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Questions tagged [absolute-value]

For questions about or involving the absolute value function.

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2answers
82 views

Which are differentiable for all x?

$F_1:$ \begin{cases} x^2/|x|, & \text{$x \neq 0$} \\ 0, & \text{$x=0$} \end{cases} $F_2$:$$|\sin(x)|^2$$ $F_3$:$$|\cos(x)|$$ I know that the absolute value of $\cos (x)$ would not be ...
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3answers
34 views

Absolute value limits

How would I calculate the limit $$\lim_{x \to 1} \frac{|x^2-1|}{x-1}?$$ I really have no idea. I know that $$|x^2 - 1| = \begin{cases} x^2 - 1 & \text{if $x \leq -1$ or $x \geq 1$}\\ 1 - x^2 ...
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1answer
21 views

Question regarding absolute equalities vs. absolute inequalities

If I have $|3x| < x + 4$, I break it into two cases and get that $x<2$ and $x>-1$ and the question is done (I think). In my solutions I have another problem that asks me to solve $|2x+3| = 2-...
1
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1answer
50 views

Prove that for all $x\in\mathbb{R}\enspace \lvert x\rvert + \lvert x-6 \rvert\geq 6$

I am beginning proofs in analysis. I am reading Kane's book, but I am not sure if this proof counts as a proof in analysis. I have tried proving by contradiction, but I failed. I also tried using the ...
1
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2answers
68 views

Proof of inequality $|x \sin \alpha + y \cos \alpha| \leq \sqrt{x^2 + y^2}$ [duplicate]

Just like in the title, I'm asking for any hints for proving (propably simple) inequality: $$ |x \sin \alpha + y \cos \alpha| \leq \sqrt{x^2 + y^2} $$
0
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1answer
33 views

Absolute Value Rules

Why can $\lvert \cfrac{3}{x} - 3 \rvert$ be turned into $\cfrac{3}{\lvert x \rvert} \lvert x-1 \rvert$? Where can I find more rules/tricks like this?
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2answers
29 views

Explaining The Shape Of An Absolute Value Graph

I don't understand why the shape of the graph would be like this, for the give equation. Why wouldn't the shape resemble that of a normal quadratic graph $(abs(x)+3)^2$?
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2answers
55 views

$(1-x)^{3}|1-xe^{iy}|^{4}|1-xe^{2iy}|^{2}<1$ for real $x,y$ with $x\in(0,1)$

For real $x,y$ with $x\in(0,1)$ we have $$(1-x)^{3}|1-xe^{iy}|^{4}|1-xe^{2iy}|^{2}<1$$ My attempt : Obviously, we have the identity $(1-x)^{3}|1-xe^{iy}|^{4}|1-xe^{2iy}|^{2}=(1-x^{3})|1-x\cos y-...
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0answers
30 views

Solving Modulus Inequalities: When to use and not use distributive property

I'm going to use examples to explain what I mean. Solve $|d| + 6 \leq 10$. \begin{align} |d| + 6 \leq 10 \implies d + 6 \leq 10 \implies d \leq 4 \end{align} or \begin{align} -d-6 \leq 10 \implies ...
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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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1answer
49 views

Normal distribution - absolute value solved

The random variable Y is normally distribution with mean = 8 and S.D = 5. Show that, P(|X−8|<6.2) = 0.785 What I did: (6.2 + 8)/5 = 2.84. The value of 2.84 is 0.9977 in the table. Normally, I ...
0
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1answer
44 views

Value of $\int_{-1}^{t} (1.5-2|x|)\,dx$

What is the value of $$\int_{-1}^{t} \left(\frac{3}{2} - 2 \, |x| \right) \,dx? $$ I get confused integrating with the absolute value. I forgot to mention $t>0$.
4
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2answers
436 views

