For questions about or involving the absolute value function.

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0answers
49 views

How does this textbook determine when this function diverges?

The question is "state the limits of x for which this binomial expansion is valid." $\frac{1}{(1+2x)^3}=1-6x+24x^2-80x^3+...$ Obviously, it's not valid where $(1+2x)^3=0$, so $(1+2x)^3\ne0$ ...
1
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1answer
32 views

Ordered Field: $|x|\le y$ iff $-y\le x\le y$

I had a question regarding this part of a theorem that describes the inequalities of the absolute value function for order field $\mathbb{F}.$ Here is the theorem: Theorem: Let $\mathbb{F}$ be an ...
2
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2answers
37 views

Forcing an absolute value of x after a square root operation

Given the following two equations: $$ f(x) = x \\ g^2(x) = 2x $$ I need to find the $(x,y)$ coordinates for when they meet. So after performing the square root operation, we have: $$ f(x) = x \\ g(x) ...
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3answers
68 views

Find the set of complex numbers $z$ which satisfy: $\left\lvert\frac{z-3}{z+3}\right\rvert=2$

Find the set of complex numbers $z$ which satisfy $$\left\lvert\frac{z-3}{z+3}\right\rvert=2\text.$$ I need help on that one. Thank you.
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1answer
19 views

Using the triangle inequality to show that if $|x| < 4$ then $|x^2-2x+3| < 27$

I'm starting school soon and doing some review problems to prep for Calculus. I'm a bit stuck on this problem: Show that if $|x| < 4$ then $|x^2-2x+3| < 27$. I know that I have to use the ...
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0answers
48 views

Proof of nearest integer equality

Let $N(n)$ be the nearest-integer function undefined on half-integers. There are many valid ways to define $N(n)$, I like to choose $N(n) =\arg \min_{z \in \mathbb{Z}} |n-z|$. Consider the function ...
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2answers
36 views

differentiation of $g(x) = \lvert f(x)\rvert$ where $f(x)$ and $D(f(x)) = 0$

I'm really stumped on this problem and don't know how to go about it. It says $g(x)$ = $|f(x)|$ and to show that if $f(c) = 0$ and g is differentiable at c, then one must have $D(f)(c) = 0$. ...
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2answers
65 views

Integrate a periodic absolute value function [duplicate]

\begin{equation} \int_{0}^t \left|\cos(t)\right|dt = \sin\left(t-\pi\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor\right)+2\left\lfloor{\frac{t}{\pi}+\frac{1}{2}}\right\rfloor \end{equation} I ...
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1answer
39 views

Inequality which involves complex numbers and absolute values

How can I solve the following inequality: $|\frac{(1+(1-\theta)z)}{1-\theta z}| \leq 1$ ? $z$ is a complex number. I have to find the values of $\theta$ for which the inequality is satisfied.
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1answer
32 views

how to solve absolute inequality functions

I have noticed in the past while solving inequality functions that when you want to change the inequality symbol you need to switch the $+$ or $-$ signs of the function itself. How do I solve this ...
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1answer
27 views

Finding an absolute value

Instruction says Find |3 - $\sqrt{10}$| Given answer is $\sqrt{10}$ - 3 I cannot give an explanation.
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0answers
10 views

X numbers that when subtracted will produce the same absolute value

Let's say I have X unique numbers and I choose one number y out of this set. Is it possible to create these X numbers such that the absolute difference between y and any other number in X will always ...
0
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3answers
42 views

Why does $-b < a < b \implies |a| < b$ (and also the converse)?

I don't have any intuitiom for this because it's just something I memorized. I only understand that $|a| = a$ if $a$ is already positive (or $0$), and $|a| = -a$ if $a$ is negative since we want to ...
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0answers
23 views

integrate an absolute value periodic function

$$ \int_{-\frac{\pi}{2}}^{t} |\cos{t}|dt = \sin(t-\pi\lfloor(\frac{t}{\pi}+\frac{1}{2})\rfloor)+ 2\lfloor(\frac{t}{\pi}+\frac{1}{2})\rfloor $$ In know that this integral holds. It can be obtained by ...
0
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1answer
57 views

Value of Pi derivation

Derive the value of Pi. I want with explanation. Is there any possible way? How do scientists calculate it?
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6answers
44 views

Compute: $|z + 2| = |z − 3i|$

Find all complex numbers that solve this equation: $|z + 2| = |z − 3i|$ How would I go on about solving this one? 4 times? Like this? $I. z+2=z-3i$ $II. z+2 = -(z-3i)$ $III. -(z+2) = z-3i$ $IV. ...
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1answer
26 views

Find image of two variables function

I have problem with proving using double inclusion that it's an image of function where we have open interval for instance: Find image $f[A]$ where $A=(0,2) \times (1,3)$ of $f(x,y)=|x-y|$. My try: ...
0
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1answer
16 views

Absolute Value proof involving epsilon and delta

The question is: If $x,z$ $ϵ$ $R$, show that for every $ϵ > 0$ there is a $ δ > 0$ such that if $y$ $ϵ$ $R$ satisfies $|y-x| < δ$ then $|zy-zx| < ϵ.$ I honestly have no idea how to go ...
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2answers
38 views

How to show that $\int_0^x |z|\,dz = \frac12 x|x|$?

