What is Absolute values: Definition and 77 Discussions
In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero.
Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.
Hello,
According to https://www.fico.com/fico-xpress-optimization/docs/latest/mipform/dhtml/chap2s1.html?scroll=ssecabsval the formula for absolute values are :
y = | x1 - x2| for two variables x1, x2 with 0 ≤ xi ≤ U
Introduce binary variables d1, d2 to mean
d1 : 1 if x1 - x2 is the positive...
I was given a problem to solve that goes like this ##\frac{3}{|x+3|-1}\geq |x+2|## . I got the correct solution for all possible cases and here they are; for ##|x+3|\geq0## and ##|x+2|\geq## i got ##x\epsilon <-2, -2\sqrt{3} ]## and for ##|x+3|\leq0## , ##|x+2|\leq0## I got ##x\epsilon [-5...
Problem Statement : I copy and paste the problem as it appeared in the text to the right.
Attempt (mine) : I copy and paste my attempt using Autodesk Sketchbook##^{\circledR}## below. I hope the writing is legible.
My answer : I have three answers and confused as to which of them hold...
Problem statement : Let me copy and paste the problem to the right as it appears in the text.
Solution attempt (mine) : There are mainly three cases to consider.
(1) ##\boldsymbol{x\ge 3\; :}## Using the relevant equations given above, the problem statement reduces to $$x-3+x-2 = 1\Rightarrow...
See attachment.
I don't understand the solution given by David Cohen. I am sure this is a shortcut explanation. I don't like shortcut explanations.
1. What in the problem indicates that x > 1?
2. What in the problem indicates that x < 2?
In some derivations of the CHSH inequality, https://en.m.wikipedia.org/wiki/CHSH_inequality, the following arises :
$$CHS=\int A(a,l1)B(b,l1)dl1-\int A(a,l2)B(b',l2)dl2+\int A(a',l3)B(b,l3)dl3+\int A(a',l4)B(b',l4)dl4\\
=\int A(a,l)B(b,l)dl1-A(a,l)B(b',l)+A(a',l)B(b,l)+A(a',l)B(b',l)dl$$
1)...
Hi all,
There's this proof that I've been trying to wrap my head around but it just doesn't seem to sink in. I've attached a screenshot below. Many thanks in advance!
Consider Case 1. There is a step that goes
$$\text{Then} \ |r| = r$$
$$Then -|r| \leq |r| \ \text{and} \ r \leq |r|$$
Why is...
Homework Statement
Homework EquationsThe Attempt at a Solution
2sin3x=1 OR 2sin3x= -1
sin3x=1/2 sin3x= -1/2
From the unit circle and in accordance with the domain
there are 3 solutions (B)
But the answer is (C)
HOW?
Hi there,
I'm having trouble understanding this math problem:
|x| + |x-2| = 2
The answer says its: 0<=x<=2
I understand you need different "cases" in order to solve this. For example, cases for when x is less than 0, when x-2 is less than 0, etc.
Thanks,
blueblast
Homework Statement
Determine the Fourier-transfroms of the functions
\begin{equation*}
a) f : f(t) = H(t+3) - H(t-3) \text{ and } g : g(t) = \cos(5t) f(t)
\end{equation*}
and
\begin{equation*}
b) f : f(t) = e^{-2|t|} \text{ and } g : g(t) = \cos(3t) f(t)
\end{equation*}Homework Equations
The...
I know that \sqrt{f(x)^2} = |f(x)| However...
I've just noticed that integrals of expressions like this are usually assumed to be equal to the integral of f(x) without the absolute value. I'd like to know how that's possible.
Is weird for me to consider those expressions; specially because of...
Homework Statement
Write F(x)= x2-5|x| as a piecewise function
Homework EquationsThe Attempt at a Solution
I was writting it out and came to
Fx= x2-5(x) and x2-5(-x)
but my book says that it comes out to be
x2-5
x2-5(-x)
I imagine there is a very simple reason why the x in the first one...
How does one solve an equation with two absolute value functions as below
My algebra book does not show how to solve with two abs functions.
2|4x-1| = 3|4x+2|
I thought this might work..
|4x-1|/|4x+2| = 3/2 then
|(4x-1)/(4x+2)| = 3/2 and solve the normal way..
This is actually a physics problem, but since my question is really about the math involved, I decided to post it in the calculus subforum.
I'm supposed to get from the term:
$$\lim_{\Delta t → 0} |\vec{v}_r (t + \Delta t)| \frac{\sin \Delta \theta}{\Delta t}$$
To:
$$v_r (t) \frac{d\theta}{dt}$$...
When solving a separable differential equation, my textbook says this:
ln|v-49|=-t/5+C→
|v-49|=e-t/5+C→
v=49+ce-t/5
What happened to the absolute values? I think it has something to do with the exponential always being positive.
