For questions about or involving the absolute value function.

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1answer
215 views

Intersection of a point and absolute value function contained within a circle

I'm attempting some crazy ideas while programming a game and ran into the following math problem that has been bugging me for a few days: Given a unit circle and a random point $P$ within the circle, ...
0
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1answer
136 views

Absolute value of infinite sum smaller than infinite sum of absolute values

A question emerging from an exercise in Ok, E. A. (2007). Real Analysis with Economic Applications. Princeton University Press. The exercise consists in showing that if $\sum_{i=1}^\infty x_i$ ...
3
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2answers
277 views

Prove the monotonicity of the expectation of a binary random variable function

Consider $R$ independent binary random variables $y^1, \ldots, y^R$ over the space $\{-1, +1\}$ such that $\Pr(y^j = 1) = p^j \geq 0.5$ and $\Pr(y^j = -1) = 1 - p^j$, $\forall j = 1,\ldots,R$. ...
3
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1answer
145 views

Question based on Triangle Inequality $\displaystyle |x+y|\leq |x|+|y|$

If $x,y,z\in \mathbb{R}-\left\{0\right\}$. Then prove that $\displaystyle 1\leq \frac{|x+y|}{|x|+|y|}+\frac{| y+z|}{| y |+| z |}+\frac{| z+x|}{| z |+| x |}\leq 3$ My Try:: Using Triangle Inequality ...
2
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1answer
66 views

Simplifying $\left|\left|\sqrt{-x^2}-1\right|-2\right|$

How do we simplify the expression $\left|\left|\sqrt{-x^2}-1\right|-2\right|$? This is very confusing. Do they cancel out and become just simply $\sqrt{-x^2}-1-2$?
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0answers
94 views

Calculation of the sub gradient of the first norm of a matrix

Lets say I have a matrix X and its first norm $||X||_1$. How do I calculate the subgradient of this norm with respect to matrix X itself.
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1answer
296 views

How to graph an absolute value equation?

How would you graph: $|x+y|=1$ ? I can do the normal $y=|x+1|$ and all that. But how would you do a question with two of these unknowns in the absolute value? Any help would be greatly appreciated, ...
2
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1answer
75 views

Is any norm on $\mathbb R^n$ invariant with respect to componentwise absolute value?

Given $\mathbf{x}=(x_1,...,x_n) \in \mathbb{R}^n$ , define $ \mathbf{x}'=(|x_1|,...,|x_n|) $ . Then, is it $||\mathbf{x}'|| = ||\mathbf{x}||$ for every norm on $ \mathbb{R}^n $ ? NB: The answer ...
0
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1answer
163 views

How to linearize the following LP

I want to minimize $|d_1-d_2|+e1+e2+e3$ where $d_1,d_2,e_1,e_2,e_3>=0$ and $|.|$ denotes the absolute value, for some linear constraints. Is there any way I can linearize the objective function?
5
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2answers
204 views

Absolute values in $\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$

in my math class we were given a list of indefinite integrals, and one of them was: $$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}$$ My working: $$\int \frac{dx}{(x+2)\sqrt{(x+1)(x+3)}}=\int ...
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1answer
466 views

how to find absolute value for complex fraction

I have a Fourier transfer equation $H(jw) = \frac{jwL}{(jw)^2LC+jw\frac{L}{R}+1}$, and I need to find frequency to make $|H(jw)|$ is max. I know I should take the derivative of $|H(jw)|$ then find ...
7
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2answers
551 views

How does one DERIVE the formula for the maximum of two numbers

I want to derive (not prove that this is true) the formula $\max (x,y) = \dfrac{x + y + |y-x|}{2}$ I was reading a proof (which they have the result ahead of time already) that we do cases and then ...
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2answers
67 views

Prove That $|a +b| = |a| +|b|$ if $a$ and $b$ Have Same Signs, And $|a +b| < |a| + |b|$ if $a$ and $b$ Have Opposite Signs (Proved Differently) [duplicate]

My Proof: This problem has mainly four cases, they are as follows: 1) $a, b > 0$ 2) $a, b < 0$ 3) $a > 0 > b$ 4) $a < 0 < b $ Let suppose that the sum of the real numbers $a ...
1
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1answer
89 views

What is the modulus of a number?

