For questions about or involving the absolute value function.

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2
votes
2answers
110 views

solving the inequality

I'm looking for hints on how to efficiently solve this inequality: $$\left( \frac {|x|-|1-x|}{|x|} \right)^{2x-1} \gt \left(\frac {|x|-|1-x|}{|x|} \right)^{8-x} $$
-2
votes
0answers
28 views

Definite Integration.Trigonometric function [on hold]

How to integrate $$3\sqrt { \cos ^{ 2 }{ \left( t \right) \sin ^{ 2 }{ \left( t \right) +\sin ^{ 4 }{ \left( t \right) \cos ^{ 2 }{ \left( t \right) } } } } } $$ for $t\epsilon \left[ 0,2\pi ...
1
vote
2answers
26 views

Where did I go wrong with this inequality involving absolute value function?

Question: Find all $x \in \mathbb R$ such that the inequality $4<|x+2|+ |x-1|<5$ is satisfied. This is my attempt at solving the problem: Case (i): If $x+2 \geq 0 $ and $ x-1\geq0$, then ...
-4
votes
2answers
52 views

SAT question stuck? [on hold]

I am preparing for sat and this question, I have no idea how to solve it. Please provide step wise solution also. If $2|x+3|=4$ and $\frac{|y+1|}{3}=2$, then $|x+y|$ could equal of the following ...
0
votes
2answers
52 views

$|x| - |y| \leq |x-y|?$

Is there a clever way to show that $$|x| - |y| \leq |x-y|$$ I believe I can think of a way to solve this using cases, but the book I'm working out of said that "A very short proof is possible if you ...
0
votes
3answers
31 views

I do not quite understand this difference in limits

According it my study material: $\lim_{x\to 0^-}\frac {x}{|x|}= -1$ and $\lim_{x\to0^-} \frac {1}{|x|}= \infty$ Why does $\lim_{x\to0^-} \frac {1}{|x|}\ne -\infty$ as 1 still devided by a negative ...
1
vote
3answers
57 views

Proving a limit exists - solving for epsilon with absolute values

I have the equation that I want to prove the limit goes to 1: $$\lim_{n \to \infty} \frac {(n+8)(n+1)}{n(n-10)} = 1$$ Using definition of limit, I get this equation: $$ \left | \frac ...
2
votes
3answers
60 views

For which $a$ does $|x+1|+|2-x|=a^2 -1$ have exactly two solutions?

If it is not a problem, I would really appreciate if someone could explain to me how to solve and graph the following equation: For which real numbers $a$ does the equation $|x+1| +|2-x|=a^2 -1$ ...
2
votes
1answer
31 views

Does “Expected Absolute Deviation” or “Expected Absolute Deviation Range” exist in stats and have another name?

So everyone is familiar with Variance and Standard Deviation from high school, but it seems no one has any familiarity with a philosophical justification for such weird, seemingly arbitrary measures. ...
0
votes
4answers
55 views

How to solve Absolute Value Inequality: |x-1| ≥ 3-x

I am learning the topic of solving absolute value inequality question. I had mostly understood the steps in order to solve for an inequality. However, I'm still clueless of a step to solve the ...
0
votes
2answers
29 views

Spivak Absolute Value Problem (Prologue 9-v)

I'm working on the following problem Express the following with at least one less pair of absolute value signs $$|(| \sqrt2 + \sqrt3| - |\sqrt5 - \sqrt7|)|$$ Now I can see that the ...
0
votes
4answers
54 views

solving the system

solve the system : $$ y+|x-2|=3 $$, $$ |x+y|= m $$ graphicly when $m$ equals $6$. I can easily (realtively) skecth the first graph , however, how the bloody hell do you sketch $|x+y|= 6$??
5
votes
1answer
49 views

Maximum value problem

A function $\hspace{0.1cm}$$f:[0,1]\to[-1,1]$$\hspace{0.1cm}$ satisfying$\hspace{0.1cm}$ $|f(x)|\leq x$$\hspace{0.1cm}$ $\forall x\in[0,1]$. Then find the maximum value of: ...
1
vote
2answers
37 views

Restore the signum of abs(sinc(x))

Is it possible, by any means, to restore the signum of sinc(x) after being transformed to its absolute value, abs(sinc(x))? How it got to abs() is irrelevant, I only want to know if the reverse is ...
2
votes
2answers
24 views

Concerning Rules of Exponents & Absolute Value

I understand that one of the accepted definitions of the absolute value function is $\left| x \right| = \sqrt{x^2}$. However, I do not understand why if I substitute $-5$ in for $x$ that I can't do ...
0
votes
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
100 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
239 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 ...
1
vote
1answer
27 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
178 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
77 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? $$x^2 - 3x - 28 = 0$$ $$(x+4)(x-7) = 0$$ ...
1
vote
1answer
40 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 ...
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
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 ...
1
vote
3answers
47 views

How do you solve two equal absolute value expressions?

I'm having trouble understanding how the following is solved. $$|x+1| = |x-2|$$
3
votes
4answers
163 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
2answers
23 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
17 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 ...
2
votes
1answer
90 views

Solving absolute inequality

I have the following inequality: $$|4 - k^2| > |10 + 13k|$$ So how to solve this ?
2
votes
5answers
110 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
votes
2answers
54 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
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.
-1
votes
2answers
56 views

Sketching a set of complex numbers and deducing the value of $|z +1 - i|$ for such numbers

The point $P$ represents the complex number $z$. a) Given that $\arg(\frac{z-2i}{z+2}) = \frac{\pi}{2}$ , sketch the locus of $P$. Ok so I've sketched this and this is what it looks like : b) ...
5
votes
5answers
174 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 ...
1
vote
2answers
42 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
51 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 ...
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
86 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
votes
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
28 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 ...
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
votes
1answer
23 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 ...
7
votes
4answers
273 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 $n$ ...
1
vote
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
29 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 $x\geq6$ ...
1
vote
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 ...
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 ...
2
votes
2answers
90 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
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 ...