $2$ or $3$ as the indices, I know that it all works out arithmetically (except for $\sqrt[2]{-64}$ which has no real solutions) but I'm not sure how ...
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How to deal with functions like $|f(x) + x| = px$ when $f(x)$ is a cubic?
The problem is :
There is a positive real number $p$, and a cubic function $f(x)$ with its leading coefficient $1$. Every real roots of an equation $f(x) = px$ is $\alpha - \beta$, $\alpha$, $2\beta$. ...
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Solve Inequalities to find boundary for one variable
X<0 and Y<0 : as we are having sorted array and we are picking two pairs.
arr: -7, -3 ,3, 4, 7
so I can have number in this way ...
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How do I solve inequalities like $||x|-3|\geq0$? [closed]
As the title says, how do I solve inequalities like $||x|-3|\geq0$?
P.S. If the question seems invalid please correct me.
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Three solutions to a quadratic involving absolute values. [closed]
If the equation $x^2+(\alpha-2)x=3|x|-1$ has exactly 3 distinct real solutions, then which of the following values can $\alpha$ not take? A. 3 B. -3 C. 1 D. 7
Starting off, for x>0, $x^2+(\alpha-5)...
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Can function with known absolute value be holomorphic im unit disc
Would you be kind enough to verify/reject my answer to the following question?
If $D$ is the open unit disk in $\mathbb{C}$, can there be a holomorphic $f:D \rightarrow \mathbb{C}$ such that $$\lvert ...
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Is there any other way to solve absolute value inequalities than to make cases?
To solve an absolute value inequality, we can do two things right: make a bunch of cases or graph the thing.
But for a big inequality like: $$||x|^2 - 8|x| + 15 + |x + 1| - |2x - 5| + |x|| < 2$$ we ...
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Stephen Abbott Understanding Analysis - Brief Technical Question
Surely I'm just confused (I am a rookie at this). I am reading Abbott's Understanding Analysis, and in Chapter 1.2 (Example 1.2.5) of second edition he begins describing the absolute value function ...
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Solving $|2^{x+1}-1|+|2^{x+1}+1|=2^{|x+1|}$. How do we work on absolute value of a power?
I'm trying to figure out how do we work on absolute value of a power. Like in this question.
$$|2^{x+1}-1|+|2^{x+1}+1|=2^{|x+1|}$$
What am I supposed to to do here?
I need to solve for $x$, but I'm ...
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Under what conditions is $\vert x-y \vert = \vert x \vert -\vert y \vert$?
This is a fairly basic question, however, I don't know what I am missing here. Solving the equation $\vert x- y\vert = \vert x \vert -\vert y \vert$ yields:
\begin{align}
\vert x- y\vert &= \vert ...
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I not understanding a cancellation step in Spivak proof.
There is a question about this exact same proof but the answer is not satisfying me so I'm going to run it again if you don't mind. I will post the original question after mine.
In his chapter on ...
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Proof $ f(x,y) = |x| + |y| $ has local minimum at $(0,0)$ [closed]
I'm Trying in various ways but failing , As :
$ \nabla f(x,y) = 0 $ gives me:
$f_x = \frac{x}{|x|} = 0;\quad f_y = \frac{y}{|y|} = 0,$ and both cannot be solved as assuming:
$x=0 \rightarrow \frac{0}{...
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Constant $A$: $|e^{(2m+1)\pi e^{i\theta}}-1| \geq A > 0$ for all $m\in \mathbb{N}_{>0}$, $\theta \in [0, 2\pi)$
Problem: Prove that there exists some real constant $A$ such that
$$|e^{(2m+1)\pi e^{i\theta}}-1| \geq A > 0$$
for any natural numbers $m \geq 1$ and any real number $0 \leq \theta < 2\pi$.
...
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Maximization in l1 norm
I have formulated a problem as follows.
$${max \sum_{i=1}^l\sum_{k=1}^m|\sum_{j=1}^nw_{j}*(a_{ij}-b_{kj}+r_{kij}*x_{kij})|}$$
$$s.t.w_{j}*|a_{ij}-b_{kj}+r_{kij}*x_{kij}|\leq w_{j}*|a_{ij}-b_{tj}+r_{...
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Integral of f(|x|)
How would you calculate the definite integral of $f(|x|)$?
In my case, $f(x)=x^2-x-2$, and when I tried to calculate the integral of $\int_{-3}^3{f(|x|)}dx$ using $2\cdot(\int_{2}^3{f(x)}dx -\int_{0}^...
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Verifying the existence of a derivative of an absolute value function at x = 0.
I've been struggling with this question for a while, and any advice would be greatly appreciated.
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Why is $\pm\sqrt{(11y-8)^2} = \pm(11y-8)$? - Analytic Geometry
$3x^2-7xy-6y^2-2x+17y-5=0$
My original goal here was to know whether or not this was a degenerate conic, so I isolated $x$ by applying the Quadratic Formula.
