For questions about or involving the absolute value function.

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0
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0answers
8 views

Difference in magnitude between two cross-correlations by two different way of calculations.

I think there are two ways of calculating cross-correlations for two difference random variables, X and Y. I am assuming discrete functions. 1) Multiplication $$ \sum_{m=-\infty}^\infty x[m]y[m+n] ...
0
votes
2answers
80 views

Prove: If $a\in\mathbb Z$ and $|a| > 1$, then $1/a \notin \mathbb Z$.

Prove: If $a$ is an integer and $|a| > 1$, then $1/a$ is not an integer. Hi, I need help proving this either by contradiction or contrapositive. I'm not sure where to begin
3
votes
4answers
232 views

inequality method of solution

Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$ I've tried first determining when the absolute value ...
0
votes
0answers
36 views

Sum of absolute value of cos(an+b)? [closed]

Can anybody help me on deriving this? where, a and b are constants. This converges since N is finite, and the converged values for most of a and b are close each other for sufficiently large number N. ...
1
vote
1answer
26 views

Limit of functions absolute value

$$ \begin{align} &\lim\limits_{x\to0} \frac{|3x-1|-|3x+1|}x\\ =&\lim\limits_{x\to0} \frac{(3x-1)^2-(3x+1)^2}{x(|3x-1|+|3x+1|)}\\ =&\lim\limits_{x\to0} \frac{-12x}{x(|3x-1|+|3x+1|)} = ...
2
votes
3answers
177 views

Derivative and integral of the abs function

I would like to ask about how to find the derivative of the absolute value function for example : $\dfrac{d}{dx}|x-3|$ My try:$$ f(x)=|x-3|\\ f(x) = \begin{cases} x-3, & \text{if }x \geq3 \\ ...
3
votes
3answers
74 views

Please help with absolute value $|x^2 - 3x| = 28$

Just a question about solving an absolute value equation: $$|x^2 - 3x| = 28$$ Do I just solve this as if the absolute value brackets weren't even there? $$x^2 - 3x - 28 = 0$$ $$(x+4)(x-7) = 0$$ ...
5
votes
3answers
126 views

How would I prove $|x + y| \le |x| + |y|$?

How would I write a detailed structured proof for: for all real numbers $x$ and $y$, $|x + y| \le |x| + |y|$ I'm planning on breaking it up into four cases, where both $x,y < 0$, $x \ge 0$ ...
1
vote
1answer
39 views

Do modulus and absolute value operations use the same sign?

Do modulus and absolute value operations use the same sign? If so, do we always assume that a modulus is intended when the number is complex? If an expression says $|a+bi|$, this means I should ...
4
votes
3answers
481 views

Proof for Integral Inequality $|\int f| \le \int |f|$ - is it sufficient enough?

Claim: If f is integrable, $\left|\int_a^bf(x)dx\right|\le\int_a^b|f(x)|dx$ Proof (attempt): We know $-|f|\le f \le|f|$, so $\int-|f| \le \int f \le \int|f|$.* Since, if $-b<a<b$, we say ...
0
votes
1answer
278 views

Absolute of a trig function

Consider the function $$f(x) = 1\dfrac{1}{2} - 3\sin \left(\dfrac{1}{2}x \right). $$ I need to find the absolute of this function, which to my eye would just be $$ f(x) = 1\dfrac{1}{2} + 3\sin ...
0
votes
2answers
70 views

Derivative of $f(x)=|x|$

Okay, so $\displaystyle \frac{d}{dx} |x| = \frac{|x|}{x}$. But I have trouble seeing why. Here's what I've tried: $$\frac{d}{dx}|x|=\begin{cases} \frac{d}{dx}x & \text{if }x > 0 \\ ...
0
votes
1answer
39 views

Infimum of absolute values versus absolute value of infimum

Let $A\subseteq\mathbb R$. Is there a nice proof of the inequality $\displaystyle\inf_{a\in A} |a|\le|\inf_{a\in A} a|$? The only proof I know is, though not very difficult, annoying because it ...
0
votes
3answers
92 views

Absolute values don't work

I don't understand, how absolute valued could possibly be considered well defined. As shown here, $|a| = |-a| , ||a|| = |-|a||$ So lets take $a=-2, |a| = -2 = |-a|,$ but $|-a| = |2| = 2$ But it ...
1
vote
3answers
45 views

How do you solve two equal absolute value expressions?

I'm having trouble understanding how the following is solved. $$|x+1| = |x-2|$$
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4answers
159 views

Show that $|z+1|\le|z+1|^2 +|z|$ for all $z \in \mathbb{C}$

Question: Show that $|z+1|\le|z+1|^2 +|z|$ for all $z \in \mathbb{C}$ So far I have, Suppose $1\le|z+1|$ $|z+1|\le|z+1|^2$ $|z+1|\le|z+1|^2+|z|$ Now I must show $|z+1|<1$ but this is where ...
0
votes
1answer
23 views

How do I solve the following absolute value equation?