Proving that a specific value is taken by a continuous function

Let $f:x\longmapsto \dfrac 1n \displaystyle\sum_{k=1}^n |x - a_k|$ where $a_k$ are elements of the interval $[0,1]$ satisfying $a_1<a_2<\cdots <a_n$. The question asks to prove that $f$ ...
2
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1answer
55 views

Solve a system of equation with absolute value and rotation

Solve $\begin{cases} \left|x_1-x_2\right|=\left|x_2-x_3\right|=...=\left|x_{2018}-x_1\right|,\\ x_1+x_2+...+x_{2018}=2018. \end{cases}$ I think there must be such a way to solve systems of equation ...
2
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2answers
70 views

$|z_{1} + z_{2}| = |z_{1} - z_{2}| \implies z_{1}/z_{2}$ is Imaginary

Two complex numbers $z_{1}$ and $z_{2}$ are taken such that $|z_{1} + z_{2}| = |z_{1} - z_{2}|$ and $z_{2}$ is not $0$. Show that $z_{1}/z_{2}$ is purely imaginary, i.e. it has no real part. So ...
0
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1answer
55 views

General Strategy For Solving Absolute Value Equations Involving The Addition Of Multiple Absolute Value Functions

I'm having trouble solving absolute value equations involving multiple absolute value functions added together. For example, take the problem $|x+3|-|x+1|+ x+2 =0$ If all the outputs of the ...
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3answers
31 views

A Question About Solving An Equation Involving The Addition Of Absolute Value Functions

I am trying to solve the following problem: $|4-x| \leq |x|-2$. I am trying to do it algebraically, but I'm getting a solution to the problem that makes no sense. I fail to see the error in my ...
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1answer
32 views

Proof involving absolute value and maximums

Given the definition for any real numbers a and b, the max function is $\max\{a, b\} = \begin{cases} a \text{ if } a \geq b \\ ...
4
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3answers
121 views

Prove that $ |x-a|<b \iff x∈(a-b,a+b)$

I am not sure if I am doing this correctly, my forward proof is: Since $|x-a|$ is an absolute value function, it can be defined as $|x-a|:= \max\{(x-a),-(x-a)\}$ . Let $S={(x-a) ∈ ℝ}$ and $b ∈ S, b&...
4
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0answers
91 views

Absolute value of tensor product of fields

Suppose that we have the Laurent series fields $F_1:=\mathbb F_p((X))$ and $F_2=\mathbb F_p((Y))$. Equip $F_1$ with the $X$-adic multiplicative absolute value $|\cdot|_1$, i.e. define $|X|_1=\dfrac{...
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1answer
26 views

Is it true to write $|\sum_{i=1}^N q_i x_i|\leq \sup_i|x_i|$?

My intuition tells me that the following equation is true but I can't prove it: $$|\sum_{i=1}^N q_i x_i|\leq \sup_i|x_i|$$ where $x \in \mathbb{R}$, $|x|$ is x's absolute value, $\sum_{i=1}^N q_i=1$ ...
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2answers
45 views

I'm stuck in this kind of question (Absolute Value) [closed]

How to solve $$|x-6|=x-6$$ $$|2x^2-3x|=x|2x-3|$$ and other similiar question?
2
votes
2answers
58 views

Absolute Value Inequalities (Quadratics)

I am currently struggling with the solutions of absolute value inequalities that involve quadratics. This is the example problem: $$x|x + 5| \geq -6$$ I am able to find the solutions, but I struggle ...
0
votes
3answers
36 views

Determine the set of all complex number z satisfying following conditions

I’m having some troubles of calculating complex numbers where I need to deal with absolute values and inequalities. Here is an example I’ve been working on but I get stuck Re(2/z)+Im(4/z)<1 I use ...
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7answers
88 views

What is the method for solving the inequality: $|x+1| \geq |2-x|$? [closed]

If I had a question like $$|x+1| \geq |2-x|$$ what is the method for solving it?
2
votes
2answers
45 views

How to solve this rational inequality?

$$\Big|\frac{2x - 1}{x + 1}\Big| \geq \frac{5x}{2}$$ First I attempted the positive case. I tried moving everything to one side of the equation and then factoring, but I am left with an un-factorable ...
2
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1answer
69 views

Using Supremum to prove $|x+y| \leq |x|+|y|$ vs. my proof

I saw postings on MathStackExchange for this but I need to prove it using the definition of Supremum. I proved it the way I see it. I do not see the Supremum way. I would appreciate help. $ \forall ...
2
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1answer
326 views

Weierstrass approximation theorem. Approximation of |x|.

