For questions about or involving the absolute value function.

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2
votes
2answers
64 views

Prove that $|ab+1|>|a+b|$ with $|a|<1$, $|b|<1$

Prove that $|ab+1|>|a+b|$ with $|a|<1$, $|b|<1$ $a$, $b$ are real numbers Where $|a|$ is the absolute value of $a$. Every time, I arrive to a dead-end.
1
vote
2answers
47 views

Indefinite integrals with absolute values

Which is the right way to solve indefinite integrals which contain absolute values? For example if I have $\int |2x+3| e^x dx$ Can I consider the sign function and integrate separetly? I mean doing: ...
0
votes
1answer
28 views

Prove $|1+z^{2n}|\geq1-|z|^{2n}$

i need to prove that $|1+z^{2n}|\geq1-|z|^{2n}$, I have tried use that $|a-b|\geq||a|-|b||$ but I think that it is not so, help?, $z\in D(0,1)$
0
votes
4answers
104 views

Why isn't the derivative of $|2x^2-3x|$ equal to $|4x-3|$?

I don't quite understand why this is the case? Since when differentiating $|2x^2-3x|$ you get $\frac{(2x^2-3x)(4x-3)}{|2x^2-3x|}$...... when it is $2x^2-3x$, the derivative is $4x-3$ and when it is ...
0
votes
0answers
50 views

How to determine whether expression is positive or negative?

Given expressions $|x - 3 + y|$ and $|x + 3 + y|$ how can I determine, whether are those positive or negative, and determine their value in the intervals of: $y < -x - 3$ $y \in [-x - 3, 3 - x)$ ...
0
votes
2answers
23 views

Is $ \left| \lim_{t \rightarrow 0}{\frac{f(x+ty) - f(x)}{t}} \right| \leq \lim_{t \rightarrow 0}{ \frac{|f(x+ty) - f(x)|}{|t|}}$

Suppose $X$ is a Banach space and $f:X \rightarrow \mathbb{R}$ is a continuous function. Is it true for all $x,y \in X$ that $$ \left| \lim_{t \rightarrow 0}{\dfrac{f(x+ty) - f(x)}{t}} \right| \leq ...
1
vote
2answers
23 views

Absolute value inequality verification

This is a pretty trivial question, but I'm trying to list out steps to show that if $|x-c|<1\Rightarrow |x|\leq |c|+1$. Is there a trick with the triangle inequality? Thanks
0
votes
1answer
65 views

Find the absolute maximum and absolute minimum values of f on the given interval. f(t) = 8t + 8 cot(t/2), [π/4, 7π/4]

Having a little bit of trouble figuring out this problem here. Find the absolute maximum and absolute minimum values of $f$ on the given interval. $f(t)$ = $8t + 8 cot(t/2)$, ...
2
votes
2answers
39 views

How to integrate $|x^n|$

How to integrate $|x^n|$? The answer was given as $\frac{|x^n|x}{n+1}$. How do you find this? I would also like to know how do find the anti derivative of a modulo function when interval is not given. ...
1
vote
4answers
76 views

Why if $a<b$ and $-a<b$ we can say that $|a|<b$?

Why if $a<b$ and $-a<b$, then we can say that $|a|<b$? Maybe this is trivial by I don't know how to proof it.
0
votes
2answers
68 views

How can we prove that $\sqrt{ x^{2} }$ is equals to $|x|$?

I used to use this equality at school. But now in my books of Analysis this property is not mentioned. Is this maybe incorrect?
2
votes
0answers
46 views

Reformulate absolute value as quadratic problem

I'm looking for standard approach to reformulate this objective function. The aim is to find values of $x_i$ that are close to either $y_i$ or $-y_i$ ($y_i$s are known) in a least-squares sense: ...
1
vote
1answer
95 views

Expected value of the absolute value of the difference of two random variables

I have to compute the absolute value of an estimator defined as $T_5=\frac{1}{2}E[|X_1-X_2|]$ in order to state if it is unbiased for $\sigma$, where $X$ is distributed as a $N(0,\sigma^2)$. I am ...
2
votes
1answer
50 views

Ideas for a limit calculation

The limit to show is the following: $$ \lim_{t\to \infty} \int_\mathbb{R}\left|\frac{-\sin x \sin tx}{x^2} \right|dx $$ A direct splitting of the integral into $\int_{-\infty}^0+\int_0^{+\infty}$ in ...
0
votes
1answer
19 views

Question about solving absolute value equation.

What is the sum of all possible solutions to this equations? $|x+4|^2 -10|x+4|=24$ My attempt: Since $(x+4)^2=|x+4|^2$, so I can ignore the absolute sign of the first term. So we only need to deal ...
4
votes
5answers
91 views

Solve $ \left|\frac{x}{x+2}\right|\leq 2 $

I am experiencing a little confusion in answering a problem on Absolute Value inequalities which I just started learning. This is the problem: Solve: $$ \left|\frac{x}{x+2}\right|\leq 2 $$ The answer ...
0
votes
1answer
29 views

Inequality Involving Absolute Value

Let $f$ be a differentiable function on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. If the derivative $f'$ of $f$ is also continuous on $[0,1]$, prove that: $ \int_0^1 |f'(x)-f(x)| dx \geq \frac{1}{e}$. ...
1
vote
1answer
34 views

How to expand this expression?

