For questions about or involving the absolute value function.

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-1
votes
2answers
57 views

How to solve this with a modulus [on hold]

I do not understand these challenges, how to solve them
1
vote
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.
0
votes
0answers
6 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 $Z₂=|X|+|Y|$. ...
1
vote
0answers
45 views

Antiderivative of $|x − 2| + |x − 3|$ [on hold]

Find the most general antiderivatives of the following function. $$|x − 2| + |x − 3|$$ I started with showing that the antiderivative for $|u|$ is $\dfrac{u|u|}2$. How to proceed then?
0
votes
2answers
65 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 ...
1
vote
1answer
52 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
votes
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?
0
votes
2answers
16 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 + ...
1
vote
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.
0
votes
2answers
23 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 ...
0
votes
1answer
16 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 ...
0
votes
1answer
37 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 ...
6
votes
8answers
967 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?
1
vote
1answer
26 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 ...
0
votes
2answers
17 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 ...
0
votes
1answer
29 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 ...
-4
votes
1answer
83 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
votes
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 ...
0
votes
2answers
41 views

Please help me to find absolute values of equation: $2^{|x+1|}-2^x=|2^x-1|+1$ [closed]

Please solve this: $2^{|x+1|}-2^x=|2^x-1|+1$
-2
votes
3answers
52 views

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

Help! proof that if $x,y \neq 0$ $\left|\frac{x^5+y^5}{x^4+2y^4}\right|<\left|x\right|+\left|\frac{y}{2}\right|$
1
vote
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 ...
0
votes
3answers
49 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
43 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)| ...
1
vote
2answers
57 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 $x ≤ 0$ ...
0
votes
1answer
18 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
vote
1answer
20 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 ...
0
votes
1answer
23 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
votes
2answers
46 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 $\forall a,b,c : \left|a + b\right| \space\lt \left|a + c\right|$
0
votes
3answers
30 views

Finding $f'(1/3)$ when $f(x)= |x-(2/3)| $ by using the definition of derivative.

How do I find $f'(1/3)$ given $f(x)= \left|x-\frac 23\right|$, by using the definition of derivative at $x=1/3\;?$ When I begin to solve the limit I have problems with absolute value. My function, ...
0
votes
1answer
19 views

equation with absolute values

I am stuck with solving this equation: $$ |x^2 - 4x +3 | + |x-1| + |x-2| -2x=0 $$ I tryed to raise it by power of 2 (including moving some of the factors to the right side, as well as factor the ...
1
vote
3answers
37 views

Techniques as to solving absolute value equation

Solve absolute value equation with absolute value variable one one side or even both side, without a number outsides of absolute value signs are typically easy. In my high school, I was taught to ...
3
votes
1answer
24 views

Reason behind solution in this inequality with absolute values

Solve the inequality $|3x-2|-|x+2|>x$ When $|x+2|<0$: $-(3x-2)+(x+2)>x\iff x <\frac{4}{3}$ When $|x+2|>0\land |3x-2|<0$: $-(3x-2)-(x+2)>x\iff x < 0$ When $|3x-2|>0$: ...
1
vote
0answers
20 views

Comparing function to parent function without graphing

How can I compare this function to the parent function without graphing? Where did the 5/4 come from and what steps do I need to take to solve this?
1
vote
2answers
47 views

Modulus of z^-3?

What is the result of $|z^{-3}|$ and how can one show it? I know $z = e^{i\omega T}=cos(\omega T) + i\sin(\omega T)$, but I cant go further... I would be glad if someone can explain further.
1
vote
2answers
61 views

Absolute value problem $|x-y|=|y-x|$

My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook. Page 43. Problem 1 Prove each of the following properties of absolute values. (c) ...
1
vote
2answers
30 views

$(f \circ g)(x) = \sin (x^{1/2})^2, \; (g \circ f)(x) = | \sin x |$

If $(f \circ g)(x) = \sin (\sqrt{x})^2$ $(g \circ f)(x) = | \sin x |$ Find $f(x)$ , $g(x)$. I'm told there are 2 solutions. I do not have an idea of how to approach these questions. Would ...
1
vote
2answers
27 views

Absolute value proof with epsilon

I'm having trouble with this proof. any hints would be greatly appreciated! If $x$ is a positive real number, show that for some $\epsilon$ $>0, $ then $y\in \Bbb{R}$ is positive if $|(x-y)|< $ ...
0
votes
1answer
29 views

between what two disjoint sections we can do a unification in order to get this group of solutions?

