For questions about or involving the absolute value function.

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-4
votes
1answer
92 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
27 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 ...
1
vote
0answers
25 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
66 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
65 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
107 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
28 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
79 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
26 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
55 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
1answer
26 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
53 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
27 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
94 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
50 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
96 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
36 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)|< $ ...
1
vote
1answer
35 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
80 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
57 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
58 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
28 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
80 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
34 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
33 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
28 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
47 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
53 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 ...
1
vote
2answers
17 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 ...
0
votes
1answer
47 views

Determine the symmetry of $y=|x-4|$

Determine whether the graph of $y = |x − 4|$ is symmetric with respect to the origin, the $x$-axis, or the $y$-axis. A. not symmetric with respect to the $x$-axis, not symmetric with respect to the ...
0
votes
3answers
184 views

Spivak's Calculus chapter 1 problem 12 v

I am having trouble proving $$|x|-|y|≤|x-y|.$$ In the solutions it says $$|x|=|y-(y-x)|≤|y|+|y-x|, \quad \text{so} \quad |x|-|y|≤|x-y|.$$ Am I missing something here? How did he get $|x-y|$ on the ...
0
votes
2answers
90 views

Equation with logarithms and absolute value

I have this equation: $$ \ln\frac{2-|y-1|}{1-|y|} = \ln x $$ which becomes $$ \ln(2-|y-1|)-\ln(1-|y|) = \ln x. $$ Can the first term in LHS be written as ...
1
vote
1answer
92 views

Is there a simple closed form of $|\alpha(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor) + \beta(\sqrt{n}-\left\lfloor \sqrt{n} \right \rfloor)|$?

Let $d_n(x)$ denote the $n$'th digit after the decimal point in $x$. Examples: $d_8(e) = 2,\;d_5(\pi) = 9$ $\alpha(x)$ and $\beta(x)$ are defined this way: $$d_n(\alpha(x)) = \left\{ ...
0
votes
2answers
110 views

Proof related to absolute value

I was trying to prove $|x||y| = |x\cdot y|$ but do not have a clue to start. I have seen examples of |x|+|y| >= |x+y| but could not translate it to my problem. Please my fellow math geniuses, help a ...
0
votes
0answers
34 views

Why the plus-minus sign within a pseudo-Riemannian-manifold arc length integral?

Deep with the Wikipedia page on arc length, there exists the following puzzling excerpt (mathematics further marked up by yours truly for readability): Generalization to (pseudo-)Riemannian ...
0
votes
1answer
33 views

Differential Inequalities involving Absolute Values

I have to show that $|f '(x)| \leq 1, \ \forall x\in R$. The information I have been given is $|f(x)-f(y)|\leq |x-y|$ ... cauchy schwarz inequality. This is for calculus. Thanks so much.
2
votes
1answer
20 views

How to combine OR linear inequality with absolute value

I have x < -10 OR x > 15 How do I turn it into a single inequality using an absolute value? Like a < |x+b|. What are ...
0
votes
3answers
47 views

Injectivity in function $f(x)=x\cdot|x|+1$

I want to prove that $f(x) = x\cdot|x|+1$ is injective, and if it is; find the inverse of the function. $f(a) = f(b) \iff a|a|+1 = b|b|+1 \iff a|a| = b|b|$ $\begin{cases} -a^2 = b^2 \quad undefined ...
1
vote
1answer
35 views

Covariance of absolute values of random vaiables

How would I go about calculating $\operatorname{cov}(|X|,|Y|)$, if I know $f_{X,Y}(x,y)$ and $\operatorname{cov}(X,Y)$ ?
0
votes
2answers
50 views

Absolute value of a complex number proof

Ok, so I have the following proof. Let $z$ and $w$ be complex numbers. Prove $\lvert z+w \rvert ^2 + \lvert z-w \rvert^2 = 2[\lvert z \rvert^2 + \vert w \rvert^2]$. Using $\vert z \rvert^2=z\bar{z}$, ...
0
votes
1answer
40 views

Calculate the conjugate of a complex number

ok, so I have to calculate the conjuage of ${(8-2i)^4\over(4+3i)^5}$ using the properties such as $\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\bar z_1}{\bar z_2}$ and $\overline{(z_1z_2)}=\bar ...
0
votes
1answer
228 views

Absolute Value Equivalence relation inequality Question

I'm having trouble understanding what exactly to do to see if the following relation is symmetric and transitive. I've already determined that it is reflexive. Could someone please help me? For $a, b ...
-2
votes
5answers
59 views

What is the set of real solutions (x,y) that satisfy this absolute value equation?

How many real solutions (x,y) from |x-y| + |x+y| = 1 ? I really wonder how to find it. My attempt: I think I need to separate this problem into some cases: First case: for |x-y| >0 we got: x-y + ...
1
vote
5answers
75 views

How to solve this absolute value equation?

Consider the absolute value equation: |x| + |x-2| +|x-4|= 6 How to find the solution(s)? My attempt: For |x|, we got x, for x>=0 and -x, for x <0 For |x-2|, we got x-2, for x >= 0 and -(x-2), ...
2
votes
2answers
92 views

Triangle inequality problem with equality

How does one prove that, for any reals $x,y$ , there holds the equality $$|x|+|y|+||x|-|y|| = |x-y|+|x+y|\quad?$$ I have tried this using both the reverse and triangle inequalities, but I cannot get ...
3
votes
2answers
83 views

Let $f$ be a holomorphic in $D(0,1)$, with Re$\,f(z) >0$ and $f(0)=1.$ Then $\lvert\, f'(0)\rvert\leq 2$

Let $f:D(0,1) \to \mathbb{C}$ be a holomorphic function, such that $$ \mathrm{Re} \,f(z) >0\quad \text{and}\quad f(0)=1. $$ How to prove $\lvert\, f'(0)\rvert\leq 2 \ ?$ This is now a ...
1
vote
1answer
196 views

Expanding Binomial with Absolute Value

I want to expand the least-squares formula $\sum |a-b|^2$, but I can't follow the reasoning behind what I've heard is the answer: $|a-b|^2 = |a|^2 - 2|ab| + |b|^2$ Instructions or a link would be ...
1
vote
1answer
36 views

rule for the power of absolute value expressions

Is $|x^n|=|x|^n$ for any rational $n$ and for any real number $x$? If the above is true, what is the proof?
0
votes
1answer
64 views

how to write the absolute value of $| x_n - x$ | separately

$| x_n - x$ | = | $x_n$ | - | $x$ | is this right or is it less than or equal to. the equality is in fact $| x_n - x$ | less than or equal to 3 how does this mean that $| x_n |$ smaller or equal ...