I'm studying a proof of the Weierstrass approximation theorem that requires an uniform approximation using polynomials of the function |x| in the interval $[-1,1]$ i.e. we need a sequence of ...
0
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2answers
39 views

What is the range of $x$ satisfying the inequality $-x|x| > 4$ where $x$ is an integer?

The range according to me is x<-2. But I got to know that x>2 is also a possible solution. How can that be correct? P.S. I know that this is a simple question. But I couldn't find any existing ...
2
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1answer
27 views

Interpretation of a fundamental inequality

How do you interpret $0 < \vert x-a\vert < d$ with algebra? (I understand what this means geometrically but am struggling to understand this through algebra.)
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1answer
31 views

Why does $\int_{-\infty}^{\infty} |x-b|f(x)dx=\int_{-\infty}^b(b-x)f(x)dx+\int_{b}^{\infty}(x-b)f(x)dx$?

I am working on the following problem: Let $X$ be a random variable of the continuous type that has pdf $f(x)$. If $m$ is the unique median of the distribution of $X$ and $b$ is a real constant, ...
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1answer
19 views

Intuition about combining half-open intervals into inequality with absolute value?

I'm think about how to use the two conditions $$0\le r\lt n\\ 0\le r'\lt n,$$ to prove $\lvert r'-r\rvert\lt n,$ and the way I achieved this is by backwardly expand the result into $$-n\lt(r'-r)\lt ...
3
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3answers
262 views

Solving an absolute-value inequality: $−|x|+2 \geq 8x $

How would I go about solving the domain of this inequality? $$−|x|+2 \geq 8x $$ I can't combine the $x$'s so I don't know what to do. Could I say: $$-x + 2 ≥ 8x $$ and $$x - 2 ≥ 8x $$ and solve the ...
4
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5answers
639 views

What does it mean to have an absolute value equal an absolute value?

I have no problem reading absolute value equations such as $|x -2| = 2$. I know this means that the distance of some real number is $2$ away from the origin. Because the origin splits the number ...
0
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1answer
69 views

Veach's thesis, projected solid angle sanity check

Here's an equivalence from Veach's thesis on light transport (page 88, 3.16): $$|\cos(\theta)|\sin(\theta)d(\theta)d(\phi) \equiv \\ \sin(\theta)d(\sin(\theta))d(\phi)$$ This seems wrong in the ...
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1answer
66 views

Absolute value of a negative number

I was reading 'The method of Coordinates - Gelfand' and in the section about the absolute value of a number, it is stated what follows : if x > 0, then |x| = x, if x < 0, then |x| = -x, if x = 0, ...
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0answers
23 views

On Cauchy sequences over finitely generated modules over complete DVR

I have a question on Cauchy sequences. Let $R$ be a complete DVR with a valuation $v$, $\pi$ a uniformizer, $V$ a finitely generated $R$-module. Since $R$ is PID, $V$ is the direct sum of cyclic ...
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1answer
33 views

Doubt on piecewise function

Suppose we have a function given by: $f(x)= \begin{cases} x & x < 0\\ x+1 & x \ge 0 \end{cases}$ Then $f(|x|)= \begin{cases} |x| & x < 0 \\ |x|+1 & x \ge 0\end{cases}$ Or $...
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1answer
82 views

Explicit absolute value of $\mathbb{C}_p$ and completion of algebraic closure of $\mathbb{F}_p((t))$

By abuse of notation, I'll call a complete, algebraically closed field as an "impressive field" in the article. The smallest impressive field of characteristic 0 with Archimedean absolute value is ...
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1answer
23 views

What is the easiest and intuitive way to solve a equality system given a constraint?