Say you are integrating a simple $|z|dx$ from $0$ to $x$. How do you go about solving to get $.5 x|x|$?
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1answer
63 views

Solving messy integral with modulus and trigonometry.

If $$a\in \mathbb R,\int_{a-\pi}^{3\pi+a}|x-a-\pi|\sin(x/2)dx=-16$$ then a can be? I tried substituting $x-a=u$ and then breaking into two integrals removing modulus then used $\int \sin x=-\cos ...
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2answers
25 views

Absolute value of addition of positive real numbers great than that of subtraction?

$$∀a,b ∈ R+, |a + b| > |a - b|$$ I'm wondering if this is true? I'm not sure exactly how I could check or prove it to myself with the absolute value there. I thought I might be able to do ...
3
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2answers
34 views

Graph of $f(x)$ given, find graph of $f(|x|)$

I know the graph of $f(x)=x^2-2x$. Google calculator https://www.google.com/#q=graph+of+x%5E2-2x But how can I find the graph of $f(|x|)=|x|^2-2|x|$? What is the best method to approach here? ...
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0answers
25 views

Trivial absolute value

Let $K/L$ be a algebraic extension. Suppose that $\left|\cdot\right|$ is a absolute value in $K$ such that is trivially in $L$. Then is trivially in $K$. Thanks for anny suggestion. If is trivially ...
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1answer
20 views

Absolute value of the sume of two complex number

I have a question about the following. $|A+B|^2$, where $A, B $ is complex number. The question is , when can $|A+B|^2$ be equal to $|A|^2 + |B|^2$?
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2answers
65 views

How to solve this with a modulus [closed]

I do not understand these challenges, how to solve them
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1answer
19 views

Show that $|a|>|b|/2$ knowing that $|a−b|<|b|/2$.

I'm working on a proof and I need to show that $|a|>|b|/2$ knowing that $|a-b|<|b|/2$. I would like to do it without enumerating the different cases. Thanks for you ideas.
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0answers
11 views

Signal-extraction knowing both the sum and the sum of the absolute values of normally distributed variables

I have two normally distributed variables $X∼N(μ_{x},σ_{x}²)$ and $Y∼N(μ_{y},σ_{y}²)$. I can observe both the sum of their values and the sum of their absolute values, i.e. $Z₁=X+Y$ and $Z₂=|X|+|Y|$. ...
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2answers
66 views

Find the limit $\lim_{x\to 0^-}| \left( 1+x^{3} \right)^{1/2}-1-x^{5} |/(\sin x-x)$

I am studying for my first calculus exam (well, it's half an exam), and of course we have to solve limits, without using L'Hospital rule, and using asymptotic analysis. I can't solve this one ...
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1answer
57 views

Estimating the modulus of the roots of $\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$

If $θ_1,θ_2,θ_3,θ_4$ are four real numbers, then any root of the equation $$\sinθ_1z^3+\sinθ_2z^2+\sinθ_3z+\sinθ_4=3$$ lying inside the unit circle $\vert z\vert$=1, satisfies which inequality? ...
0
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3answers
51 views

Inequality (absolute value)

$$|x-4|^2 -5|x-4| +6 > 0$$ How can I get rid of the absolute value? Does it work the same way equations with absolute value work?
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2answers
17 views

inequation with complex solutions

Could somebody please help me solve this inequality: $|x-2| < x|x|$ I tried to solve it by using three different values of $x$: 1. $x < 0$ Solution : $1/2 - \sqrt{7}i/2 < x < 1/2 + ...
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1answer
33 views

If $|\alpha|\leq 1$ and $|\beta|\leq 1$, prove that $|\alpha+\beta|\leq |1+\overline{\alpha}\beta|$

Note $\alpha$ and $\beta$ are complex numbers and $\overline{\alpha}$ is the conjugate of $\alpha$. I've tried using variations of the triangle inequality and I couldn't find anything to work.
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2answers
26 views

Calculate absolute value using matrix

Let's say I have a vector a. I would like to construct a matrix or vector b such that if I multiply ...
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1answer
18 views

Continuous functions and primitives

By the fundamental theorem of calculus, we have that a continuous function always has a primitive. However, if I take f(x) = absolute value of x, that f function is continuous, but does not have a ...
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1answer
38 views