Homework Statement Find the ## lim _{x-> -1+} sqrt(x^2-3x)-2/|x+1| ##
Homework EquationsThe Attempt at a Solution
I can only solve it using l'hopital rule and would like to know the steps of solving it without using it.
## lim _{x->-1+} (2x-3)/|1|= -5/4 ##
How to input absolute values in FORTRAN77?
This was the code I used
READ *,H
PRINT *,H
The input I gave was 0.01
But the output I got was 0.00999999978.
I feel like I'm asking the weirdest questions that most people don't ask, but here it is.
Suppose we have this integral (I made it up):
$$\int \sqrt{x^4+2x^3+x^2}$$
Now, I feel most people would say the answer is simply, $\frac{1}{3}x^3+\frac{1}{2}x^2+C$. But technically, that is only true...
To find E |X| of a cauchy random variable, I need to integrate
\int_{-\infty}^{\infty}\frac1{\pi}\frac{|x|}{1+x^2}dx .
From the definition of absolute value, we have
\int_{-\infty}^0\frac1{\pi}\frac{-x}{1+x^2}dx + \int_0^{\infty}\frac1{\pi}\frac{x}{1+x^2}dx (I think).
But, the very next...
Homework Statement
I have a simple problem with roots and absolute values. When is the root of a number both negative and positive? Is only the equation of a number say f(x) = √x both the negative root and the positive root?
Homework Equations
If a = 1; b = -2, och x = a2√(ab-b2+2)
Why is x...
Homework Statement
Here is an alternative approach to handling absolute value terms as the decision variables: abs(x) is the smallest value z that satisfies x \leq z and -x \leq z. Using this,convert the following into a lp
Min 2x1 + 3abs(x2)
S.T x1 + x2 \geq 6
Homework Equations
Here is a...
Homework Statement
|x| + |y| ≤ 1
What is the region in the plane that solves this inequality?
Homework Equations
The Attempt at a Solution
I first tried graphing it by isolating the y variable
|y| ≤ -|x| + 1
Then I looked at the hint we were given, which was to assume that x and y...
Hi all,
I was working on a proof that essentially worked because:
|x-y|+|y-z| >= |x-y+y-z|
I knew this was true because, but I'm looking for a generalization in a way that I could write in a proof.
Can you say that when comparing two expressions of addition/subtraction that are...
$${ x }^{ 2 }=4\\ \sqrt { { x }^{ 2 } } =\sqrt { 4 } \\ |x|=2$$
According to my professor, in the above case, the absolute value gives two solutions: ##x=±2##
Consider the discriminant in the quadratic formula: $$x=\frac { -b±\sqrt { { b }^{ 2 }-4ac } }{ 2a } \\ Let\quad { z }^{ 2 }={ b }^{ 2...
Homework Statement
∫(0 to 3pi/2) -7|sinx|dx
Homework Equations
The Attempt at a Solution
I am not sure how to treat it as it has an absolute value
i assumed that you could remove the -7 to get
-7∫|sinx| dx
then integrate sinx into -cosx but since there is absolute...
this isn't really homework, but I was just wondering if someone could offer an intuitive reason as to why when random variables are transformed, we use absolute values of derivative of those functions, as opposed to the functions themselves?
Homework Statement
Find <x> in terms of X0 if X0 is constant and
\Psi(x) = \frac{1}{\sqrt{X_0}}e^{\frac{-|x|}{X_0}}
and
<x> = \int^{\infty}_{-\infty}{\Psi^* x \Psi}dx
where Psi* is the complex conjugate of Psi.
Since there is no imaginary component, this is effectively Psi2.
so, from...
Homework Statement
Find all real values of x that satisfy the following inequality.
Homework Equations
|x-3| > |x + 1|
The Attempt at a Solution
Splitting up the inequality into cases I get:
1. |x-3| > x + 1 and 2. |x-3| < -x - 1
1. x-3 > x + 1 or x-3 < -x - 1...
Homework Statement
Simplify.
a) \sqrt{x^6}
b) 8 \sqrt{x^7y^{10}} - 10 \sqrt{x^7y^{10}}
For b, it's y^10. I can't make it look right for some reason.
Mod note: Fixed the exponent.
Homework Equations
The Attempt at a Solution
I can simplify all of them but I don't know when or where I need to...
Homework Statement
Solve Ix+3I>2
*I is used for absolute value notation
The Attempt at a Solution
Considering both
a) Ix+3I > 0 then Ix+3I= x+3
b) Ix+3I < 0 then Ix+3I= -(x+3)
when solved this would yield to;
a) x>-3 and x>-1
b) x<-5 and x<-3
from my general reasoning i...
The variance equation basically sums up all the distances between each data value and the mean of the set. The interesting thing is that each distance and squared for a reason that I believe is to make the distance positive, but why don't the statisticians just take the absolute value of each...