What is the exact definition of the modulus of a number? As far as I know, it is the distance between the origin and the point associated with this number. So if $z=a+bi \in \Bbb ...
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5answers
113 views

Prove That $|a +b| = |a| +|b|$ if $a$ and $b$ Have Same Signs, And $|a +b| < |a| + |b|$ if $a$ and $b$ Have Opposite Signs

My Proof: $|a +b| = |a| +|b|$ ..... $(i)$ $|a +b| < |a| + |b|$ ..... $(i)$ If $'a'$ and $'b'$ have same signs: Let $a$ and $b$ be equal to $-x$. Replacing $a$ and $b$ with $-x$ in the equation ...
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2answers
95 views

Absolute Value of $|-3 -2|$

$|-3 -2|$ is the distance between the points $-3$ and $-2$. If we solve it further then, in one way I get $|-5| = 5$. But $5$ is the distance between $0$ and $-5$ in this case. In other way, $2 ...
0
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2answers
79 views

Absolute Value Problem, Solution and Method

Please check my method and also if I have solved the following problem correctly: Problem: $f(x) = |x - \frac12| + |x + \frac12|$ If $x = -1$, then: $f(-1) = |-1 - \frac12| + |-1 + \frac12|$ From ...
2
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3answers
105 views

Question about absolute value in inequalities

My book presents the following: $$7 \le x \le 9 $$ so $$ -1 \le x - 8 \le 1 $$ and $$ |x-8| \le 1$$ I usually get confused with the way that taking the absolute value of an expression works. Could ...
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3answers
60 views

finding values for absolute convergence

Find all values of real number p or which the series converges: $$\sum \limits_{k=2}^{\infty} \frac{1}{\sqrt{k} (k^{p} - 1)}$$ I tried using the root test and the ratio test, but I got stuck on ...
0
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1answer
433 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 ...
2
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2answers
94 views

Absolute Convergence of a Series

Find all values of real number p for which the series converges absolutely $$\sum_{k=2}^{\infty} \frac{1}{k\, (\log{k})^p}$$
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2answers
91 views

Calculating the integral $ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$

How do we calculate the following integral: $$ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$$
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2answers
106 views

How to evaluate the inequality $|x+1|<-1$?

Okay perhaps the title isn't specific enough, I didn't know how to word it exactly. I'm finding the interval of convergence for a power series and i know the answer to be (-2,0] I end up with the ...
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2answers
220 views

Can summations distribute across absolute values?

Can I distribute a summation as follows? $$ k\sum_{x \in X} \left| x - b \right| = \left| \left(k\sum_{x \in X}x \right) - \left( k\sum_{x \in X}b \right) \right| $$
1
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2answers
101 views

Finding absolute max and min values of function

Function given as $f(x,y) = 3x^2 + 2xy^2$. If $(x,y)$ lies in the region inside including edges of the triangle in the first quadrant given by $x\ge0, y\ge0, y\le2-x$. Reduce $f$ to a single variable ...
0
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1answer
141 views

For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$?

I came across the following question: For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$? What I have tried: Since for $ \alpha = 1 $ is clearly non-differentiable in ...
0
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1answer
66 views

Is this (or when) does this equality hold?

Let $a,b,c,d \in \mathbb{R}$ and $x,y$ are variables which are also real numbers $$|ax + by|^2 + |cx + dy|^2 + 2|ax + by||cx + dy| = (ax + by)^2 + (cx + dy)^2 + 2(ax + by)(cx + dy)$$ Is this always ...
2
votes
2answers
65 views

proving $|x - 1| < {1\over4} \Rightarrow |2x - 1| \geq {1\over 2}$

I tried solving the above, consider that: ($x \in R)$, I know it's not a complicated problem to solve though I struggle getting on with this question, What I've done far is: $|x-1|<{1\over4} ...
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3answers
70 views

Find $x$ for absolute value inequalities

I'm trying to figure out this inequality: $|x+1| + |x| \leq x^2$ I thought about trying it with two cases: $ (x = -x)$ and $(x = +x)$ but I don't seem to find out how to go through from here, ...
0
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1answer
100 views

Finding The Contour Maps Of A Function Of Two Variables

I am given the function $f(x,y) = \ln|y-x^2|$, and am suppose to find the contour maps. Let $z = c = f(x,y)$. $c = \ln|y-x^2| \rightarrow e^c = e^{\ln|y-x^2|} \rightarrow e^c = |y-x^2|$ I know I ...
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3answers
79 views

Solving $2|x+1|>|x+4|$

I'm trying to solve the following equations and inequalities for $x\in\mathbb R$: $$2|x+1|>|x+4|$$ I know I'm supposed to consider the intervals $(-\infty,-4), [-4,-1]$ and $(-1,\infty)$ but ...
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5answers
133 views

Determine all solutions to $|x+12|+|x-5|=15$

Determine all solutions to the following. $$ \lvert x+12\rvert +\lvert x-5\rvert =15.$$ How can I solve problems like this? Should I try graphing the function? Should I somehow consider various ...
1
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1answer
4k views

Finding the points of the curve where the tangent line is horizontal

The curve given is $\displaystyle y = \ln|x-2| + x + \frac{12}{x-2}$. Find the points of the curve where the tangent line is horizontal. My first stumbling block is the absolute value function. I ...
3
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3answers
405 views

How does one calculate the integral of the sum of two absolute values?

