What is Inequality: Definition and 1000 Discussions
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that
z
≤
x
+
y
,
{\displaystyle z\leq x+y,}
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
‖
x
+
y
‖
≤
‖
x
‖
+
‖
y
‖
,
{\displaystyle \|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|,}
where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in R1, and the triangle inequality expresses a relationship between absolute values.
In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either R2 or R3. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.
In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints.The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces (p ≥ 1), and inner product spaces.
Having trouble with this question:
The question is: establish the inequality
|\inteizzdz| \leq \pi(1-e-R2)/4R
on C {z(t) = Reit, t \in [0,\pi/4, R>0
When i saw the modulus of an integral i thought ML inequality.
I think the length will be R\pi/4 but I am struggling with...
Prove for all Z E C
|ez-1| \leq e|z| - 1 \leq |z|e|z|
I think this has to be proven using the triangle inequality but not sure how.
Please help. :)
thanks
Bell's theorem is generally thought to show that the world cannot be both local and real.
http://en.wikipedia.org/wiki/Bell%27s_theorem
In simplistic terms, Bell derives an inequality which allegedly must be satisfied if the world is both local and real. In practice, it is found in...
Homework Statement
Solve the inequality
(2x-3)(4x+5)>(x+6)(x+6)
Homework Equations
factoring?
The Attempt at a Solution
I got to the point where
(7x)^2-14x-51>0 I can't solve this, because it can't be factored out. So am I doing something wrong?
"Solving Inequality With Complex Numbers" Question
Homework Statement
What does the inequality pz + conjugate(pz) + c < 0 represent if |p|^2 >c ?
Homework Equations
p is a constant and a member of the set of complex numbers. c is a constant and a member of the set of real numbers...
Homework Statement
Sketch the graph
|Re(z)|>2
Homework Equations
z=x+iy
The Attempt at a Solution
|Re(z)|>2
|Re(x+iy)|>2
|x|>2
|x-0|>2, this is a circle centered at zero with radius 2
4. My question
What I'm having a hard time with is the | | notation.
Is this the absolute value, or...
Homework Statement
f(x) 2 times differentiable function on (0, \infty), and \lim\limits_{x \rightarrow \infty} f(x)=0. there is a M such that M=\sup\limits_{x>0}\vert f^{\prime\prime}(x) \vert. And also for L>0
g(L)=\sup\limits_{x>L}\vert f(x) \vert, and h(L)=\sup\limits_{x>L}\vert...
For my economics/game theory thesis I need to optimize a function subject to an inequality constraint.
maximize f(x1, x2) = 1/(x1+x2+y1+y2-w) subject to g(x1, x2) = x1+x2+y1+y2 < w
This isn't particularly important, but the x and y variables are quantity of production by a firm. The objective...
Homework Statement
Deduce that 0 ≤ ≤ 10/9 for all values of x. Homework Equations
The Attempt at a Solution
Is it possible to sketch a graph for ? How?
Or is there any methods to find the max./min. value of ?
Please enlighten me...
Homework Statement
This is a question taken from an old exam so I am not sure to which subject in calculus it's connected to...
Prove the inequality:
\frac{1}{4(ln2)^2}\leq\sum\frac{2^n}{2^(2^n)}
(sigma is from 1 to +inf, and the Denominator on the right side is (2^(2^n))
Homework...
Homework Statement
Consider any two vectors, |a\rangle and |b\rangle. Prove the Schwartz inequality
|\langle a|b \rangle |^2 \leq \langle a|a \rangle \langle b|b \rangle
Homework Equations
a basic understanding of vector calculus over \mathbb{C}...
The Attempt at a Solution
I...
Homework Statement
Solve \frac{|x^2-5x+4|}{|x^2-4|}\le1
Homework Equations
The Attempt at a Solution
as
|x^2-4|will be positive always
cross multiply and take 1 to other side of equation
solve by taking LCM
we get
|x^2-5x+4|-(x^2-4)\le0
on solving we get...