I'm having trouble solving this equation: $$|x+1| = |2x-2|$$ For $x+1 = 2x-2$ and $-(x+1) = -(2x-2)$ I received $x = 3$ and for $-(x+1) = 2x-2$ and $x+1 = -(2x-2)$ I received $x = 1/3$ I tried ...
0
votes
2answers
22 views

Filling in the derivative of the absolute value at zero

I have a function $f(x)$ such that $f(x_0)=0$ and I'm interested in the derivative $\frac{d |f(x)|}{dx}$ evaluated at the point $x_0$. I realize that this is usually undefined. However, if ...
0
votes
1answer
16 views

Modulus function (working out coordinates)

Lets say you have $y = -|3x - 1|$ when working out where it cuts the axis, particularly the x-coordinate you do the following when $y = 0, 3x - 1 = 0$ therefore $x = 1/3 $ the modulus and the ...
6
votes
4answers
257 views

Inequality for absolute values

How do you show either of the equivalent inequalities: $$2(|a|+|b|+|c|)\leq |a+b+c|+|a+b-c|+|a-b+c|+|a-b-c|$$ or $$|x+y|+|x+z|+|y+z|\leq |x|+|y|+|z|+|x+y+z|$$ Hold for complex numbers or in $n$ ...
2
votes
1answer
89 views

Solving absolute inequality

I have the following inequality: $$|4 - k^2| > |10 + 13k|$$ So how to solve this ?
2
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5answers
106 views

Finding the minimum value of a sum [closed]

Let $x,y,z$ be real numbers . Find the real number $a$ so that $S$ has a minimum value , where $$S=|x-a|+|y-a|+|z-a| .$$
0
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2answers
51 views

Does the absolute value of +3 lose its positive direction yet have its positive value? [closed]

We have no sigh with the absolute value of +3, yet its value is positive.(Wikipedia) Does this mean that the absolute value doesn’t have its positive direction (+3 is located on positive direction ...
0
votes
3answers
432 views

Integrating absolute value function

I'm working on a problem drawing phase plane diagrams in my applied mathematics course. I'm supposed to draw the phase line diagram of $x''+\vert x\vert=0.$ In the process, I get to the differential ...
0
votes
1answer
24 views

$|2- (\sqrt{n^2+4n} - n)| ≥ \frac{1}{10}$

Any suggestions how to solve the following equation: $|2- \sqrt{n^2+4n} + n| ≥ \frac{1}{10}$ Thank you in advance.
5
votes
5answers
164 views

Evaluating the following integral: $ \int \frac{x^2}{\sqrt{x^2 - 1}} \text{ d}x$

For this indefinite integral, I decided to use the substitution $x = \cosh u$ and I've ended up with a $| \sinh u |$ term in the denominator which I'm unsure about dealing with: $$\int ...
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vote
2answers
41 views

Maximal distance between points on a line

Two points A and B are on different sides of a line. Find a point Y on the line such that the absolute value of the difference from Y to A and Y to B is maximal. My thoughts are as follows. Let's ...
0
votes
1answer
50 views

Integral of absolute value = absolute value of the integral

Let $(a,b) \in \mathbb{R}^2$ and $f \in C^0([a, b] , \mathbb{C})$ Find the condition on $f$ so that $$|\int_a^b f|=\int_a^b|f|$$ My try : The function $f: t \mapsto r(t)\exp(i\theta)$ where $r$ is a ...
0
votes
1answer
60 views

Question about one of the first problems in Spivak's Calculus

It's about Chapter I, Problem 21 from Spivak's Calculus: Prove that if: $|x - x_0| < \frac{\epsilon}{2}$ and $|y - y_0| < \frac{\epsilon}{2}$ then $|(x + y) - (x_0 + y_0)| < \epsilon$ ...
1
vote
5answers
76 views

Calculate $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$

I know that $\displaystyle \lim_{x \to \frac{3}{2}} \frac{2x^2-3x}{|2x-3|}$ does not exist, because the lateral limits are different and I also know that the absolute value on the denominator has ...
1
vote
4answers
85 views

Simple question about the range of possible values for a function

So we have $2 |3-x| + 5 = k$, where $k$ is a constant. Provided this equation has two real solutions for $x$, what is the range of possible values for $k$?
2
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6answers
106 views

Solving $|\frac{x+1}{x}|< 1$

I need some help/suggestions solving the following math problem. I don't know how to continue from step 2. Find x. 1.) $\displaystyle\left|\frac{x+1}{x}\right|< 1$ 2.) ...
0
votes
1answer
24 views

Linear Programming : Alternative to summation of absolutes in constraints

I am solving a placement problem, i.e. map $integers\ i\ from\ 0\ to\ 6$ to $(x_i,y_i)\ st\ 1 \le x_i,y_i\le 3$ such that : $ \sum\limits_{i=0}^6 \sum\limits_{j=0}^6 Cost(i,j)*(|x_i - x_j | + | y_i ...
12
votes
6answers
883 views

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.

Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$. I do not understand how to go about completing this problem or even where to start.
0
votes
1answer
30 views

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $?

$x \ge |a| \leftrightarrow x \ge a \land x \ge -a $ ? WTS $x \ge |a| \rightarrow x \ge a \land x \ge -a $     Since $|a| > -a$ then we have $x \ge -a$ ...
0
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1answer
22 views

Absolutet Value Inequality with cases number line

I was wondering if anyone knows how to solve $|ax+b|<cx+d$ type questions by using cases and the number line to finish. I am personally struggling with the number line, I have half-finished a ...
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0answers
14 views

Cumulative distribution function of a model similar to the multinominal distribution

I would like to solve a problem similar to the multinominal distribution (http://en.wikipedia.org/wiki/Multinomial_distribution): For k independent trials each of which leads to a success for ...
3
votes
2answers
28 views

$5-3|x-6|\leq 3x -7$

I have this inequation: $$5-3|x-6|\leq 3x -7$$ i solved this this way: i said, for $x\geq6$ is the modulus positive, so I made 2 cases in which the modulus gives + or - : 1) for $x\geq6$ ...
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1answer
23 views

Quadratic inequality with absolute values

I've decided to study calculus on my own, and I've started working on "A First Course in Calculus" by Serge Lang, 5th edition. Now I'm just reading the chapter on preliminaries, and there is a section ...
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2answers
86 views

Why can't absolute values be expressed with negative numbers. [closed]

The answer to this question seems obvious. 'An absolute value expresses the quantity of ones between any number and 0'. But does that mean it must be positive? I took a shot at answering my ...
0
votes
2answers
31 views

Need help with this absolute value equation

I need to solve the following equation involving absolute value: $$|x-1| = 1-x$$ Looking at the term $x-1$, I thought I'd divide the interval into parts: $x < 1$ and $x \geq 1$. Now, when ...
0
votes
3answers
23 views

What's the best method to graph the following function by hand…

Here in my exercise I have to study the function and draw its graph. Can you please tell me what's the best method to do this, because I don't think that's reasonable to use the input output method, ...
0
votes
1answer
29 views

Finding best fitted value for power function. please help!

I need to find: 1. the best fitted value for $a$ in the power function 2. the best fitted value for $b$ in the power function Data given: I know that $b=bi$ and $a=e^{bo}$ --> my question is how ...
2
votes
2answers
45 views

Basic question about solving modulus equation

It common in the literature to solve the modulus equation like $|x+5|+|x-1|=8$ by dividing into cases when $x<-5$, $-5\leq x<1$ and $x\geq1$. My question is whether dividing into cases is ...
2
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3answers
2k views

Proof of triangle inequality

I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
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vote
2answers
44 views

Prove that $|x+y| \leq |x|+|y|$ [duplicate]

How to Prove the triangle inequality which says for all x (no matter how big or small) and for all y (no matter its size) in the set of irrational+rational numbers, this holds: $|x+y| \leq |x|+|y|$
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vote
2answers
51 views

Suppose f(x) is an odd function. Prove that g(x) = |f(x)| is an even function.

Suppose f(x) is an odd function. Prove that g(x) = |f(x)| is an even function. I understand that an odd function is where f(-x) = -f(x), and an even function is where f(-x) = f(x), but am struggling ...
1
vote
0answers
50 views

Absolute values nested multiple times

Is there any algorithm to quickly determine "zero points" (i.e. points with undefined derivation) of absolute values functions which are nested multiple times? I do know, that any part of this ...
3
votes
1answer
28 views

no. of positive integral solutions of ||x - 1| - 2| + x = 3

What are the no. of positive integral solutions of ||x - 1| - 2| + x = 3 ? My effort ||x - 1| - 2| = 3 - x |x - 1| - 2 = 3 - x OR |x - 1| - 2 = x - 3 |x - 1| = 5 - x OR |x - 1| = x ...
0
votes
1answer
30 views

Absolute value being an odd function

Correct me if I am wrong, but I learned that for a function to be symmetrical to the origin, it can be rotated 180 degrees and still appear the same. How is ${x^2 - y^2 = 0}$ an odd function if when ...