$3x^2+(-7y-2)x+(-6y^2+17y-5)=0$
$x = \frac{...
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Squaring complex number equation with absolute values
I don't understand how you go from the first to the second line in this problem :
$$|(a-k)+i(7-2a)|=|(a-2)+i(9-2a)|$$
$$(a-k)^2+(7-2a)^2=(a-2)^2+(9-2a)^2.$$
Firstly, squaring i should make it -1 I ...
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Show that if $0 < \frac{1}{j},\frac{1}{k} \leq \frac{1}{N} \leq 1$, then $|\frac{1}{j}-\frac{1}{k}| \leq \frac{1}{N}$. ($j,k,N \in \mathbb{N}^+$))
I am working through Tao's analysis I book and I am trying to prove that the sequence $a_n = \frac{1}{n}$ is a Cauchy sequence (Proposition 5.1.11). I understand the proof, but I am having trouble ...
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If $|x|+|y|>|x+z|$, then is $y > z, y<z$, or $y=z$, or not determinable?
If $|x|+|y|>|x+z|$, then is $y > z, y<z$, or $y=z$, or not determinable?
I thought about cases, $x>0,x<0,y>0,y<0,z>0,z<0.$ Plugging numbers is unconvincing to me.
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If $|28 - x^2| < |3x|$ ,then what is the product of all possible integer values of $x$?
$|28 - x^2| < |3x|$
$\Rightarrow |x^2-28| < |3x| $
Case $1$ when $3x \geq 0 \Rightarrow x \geq0$ :-
$|x^2-28| < 3x$
$\Rightarrow -3x < x^2-28 < 3x $
$\Rightarrow x \in (4,7) \Rightarrow ...
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Maximizing $f(x)=\frac{1}{1+\left|x\right|}+\frac{1}{1+\left|x-1\right|}$
Find the maximum value of the function
$$f(x)=\frac{1}{1+\left|x\right|}+\frac{1}{1+\left|x-1\right|}$$
For $1<x$, when $x$ increase both fractions decrease hence $f(x)$ decrease. Similarly for $x&...
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Why do we take $\sqrt{f^2(x)} = f(x)$ when integrating by substitution?
I've posted a similar question Confusion in finding derivative of $\sqrt{\frac{1-\cos(2x)}{1 + \cos(2x)}}$.
Consider the following integral: $$\int\sqrt{1 - x^2}\ dx.$$
Putting $x = \sin(\theta):$
$$=...
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Method checking to show the given inequality to be true
Given quadratic polynomial $f(x)$ satisfies $\lvert ax^2 + bx + c \rvert \leq \lvert x \rvert$ for all $x \in [-1,1]$. Show that $\lvert a \rvert + \lvert b \rvert \leq 1$.
My approach was graphical ,...
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weak star continuity for the absolute value
Let $X$ be a normed space, and consider the weak-star topology for its dual. We know that the $w^*$continuous functionals are the evaluating ones, those are $\hat{x}: X^* \rightarrow \mathbb{K}$ ...
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Is my attempt at a proof valid?
Warning: I have no formal education in these, so sorry if my attempt is horrible.
I've recently learnt the concept behind a limit, i.e. the epsilon-delta definition and it seems really cool. I've ...
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Investigating continuity for function at given point
Given the function
$$
f(x,y)=\begin{cases}\big|1+xy^2\big|^\dfrac{1}{x^2+y^2} & \quad\hfill (x,y)\neq(0,0)\\\\ 1 &\quad\hfill (x,y)=(0,0)
\end{cases}
$$
investigate whether the function is ...
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Fraction Field of Non-Archimedean Valuation Ring and Relate the Divisibility to the Comparison of Absolute Value for Elements in this Ring
Let $K$ be a field with $|\ \ |$ a non-archimedean absolute value on it. Define the valuation ring of $|\ \ |$ to be $$R:=\{a\in K:|a|\leq 1\}.$$ I want to prove the following two claims:
$K$ is the ...
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Show $|\sqrt{|x|}-\sqrt{|y|}|=\frac{| |x|-|y| |}{\sqrt{|x|}+\sqrt{|y|}}$. [closed]
In a proof in class we used that for any real $x$ and $y$ we have $$\left|\sqrt{|x|}-\sqrt{|y|}\right|=\frac{| |x|-|y| |}{\sqrt{|x|}+\sqrt{|y|}}.$$However I'm not quite sure how one would show this ...
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If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$?
If a real-valued $f(t)$ is absolute continuous on a domain $[a,\,b]$, Does it imply it is also absolute integrable $\int_a^b |f(t)|dt < \infty$?
If is not true in general, please give some counter-...
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How do you find the range of $f(x)=\frac{x+3}{|x-2|}$?
I know how to find the Range when the Modulus/ Absolute-Value function is in the Numerator.