How to express this expression $\frac{z}{2}<|y|<z$. Is it correct to expand it as following $-z<-\frac{z}{2}<y<\frac{z}{2}<z$
1
vote
2answers
41 views

When solving an equation with absolute value on both sides, how to choose the side to work with?

When solving an equation with absolute value on both sides, such as $$|2x-1|=|4x+3|$$ how to choose one side of which to use the definition of absolute value? For example, if we apply absolute value ...
0
votes
1answer
37 views

How to find the minimum of $c|1+x|^n+|1-x|^n$

How to find the minimum of the \begin{align} f(x)=c|1+x|^n+|1-x|^n \end{align} for $n \ge 1$ and $c > 0$. If we take the derivative of $f(x)$ we get \begin{align} f'(x)=-c {\rm sign}(1+x) ...
0
votes
0answers
29 views

Cauchy Determinant with Absolute Values

This is perhaps a straightforward question but I'm a little confused. An $n\times n$ Cauchy matrix $A$ is a matrix with entries $$a_{i,j}=\frac{1}{x_i-y_j}$$ for $1\le i,j\le n$, where $x_i$ and $y_j$ ...
0
votes
1answer
69 views

if |f| is periodic then f is periodic [duplicate]

Decide whether the following statement about a function f: R -> R is true. If |f| is periodic, then f is periodic. Give a proof or counterexample.
0
votes
2answers
61 views

How do I solve a quadratic inequality with absolute value using cases?

$$\left| x^{ 2 }-5x+5 \right| \le x$$ Steps I took: Using the quadratic formula, I split the solutions up into: Case 1: $$ x^{ 2 }-5x+5\le 0$$ $$x\le \frac { 5+\sqrt { 5 } }{ 2 } and\quad x\ge ...
0
votes
1answer
96 views

Prove or disprove: If $ f $ is periodic, then $|f|$ is also periodic.

I started with the definition of periodic function and absolute value function. And I do it with discussing different cases of $x$ and $p$. But I got stuck with when $ -p\leq x <0 $ , I want to ...
1
vote
1answer
31 views

Meaning of abs. value of a sigma field?

In a hand out I saw the a notation which looks to be the absolute value of a sigma field F, |F|. I googled it but I could not really find what it means and the notation confuses me. Anyone that might ...
1
vote
1answer
30 views

Can one realize the real part of every entire function $f$ as $\ln| g|$ with $g$ entire?

Let $\Re$ denote real part and $|\cdot|$ absolute value. Does there exist, for every entire $f$, an entire $g$ such that $\Re f = \ln |g|$ ?
0
votes
0answers
56 views

Use axioms to solve inequality $| x-2| +| x-4| < 1$

I have a feeling that the inequality is false for all values of x, but I don't know at which point that should have become clear to me. I am supposed to solve the inequality using $x < a$ ...
3
votes
2answers
35 views

Solve $[\frac{x^2-x+1}{2}]=\frac{x-1}{3}$

How can one solve the equation : $[\frac{x^2-x+1}{2}]=\frac{x-1}{3}$ ? Such that $[x]$ is the integer part of $x$. By definition : ...
3
votes
5answers
98 views

How to solve $\lvert{x}\rvert - \lvert{2+x}\rvert = x$?

How do I solve the following equation? $$\lvert{x}\rvert- \lvert{2+x}\rvert= x$$ I was thinking about dividing it into 4 cases: plus plus, plus minus, minus plus and minus minus. What is the best way ...
1
vote
2answers
40 views

Integral of absolute value: $\int_{-\infty}^\infty {e^{-\frac{2}{b}|x - \mu |}}dx$

I am stuck trying to integrate $$\int_{-\infty}^\infty {e^{-\frac{2}{b}|x - \mu |}}dx$$ Incidentally, I'm interested in solving equation (5) in this paper using the Laplace distribution. I just got ...
2
votes
2answers
58 views

Solution of integral involving exponential and absolute values [closed]

I want to solve this integral $\int_0 ^\infty e^{-iwx}e^{-α|x|} dx$ Any ideas on how to solve it?
0
votes
1answer
28 views

Why does my book say that "since a major change takes place at (1,1), the expression in the absolute value should equal to zero at x=1?