Homework Statement
I am wondering if the general approach to these proofs involving absolute values and inequalities is to do them case-wise? Is that the typical approach (unless pf course you see some 'trick')? For example, I have:
Prove that if
|x-xo| < ε/2 and Prove that if |y-yo| <...
Homework Statement
I'm currently implementing an algorithm in Matlab, however, I've hit a bump, I'm trying to solve the following system of equations:
Known variables = a, b, c, d[complex]
Unknown variabls = fs, fd, fv, alpha
My problem is what to do with the |alpha| since I can't get...
Homework Statement
Determine the values of z \in \mathbb{C} for which |z+2| > 1 + |z-2| holds.
Homework Equations
Nothing complicated I can think of.
The Attempt at a Solution
For real values this holds for anything greater than 1/2. If I could figure out the boundaries of the...
Homework Statement
Consider a simple first-order linear differential equation, such as
y' + \tan x y = 0
With initial condition y(0)=C for some constant C. Find all solutions y which satisfy this differential equation on the entire real line.
Homework Equations
General method for...
Homework Statement
How to solve x for these inequality?
Homework Equations
|x-2|/|x+3|> (x+2) / (x+1)
The Attempt at a Solution
(x - 2)/(x + 3) > (x + 2) / ( x+1)
the left side holds the condition that is x >= 2
however, I wonder the next step. should I crossly multiply so...
Homework Statement
solve the integral [abs(x+1)(3+abs(x))]/(x+1) between -3 and 1
Homework Equations
The Attempt at a Solution
when x<-1 then [abs(x+1)(3+abs(x))]/(x+1) = [-(x+1)(3-x)]/(x+1) = -(3-x)
when -1<x<0 then [abs(x+1)(3+abs(x))]/(x+1) = (x+1)(3-x)/(x+1) = 3-x
when x>0...
Homework Statement
if f(x)=abs(x-2) and g(x)=abs(x), then solve the integral from -1 to 3 of abs(f(x)-g(x))dx
Homework Equations
The Attempt at a Solution
resolved absolute values:
when x<0, abs(x-2)-abs(x) = -x-2+x = 2
when 0<x<1, abs(x-2)-abs(x) = (-x+2)-x = 2-2x
when 1<x<2...
Homework Statement
Solve the IVP
(x^2)y'' + 4xy' - 40y = x^6
for y(1) = 10, y'(1) = 1Homework Equations
not so much "equations" but here I try to use variation of parameters to get the particular solution.The Attempt at a Solution
FOR THE HOMOGENEOUS SOLUTION:
using the substitution y = x^r...
Homework Statement
I have this equation
|2x+7| - |6-3x| = 8.
The step I did is to replace the || with () and then solve the equation
2x+7-6+3x = 8
X = 7/5
But how do a go about solving for the second solution?
With one absolute value I would
|2x + 7| = 8
2x + 7 = +-8
2x = -7...
Hey Guys! I've frequently come by this forum and have finally joined it in hopes that I could get some more conceptual insight in understanding math.
One thing that I have trouble with is absolute values. I understand that:
|x|= \sqrt{x^2} .. and how it can be defined given restrictions on...
Hello all,
I am trying to solve a problem based on some computer programming task I am trying to solve, and I have encountered a situation I am having trouble continuing..
Given a function f(x)=|1-x| + |0.5-2x| ...
How can I find it's minimum efficiently? This sum may extend to 4 or 5...
Homework Statement
-F is a continuous function on [0,1], so let ||f|| be the maximum value of |f| on [0,1]
a. Prove that for any number c we have ||cf|| = |c|\ast||f||
b. Prove that ||f + g|| \leq ||f|| + ||g||.
c. Prove that ||h - f|| \leq ||h - g|| + ||g - f||
Homework Equations
Based...
Homework Statement
Draw |z| on a complex plane, where z = -3+4i
Homework Equations
N/A
The Attempt at a Solution
[PLAIN]http://img530.imageshack.us/img530/1786/aaakr.jpg
Can anyone please tell me which answer is correct?
Both of them have a moduli of 5.
So should the circle...
My calc book rewrites this equation:
|y|=e^c|x|
As this:
y=\pm e^cx
But that doesn't really make any sense to me. I know I should understand why we're allowed to do that, but I don't. Could someone please try to explain it to me?
I really appreciate your help, thanks!
Homework Statement
I saw this in my real analysis textbook and I have been trying to understand how this equation \left | x-c \right |< 1
you can get this: \left | x \right |\leq \left | c \right | + 1
Homework Equations
I wanted to know what steps made this possible ...
Homework Statement
Let f(x, y) = |xy|.
I want to prove that f is not continuous at (0,0).
The Attempt at a Solution
To prove that f is not continuous at (0,0) I think I need to show that
\lim_{(x, y) \to (0, 0)}|xy| \neq 0
I'm a little confused about the |absolute value|...