between what two disjoint sections we can do a unification in order to get this group of solutions? $0<|x+6|\leq{0.4}$ in other words, in what values should I fill the blankets: (____,____) ...
1
vote
1answer
28 views

Tricky logarithm problem

I having a problem in this logarithm problem involving modulus- Solve for x |x-1|^((log(x))^2-2log(x))=|x-1|^3 Bases same so powers equal. If I take log x as a then I get the following quadratic- ...
1
vote
2answers
33 views

limit of absolute value

$$ \lim_{x \to 0} \frac{\lvert2x-1\rvert - \lvert2x+1\rvert}{x} $$ Defining the function piecewise reveals the limit is in fact, continuous about 0 However when I go to solve it in a normal ...
4
votes
4answers
51 views

How to solve a convoluted absolute value inequality?

$$ \lvert \lvert x-2\rvert -3\rvert \lt 5 $$ How can I attack this the best way? I see that both sides are positive. Squaring yields: $$ \lvert x-2\rvert ^2 -6 \lvert x-2\rvert +9\lt 25 $$ $$ ...
0
votes
1answer
17 views

Solving absolute value equation, different methods.

I'm interested to know how people solve absolute value equations differently and how many methods there are out there. For example, say I wish to solve $|x-2|-|x-3|=|x+4|$. How would you solve it ...
0
votes
1answer
42 views

How can the integral of $|\sin(x)|$ be $-\cos(x)\text{sgn}(\sin(x))$?

Wolfram|Alpha tells me that $\int|\sin(x)| = -\cos(x)\text{sgn}(\sin(x))$ (which happens to also be its derivative), but I don't understand how this is possible, because the resulting function jumps ...
2
votes
1answer
31 views

Inequalities with more than one absolute value

I saw a question which asked to find all the solutions to: $|x+2|+|x-5|=7$ For $x\leq -2$, the answer is $-2$. For $-2< x <5$, the answer is $R$. For $x>5$, the answer is $5$. First I ...
0
votes
1answer
26 views

Finding $\lim_{t\to 0}\frac{|t-2|}{t}$ and $\lim_{t\to \infty}\frac{|t-2|}{t}$

Find $$\lim_{t\to 0}\frac{|t-2|}{t}$$ and $$\lim_{t\to\infty}\frac{|t-2|}{t}$$ Usually I would simply the top and bottom but I'm not sure what to do for absolute values. Any help would be ...
0
votes
1answer
18 views

Let $G$ be a group of order $36$ and $H$ be a subgroup of $G$ with order 4. Then which is/are true?

Let $G$ be a group of order $36$ and $H$ be a subgroup of $G$ with order 4. Then (1) $H\subset Z(G)$ (2) $H=Z(G)$ (3) $H$ is normal in $Z(G)$ (4) $H$ is abelian group Can I tell $H$ abelian, ...
0
votes
1answer
40 views

Trying to prove an absolute value inequality $\left | a\sqrt{2} -b \right | > \frac{1}{2(a+b)}$

I am trying to prove that: $$\left | a\sqrt{2} -b \right | > \frac{1}{2(a+b)}$$ I was given that $a$ and $b$ are any positive integers. Can someone please help me? Thanks.
4
votes
2answers
47 views

Show that if $a,b \in \Bbb R$ then [duplicate]

$\max\{a,b\} = \frac12(a+b+|a-b|)$ and $\min\{a,b\} = \frac12(a+b-|a-b|)$ how would you go about solving this? I started with suppose $a \leq b$ Also, show min{a,b,c} = min{min{a,b},c}. How would ...
0
votes
0answers
27 views

Determine and sketch the pairs $(x,y)$ in $\mathbb{R} \times \mathbb{R}$ that satisfy some inequality

a) $|x| \leq |y|$ Continue my explanation below... If $y \geq 0$, then $-y \leq x \leq y$ and we get the region in the upper half-plane on or between the lines $y = x$ and $y = -x$
1
vote
2answers
14 views

Which conditions imply $\sup_n |\ln x_n| < \infty$?

I want to find conditions which imply that $\sup_n|\ln x_n| < \infty$. Intuitively I think that $\inf_n x_n > 0$ and $\sup_n x_n < \infty$ should be enough, but I don't know how to write it ...