Question I am referring to: Why does the absolute value portion of the expression of the function this graph corresponds to has to be 0 when x=1? Here are the five choices:
0
votes
1answer
61 views

absolute value inequality with complex number

Strangely, I don't find easily on the internet sources about inequalities with complex numbers. In this moment, I am interested to absolute value inequalities with complex numbers but would be good ...
1
vote
3answers
64 views

Different way to solve $|x-3|<|2x|$

I know two ways I can solve $|x-3|<|2x|$ By squaring both sides By interpreting the inequality as a statement about distances on the real line. Question: How can I solve this inequality ...
2
votes
1answer
58 views

Solve the equation within 'floor function'

I added my solution, but I'm not sure I've got it right. I'd like to know what you think. The question: Solve the equation: $$\lfloor |x+1|-|x| \rfloor \ge x^2.$$ the left and right symbols ...
0
votes
3answers
56 views

Is it possible to find the absolute value of an integer using only elementary arithmetic?

Using only addition, subtraction, multiplication, division, and "remainder" (modulo), can the absolute value of any integer be calculated? To be explicit, I am hoping to find a method that does not ...
3
votes
1answer
152 views

If $|ax^2+bx+c|\le 1\ \forall |x|\le 1$, then what is the maximum possible value of $\frac 83a^2+2b^2$? [closed]

Let $f(x) = ax^2 + bx + c$ ; $a,b,c\in\mathbb R$ It is given that $|f(x)| \le 1$ $\forall |x| \le 1$ Q1) The possible value of $|a+c|$, if $\displaystyle \frac{8}{3} a^2 + 2b^2$ is maximum, is ...
1
vote
0answers
29 views

Is this a misprint or am I missing something?

What I'm given is this: Evaluate: x = 5, |x| -2 I'm thinking they probably mean |x|=-2, in which case the evaluation would be false. But then again I second ...
2
votes
3answers
55 views

Definition of limit with$ f(x)=|x^3|$

Using the definition of the limit I tried to find the derivative of $f(x)=|x^3|$. I came up with: $$f'(x)=\frac{3x^5}{|x^3|}$$ Question: Why is the derivative (according to this answer) not defined ...
0
votes
0answers
25 views

Absolute value with inequalities

Can we solve the following $ |f(x)| + |g(x) | < b$ by taking the intersection of the solutions for $f(x) + g(x) < b$ $-f(x) - g(x) < b$ $f(x) - g(x) < b$ $-f(x) + g(x) < ...
1
vote
0answers
31 views

Show that $|ax^2+bx+c|<1$ gives us $|c|<1$

$a,b,c \in \mathbb{R}$ and for all $-1<x<1$ Show that if $|ax^2+bx+c|=<1$ So : 1) $|c|=<1$ 2) $|a+c|=<1$ 3) $a^2+b^2+c^2=<5$ For the first one ; if I choose x=0 So |c|=<1 ...
0
votes
0answers
17 views

Graphing an Absolute Value Equation

How would I graph the equation: abs(x)+abs(y)=1+abs(xy). I have tried to consider cases, but am not sure if I need to graph it by considering a piecewise-defined function.
0
votes
1answer
25 views

how to rearrange for y for: $\ln{|y|}= \frac{x^3}{3}$?

Is it $y= \pm e^{\frac{x^3}{3}}$? I am not sure about whether the plus minus sign is correct. Just here to look for confirmation or correction! Thanks!
9
votes
2answers
204 views

Modulus Equations

$$ |x + 1| + |x − 1| = x + 4$$ The only way I can solve this equation is to graph it...Through graphing, I get the following solutions: $$x = -\frac{4}{3}, 4$$ Is their a general algebraic method ...
3
votes
3answers
89 views

How to solve this absolute value inequality? $ |x| + |x - 2| \gt 5 $

I'm not sure how to solve this inequality. Can someone please explain step-by-step? Thanks! $ |x| + |x - 2| \gt 5 $
1
vote
2answers
29 views

Finding a Real Number using epslion

Fix a real number $x$ and $\epsilon>0$. If $|x-1|\le \epsilon$ show $|2-x|\ge 1- \epsilon$ I think we were supposed to use the triangle inequality to show this. If we use the triangle inequality ...
1
vote
0answers
43 views

Compare absolute values of two expressions!

I have two sets of numbers as follows: $$X = \{x_1, x_2, ..., x_n\}\\ Y = \{y_1, y_2, ..., y_n\}$$ And a number $r$. Let $x^\ast$ and $y^\ast$ is average values of set $X$ and $Y$ respectively. ...
0
votes
1answer
56 views

Max function is a metric?

I was wondering if the max function is a metric or if, in particular, $\max(|x + y|, 1)$ is equal or less than $\max(|x|, 1) + \max(|y|, 1)$ with $x$ and $y$ belonging to $R$.
0
votes
0answers
24 views

Turning points of an absolute value equation

Given the equation $$\left| x\; -\; c \right|\; +\; \left| x\; +\; c \right|\; -\; \left| y \right|\; -\; \left| \left| x\; -\; c \right|\; -\; \left| y \right| \right|\; = b \text{,} -c, c, b \in ...
0
votes
1answer
99 views

Absolute Value equations in 2 variables (both x and y ) (a relation based on absolute)

I came across equations in a math book that contains absolute of both x and y. I have done many complicated absolute equations and inequalities, but with only the x being in absolute bars. I don't ...