Is there any noncontinuous function f(x), such that the absolute value of f(x) is continuous? [duplicate]

I am trying to find such a function or a proof, which shows that there is no such function in general. I know, that the other direction of this statement is true. (I prooved it using only the ...
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8answers
973 views

Why is the absolute sign needed in the definition of a bounded function

A function $f$ is bounded if there exists a real number $M$ such that $|f(x)| \le M$ for all $x \in \operatorname{dom}(f)$. Why is the absolute sign needed?
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1answer
27 views

Inequality with absolute value and a parameter inside it

I've been stuck on this problem for quite a while even though it seemed trivial to me at first. Basically, I have this: $$\lvert ax+4 \rvert>\frac1x$$ It is quite easy to conclude that only ...
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2answers
19 views

GRE Quantitative Comparison: Determining range of values satisfying equation involving absolute values

Consider the equation \begin{equation} |2a-1|+|3b+2|=0 \end{equation} Which of the following is true: $a>b$ $a<b$ $a=b$ The relationship cannot be determined. How can one solve for the ...
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1answer
30 views

Hassle with Absolute Value and Square Root

Are my questions invalid or difficult cause I'm not getting answers since many days? Question 1:      By definition absolute value gives just no of units and does not indicate any ...
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1answer
85 views

limit of function at $x \rightarrow 2$

ok, so this is a very basic question, i'm trying to find the limit of the following function at $x \rightarrow 2$: $|x^2 + 3x + 2| / (x^2 - 4)$ what i had previously done was simply plug in 2 for ...
0
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1answer
21 views

An Integral Inequality Question

We have the functions $f$ and $g$ such that, $$f:\mathbb{[0,1]}\rightarrow\mathbb{R}$$ $$g:\mathbb{[0,1]}\rightarrow\mathbb{R}$$ and both $f$ and $g$ are bounded and continuos ...
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2answers
73 views

Need help with this proof of inequality and absolute value. [closed]

Help! proof that if $x,y \neq 0$ $\left|\frac{x^5+y^5}{x^4+2y^4}\right|<\left|x\right|+\left|\frac{y}{2}\right|$
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0answers
19 views

How to to minimize a sum by changing summation order

I have two vectors $(x_1,\dots,x_n),(y_1,\dots,y_n) \in \mathbb{R}^{n}$. I want to find a permutation $\sigma$ such that $$ \sum_{i=1}^n |x_i -y_{\sigma(i)}|^2$$ is minimized. Is there a better way ...
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3answers
50 views

Derivative of absolute value function

What is $f'(x)$ and $f''(x)$ of $f(x) = x^{1/3}\vert 4-x \vert$? Do you use two cases or can it be solved a different way?
2
votes
3answers
44 views

Prove: Use the triangle inequality to prove that for all $x, y, z, | x − z | ≤ | x − y | + | y − z |$ [duplicate]

Prove: Use the triangle inequality to prove that for all $x, y, z, |x-z|≤|x−y|+|y−z|$ Is my proof correct? Proof: Let $a = x-y$, and $b=y-z$. We can say that $|a+b| = |(x-y) + (y-z)| ...
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2answers
59 views

Prove: Use the triangle inequality to prove that for all $x, y, | |x| − |y| | ≤ |x − y|$ [duplicate]

Prove: Use the triangle inequality to prove that for all $x, y, | |x| − |y| | ≤ |x − y|$ Proof: If $x ≥ 0$ and $y ≥ 0$, then both sides of the inequality are the same. Also if $x ≤ 0$ ...
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1answer
19 views

Inequalities finding the set of solutions

Find the set of solutions to this inequality? $|x − 3| + |x − 6| < 5$ I have been taught to do it by treating $x$ in $3$ separate cases however I am not getting the correct answer. The answer is ...
1
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1answer
24 views

Getting rid of absolute value in integrating factor

If I have this equation $$|I|=e^C |x^3|$$ where $C$ is a constant, yet to be determined. Is it allowed to say: let $A$ be a constant such that $$\begin{cases} A=-e^C \space\space\space ...
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1answer
24 views

Absolute value equality on $4$ integers

For all $a,b,c,d \in \mathbf{Z},\\a<b<c<d.$ Prove $\left|10-a-b\right|+\left|10-b-c\right|+\left|10-c-d\right|\space = \left|10-a-c\right|+\left|10-a-d\right|+\left|10-b-d\right|$ Is this ...
0
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2answers
48 views

Proof of very simple absolute value inequality [closed]

I was wondering how to prove this. It always appears to be true when I plug in values. $a,b,c \in \mathbb{R}\\a\lt b\lt c$ Prove $\forall a,b,c : \left|a + b\right| \space\lt \left|a + c\right|$