I know how to find the integral of just one absolute value, but this problem presents the integral of the sum of two absolute values. Help! I want to evaluate: $$ \int_a^b{(|x-1| + |x+1|) dx} $$
5
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2answers
146 views

Prove $||a| - |b|| \leq |a - b|$ [duplicate]

I'm trying to prove that $||a| - |b|| \leq |a - b|$. So far, by using the triangle inequality, I've got: $$|a| = |\left(a - b\right) + b| \leq |a - b| + |b|$$ Subtracting $|b|$ from both sides yields, ...
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2answers
2k views

Question regarding usage of absolute value within natural log in solution of differential equation

The problem from the book. $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6 -y$ I understand the solution till this part. $\ln \vert 6 - y \vert = x + C$ The solution in the book is $6 - Ce^{-x}$ ...
3
votes
3answers
319 views

Exposition On An Integral Of An Absolute Value Function

At the moment, I am trying to work on a simple integral, involving an absolute value function. However, I am not just trying to merely solve it; I am undertaking to write, in detail, of everything I ...
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4answers
226 views

Truth set of $-|x| \lt 2$?

An exercise in my Algebra I book (Pearson and Allen, 1970, p. 261) asks for the graph of the truth set for $-\left|x\right| \lt 2; x \in \mathbb{R}$. I've re-stated the inequality in the equivalent ...
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3answers
5k 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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2answers
7k views

How to find critical points of an absolute values function

I am asked to find How many critical points does the function $g(x) = |x^2 − 4|$ have? I know that the result is $3$ but I can only find $2$. What I do, is to equal the equation to $0$, so $x^2-4=0$ ...
2
votes
3answers
225 views

Absolute ratios

I'm curious about the following idea: suppose we have two values $P$ and $Q$, and the magnitude of the ratio $\frac{P}{Q}$ is between $0$ and $\infty$. If $P$ is smaller, then it's between $0$ and ...
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3answers
304 views

Adding equations in Triangle Inequality Proof

Inequality to prove: $|a+b|\leq |a| + |b|$ Proof: $-|a| \leq a \leq |a|$ $-|b| \leq b \leq |b|$ Add 1 and 2 together to get: $-(|a|+|b|)\leq a+b\leq|a|+|b|$ $|a+b|\leq|a|+|b|$ What is the ...
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1answer
104 views

Integration Involving the Absolute Function

How do I integrate the double integral of the form $|x^2-y|$ with the boundaries $-1\leq x\leq 1$ and $-1\leq y\leq 1$?
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2answers
115 views

Solving $| ax + b | \gt c$

Does $| a x + b | > c$ always result in two solutions, $x \gt \dfrac{c - b}{a}$, and $x \lt\dfrac{-c - b}{a}$? If I understand correctly, the first solution, $x > \dfrac{c - b}{a}$, is only ...
1
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1answer
47 views

minimum of absolute value

If we consider the following problem $$ \mathbb{E}[(Y-y)^2 | X=x] $$ I can easily show that the minimum with respect to $y$ occurs at $$ y=\mathbb{E}[Y |X=x] $$ How can I find the minimum of $$ ...
0
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3answers
223 views

Graphing Absolute Value Functions

Given: $y = -|2x + 1|-3$ I came up with the graph of... $1, -7$ $0, -5$ $-1, -3$ $-2, -1$ $-3, 1$ If you were to graph this, it would turn out to be an entirely straight line. This is an ...
5
votes
2answers
455 views

Is there a lower-bound version of the triangle inequality for more than two terms?

The triangle inequality $|x+y|\leq|x|+|y|$ can be generalized by induction to $$|x_1+\ldots+ x_n|\leq|x_1|+\ldots+|x_n|.$$ Can we generalize the version $|x+y|\geq||x|-|y||$ to $n$ terms too? I need ...
1
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2answers
707 views

Log laws and modulus

If you have the log of a modulus, (like after integration), how do the log laws work? So if you have $a\ln\left|2x-3\right|$ does it become: $\ln\left|(2x-3)^a\right|$ or $\ln(\left|2x-3\right|)^a$, ...
0
votes
1answer
339 views

Absolute function continuous implies function piecewise continuous?

I have a simple true/false question that I am not sure on how to prove it. If $|f(x)|$ is continuous in $]a,b[$ then $f(x)$ is piecewise continuous in $]a,b[$ Anyone that can point me in the ...
0
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2answers
224 views

True/false question: limit of absolute function

I have this true/false question that I think is true because I can not really find a counterexample but I find it hard to really prove it. I tried with the regular epsilon/delta definition of a limit ...