I have tried graphing, but I had too many equations. Solving algebraically is long work. The problem goes as follows - Find the only positive value of x that is part of a real solution to the ...
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votes
3answers
66 views

Textbook has a strange section on absolute value that I don't quite understand [duplicate]

My textbook says $|a| = -a$ for $a \le 0$. What does it mean by this? I'm confused and think that the absolute value of a negative number like $|-5|$ would be positive $5$?
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0answers
20 views

Find $x \in \mathbb{Q}(i)$ with $ |x - 1|_{2+i} < \frac{1}{\sqrt{5}} $, $|x+1|_{2-i} < \frac{1}{\sqrt{5}}$ and $|x|_{7} < \frac{1}{7} $

I wanted to try some examples with adeles and strong aproximation. Let $\mathfrak{p}_1 = 2+i$ and $\mathfrak{p}_2 = 2-i$ and $\mathfrak{p}_3 = 7$. Can we a single number $x \in \mathbb{Q}(i)$ that's ...
1
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1answer
35 views

Can I Multiply Absolute Value Expressions

I know that for any two real numbers $a$ and $b$, that $$|a||b| = |ab|$$ And so I'd imagine that, assuming $x\in\mathbb{R}$, $$|x-a||x-b|=|(x-a)(x-b)|$$ since what is inside the absolute values are ...
2
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3answers
69 views

Does $\left|\log^2{x}\right|=\log^2x$ hold true?

I have a question about absolute values. Does the following hold true? $$\left|\log^2{x}\right|=\log^2x$$ In one problem my textbook removes the absolute from $\left|\log^{2/3}x\right|$. In theory ...
4
votes
2answers
79 views

How do I expand $|z_{1} - z_{2}|^2$?

I'm trying to expand $|z_{1} - z_{2}|^2$ where $z_{1}$ and $z_{2}$ are complex numbers. Is it exactly the same as with real numbers $z_{1}$ and $z_{2}$?
4
votes
2answers
162 views

Deciding when to drop the absolute values in differential equation?

I am currently solving the following differential equation (link is to another post): $\dfrac{dr}{d \theta}+r\tan \theta =\frac{1}{\cos \theta}$ The following is in standard form (i.e. $\dfrac{dr}{d\...
17
votes
3answers
299 views

Quadratic inequality puzzle: Prove$ |cx^2 + bx + a| ≤ 2$ given $|ax^2+bx+c| ≤ 1$

I came across this problem as part of a recreational mathematics challenge on university: Suppose $a, b, c$ are real numbers where for all $ -1 \le x \le 1 $ we have $|ax^2 + bx + c| \le 1$. Prove ...
2
votes
1answer
75 views

discrete multiplicative subgroup of the reals are cyclic

I'm reading a Dwork's book about G-functions and I'm stuck in one part who assert the following: If $\pi$ is an element of $K$ such that $|\pi|=\max\{z\in G_K: z<1\}$ then $G_K = \langle|\pi|\...
0
votes
2answers
66 views

AIME: I'm not sure what the question is asking for

I've encountered this problem: Let $f(x)=|x-p|+|x-15|+|x-p-15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \leq x\leq15$. I'm not sure what ...
1
vote
3answers
69 views

Prove that $| xy-\sqrt{(1-x^2)(1-y^2)}|\leq1$ where $|x|\leq1$ and $|y|\leq1$ [duplicate]

Prove that $| xy-\sqrt{(1-x^2)(1-y^2)}|\leq1$ where $|x|\leq1$ and $|y|\leq1$ I tried: $x=\sin\alpha$ and $y=\cos\beta$ $\sqrt{(1-x^2)(1-y^2)}=\sqrt{\cos^2\alpha\sin^2\beta}$ but if I write $\sqrt{\...