But how do I solve it when a modulus function is in the denominator?
Can you please explain it with this ...
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How can I prove the triangle inequality and the given corollaries? [duplicate]
How can I prove the following 3 claims? Or, if possible, could you provide suggestions for how I could prove them?
For real numbers $x$ and $y$:
Claim 1. $|x| − |y| ≤ |x − y|$
Claim 2. $|x-y| ≤ |x|+|...
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Existence of positive real $R$ such that $|f(z)| > M$ for all $|z| > R$
Exercise: If $f(z)$ is a polynomial of degree $n \geq 1$ and $M$ is an arbitrary positive real number, show that there exists a positive real number $R$ such that $|f(z)| > M$ for all $|z| > R$.
...
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Convergence of a seq
I would like to know if I can say (and how to justify it) that :
Let $(u_n)_{n\in\mathbb{N}}$, and let $(v_n)_{n\in\mathbb{N}}$ be a sequence defined by $v_n = | u_n - k|$ with $k$ a real. If $v_n$ ...
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$|f(z) - g(z)| < |f(z)|$ vs $|f(z) - g(z)| < |g(z)|$ in Rouche's theorem?
I was reading Marsden's Basic Complex Analysis and noticed that Rouche's Theorem was formulated as requiring $|f(z) - g(z)| < |f(z)|$ (for all $z$ on closed $\gamma$, etc.), and the statement was ...
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Spivak absolute value Problem: abs(abs(a + b) + abs(c) - abs(a + b + c)) [SOLVED]
Currently I'm working on Spivak's Calculus but I'm stuck at the following example, where one should drop at least one pair of absolute value signs:
$\mid(\mid a+b\mid + \mid c \mid -\mid a+b+c \mid) \...
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Removing common factors when an expression is within absolute value bars
I'm working through a derivative problem involving inverse trig functions from a calculus course on Udemy. The part of this problem I'm having trouble with is the simplification step at the very end. ...
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Linearisation of absolute values in the objective function with sign change
I hope you can help me.
Basically, I am aware of how I can implement, for example, $Min |X -1|$ as an absolute value in the objective function of an LP.
However, I have the following problem:
I have a ...
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Why exponentiate valuations (to become absolute values), and why intuitively are they so important to global fields?
I think (discrete) valuations come up very naturally in the theory of meromorphic functions/Laurent series, and by analogy the theory of $p$-adic numbers, and generalizations of those. One can then ...
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Approximating the absolute value function with a polynomial, starting from the Taylor series of $\sqrt{1-x}$
This is Exercise 6.7.7 from Abbott's Understanding analysis. For context, the section is about the Weierstrass approximation theorem, but this is a step towards proving it, so we cannot use WAT to ...
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Most concise way for derivation of$~|x+y|\geq|x|-|y|~$
I want to derive the following inequality.
$$\left|x+y\right|\geq\left|x\right|-\left|y\right|\tag{1}$$
My tries for it are as following.
$$|x+y|\leq|x|+|y|~~\leftarrow~~\text{I omit derivation of it ...
- 1,287
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minimum value of sum of absolute diferences
i was thinking about finding a $x$ that minimize the function $f(x)=\sum_{i=0}^{n}|x-a_i|$. I plot this function in geogebra (here is the url) and i saw that the function have an x that minimize it ...
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Why isn't there any Anti-Modulus Function?
Why isn't there anything which works like anti modulus? That is, a function which gives negative of absolute values of the number?
Simply, if modulus function |x| is:
when x≥0 then |x|= x and when x&...
2
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Inequality searching for an upper-bound
Suppose that there exists a constant $C$ such that the following relation
holds for all $G$:
\begin{equation*}
\vert T(F)-T(G) \vert \le C \sup_y \vert F(y)-G(y) \vert
\end{equation*}
Suppose that ...
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How did my textbook remove the absolute value from this linear differential equation problem?
The question is to solve the inital-value problem $xy'=y+x^2\sin{x}, y\left(\pi\right)=0$. I got to the same answer as the textbook except I have an x in absolute value. Here's what I did:
$$y'-\frac{...
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Solving 2nd-order DE with Abs[ ] and inverse power
I am trying to solve the following differential equation
\begin{equation}
|y''(t)|-\dfrac{2C}{y(t)}=2
\end{equation}
for some $C>y(t)$ and with the initial condition $y(0)=y_0$. The condition for $...
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Two basic questions
Here are two basic questions to which I currently do not know the answer:
(1) If $a > 0$ and $b$ are integers (where $b$ is negative), then if there exists a (necessarily) negative integer $c$ ...
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Maximum of absolute differences inequality
Let $x_1 < x_2 < \cdots < x_q$ and $y_1 < y_2 < \cdots < y_q$ be two monotonic sequences of real numbers. Then, I want to show
\begin{align}
\max_k |x_k - y_{\sigma(k)}| \geq \...
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