The absolute-value tag has no wiki summary.
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2answers
59 views
Express $y=|-x^2+1|$ as a piecewise function.
I'm unsure of how to start this problem. Any help would be greatly appreciated.
3
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2answers
54 views
Solving two greatest integer function equations
If $$x\lfloor x\rfloor =39 \quad \text{and}\quad y\lfloor y \rfloor=68.$$
What is the value of:
$$\lfloor x\rfloor+\lfloor y \rfloor $$
I don't know how to solve such problems.
I would appreciate ...
4
votes
1answer
38 views
Does the triangle inequality follow from the rest of the properties of a subfield-valued absolute value?
(This is a much more specific version of my earlier question from over a year ago.)
Let $F$ be a field, let $E$ be an ordered subfield of $F$, and let $\;\; |\hspace{-0.03 in}\cdot\hspace{-0.03 in}| ...
0
votes
4answers
176 views
Please help me to prove this inequality: $|x|+|y|≥|x+y|$
Please help me to prove the following inequality:
$|x|+|y|\geq|x+y|$
in which $x$ and $y$ are real numbers.
Any help or hint would be appreciated. Thanks :)
1
vote
2answers
35 views
Question involving absolute function
I saw this interesting problem in a math puzzle forum:-
Find all integral values of $t$ such that the equation $|s-1| - 3|s+1| + |s+2| = t $ has no solutions.
How does one approach these kind of ...
5
votes
1answer
43 views
Solving equation with absolute value signs
Can someone see why there is only get one solution when solving following equation in this way:
The equation $|x+1|+|2x-3|=|x-5| $
$$|x+1|+|2x-3|=|x-5| $$
$$\pm (x+1) \pm(2x-3)=\pm(x-5)$$
$$\pm x ...
1
vote
1answer
28 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
26 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
87 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$. ...
2
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1answer
49 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
39 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
39 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.
0
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1answer
24 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, ...
1
vote
1answer
43 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
24 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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1answer
68 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 ...
1
vote
1answer
36 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 ...
0
votes
2answers
52 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
vote
1answer
43 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
77 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 ...
1
vote
2answers
73 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
36 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
votes
3answers
40 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 ...
0
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3answers
42 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
votes
1answer
46 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
$$ f(x) = 1\dfrac{1}{2} + 3\sin ...
2
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2answers
45 views
Absolute Convergence of a Series
Find all values of real number p for which the series converges absolutely
$$\sum_{k=2}^{\infty} \frac{1}{k\, (\log{k})^p}$$
0
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2answers
77 views
Calculating the integral $ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$
How do we calculate the following integral:
$$ \int_{0}^{5} { \frac{|x-1|}{|x-2| + |x-4|} } dx$$
1
vote
2answers
87 views
How to evaluate the inequality $|x+1|<-1$?
Okay perhaps the title isn't specific enough, I didn't know how to word it exactly. I'm finding the interval of convergence for a power series and i know the answer to be (-2,0]
I end up with the ...
1
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2answers
44 views
Can summations distribute across absolute values?
Can I distribute a summation as follows?
$$
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|
$$
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0answers
80 views
Continuous, differentiable, continuously differentiable
I came across the following problem:
Let $\alpha \in \mathbb R$. Where is the function continuous, differentiable, continuously differentiable?
$$f(x) =
\begin{cases}
x|x|^\alpha & ...
1
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2answers
49 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
85 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
26 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
60 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} ...
1
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3answers
55 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
votes
1answer
60 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
67 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 ...
-1
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3answers
103 views
Determine all solutions to $|x+12|+|x-5|=15$
Determine all solutions to the following.
$$ \lvert x+12\rvert +\lvert x-5\rvert =15.$$
1
vote
1answer
221 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
114 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:
$$ \int_a^b{(|x-1| + |x+1|) dx} $$
4
votes
2answers
60 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,
...
5
votes
2answers
104 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
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3answers
114 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 ...
1
vote
4answers
173 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 ...
2
votes
3answers
218 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 :)
2
votes
2answers
311 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 $x^2-4=0$ ...
-1
votes
1answer
38 views
Integral of a mode squared [closed]
Hi could someone show me how to calculate $\psi_0$ out of the equation below?
$$\int \limits^{}_{V} \big|\psi_0 \sin (\omega t - kx) \big|^2 \, \textrm{d} V = 1\\$$
1
vote
2answers
51 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 ...
1
vote
3answers
109 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 ...
1
vote
1answer
66 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 $-1\leq y\leq 1$?