Homework Statement
Solve the inequation (x^2+3x+1)(x^2+3x-3)\ge 5The Attempt at a Solution
on opening the brackets i got
x^2(x^2+3x-3)+3x(x^2+3x-3)+1(x^2+3x-3)\ge 5
x^4+3x^3-3x^2+3x^3+9x^2-9x+x^2+3x-3-5\ge 0
x^4+6x^3+7x^2-6x-8\ge 0
am i right??
after that what should i do??
Solve for x-- the inequality of quadratic
Homework Statement
Solve \frac{2x}{x^2-9}\le\frac{1}{x+2}
The Attempt at a Solution
x^2-9\not=0
.'. x\in R-\{-3,3\}
and
x+2\not=0
.'. x\in R-\{-2\}
then converting the original inequality to
(2x)(x+2)\le(x^2-9)...
Homework Statement
∫(from pi/4 to pi/2)sin x/x ≤ 1/√2.
Homework Equations
The Attempt at a Solution
I know the pi/4≤x≤pi/2 and so 1/√2 ≤ sin x ≤ 1 and i have tried to manipulate this to no end and it has annoyed the living daylights out of me
Homework Statement
Part of an \epsilon-\delta proof about whether or not f + 2g is continuous at x = a provided that f and g are continuous at x = a
The Attempt at a Solution
I've got the proof (I hope), but I'm uncertain about whether I can do the following...
Homework Statement
Prove that if x < y, and n is odd, then x^{n}< y^{n}
The Attempt at a Solution
My attempt was to solve three different cases:
Case 1: If 0 \leq x < y, we have
y-x > 0
y*y*...*y > 0 (closure of the positive numbers under multiplication)...
Homework Statement
Solve for x
Homework Equations
x^3 < x
The Attempt at a Solution
x^3 < x
x^3 - x < 0
x(x^2 - 1) < 0
x(x+1)(x-1) < 0
For the expression on the left to be less than zero, it has to be two positives + negative or three negatives right? I've tried setting...
I saw this rather odd symbol of the the greater sign on top of the less sign in my lecture notes. I am wondering if there is a name for this symbol and if signifies 'equal to' as well?
Hello
I need to prove this inequality:
http://img6.imageshack.us/img6/2047/unledwp.jpg
Uploaded with ImageShack.us
where y=im(z) ,x=Re(z).
I used the triangle inequality but I got stuck.
Can someone show me how to do it? specially the left side of the inequality.
thanks
I don't understand how it is possible to show using the Minkowski's Inequality that
(\sum x_i )^a \leq \sum x_i^a where x_i \geq 0 \forall i and 0<a<1 .
I also tried to prove this without using Minkowski, but to no avail.
This is driving me crazy although it seems to be trivial in...
Homework Statement
I'm actually only concerned here with proving equality. I would like some review of my proof before I crawl back to my professor again with what I think is a valid proof.
The Attempt at a Solution
Show:
\frac{x_1+x_2+...+x_n}{n}=\sqrt[n]{x_1x_2\cdots x_n} \Leftrightarrow...
Homework Statement
Suppose that a and b are nonzero real numbers. Prove that if a<1/a<b<1/b then a<-1.The Attempt at a Solution
So after a while I realized that I could prove that a<-1 by contradiction but first I have to prove that a<0. I figured out how to prove it but I'm not sure if my...
Hi,
Quick question here: I know that C-S inequality in general states that
|<x,y>| \leq \sqrt{<x,x>} \cdot \sqrt{<y,y>}
and, in the case of L^2(a,b)functions (or L^2(R) functions, for that matter), this translates to
|\int^{b}_{a}f(x)g(x)dx| \leq \sqrt{\int^{b}_{a}|f(x)|^2dx} \cdot...
Homework Statement
prove the inequality.
Homework Equations
(in the attached file.)
The Attempt at a Solution
The derivative of (1/2)^m where m=2^n is exactly what I need. But I can’t find the sum of (1/2)^m (because as it’s not geometric series, m doesn’t run from 0 to inf).
by...
Prove that
R(p,q) \leq \left(\stackrel{p+q-2}{p-1}\right)
where p and q are positive integers
I'm supposed to use induction on the inequality R(p,q) \leq R(p-1,q) + R(p,q-1) , but I'm having difficulty there.
How do I go about doing this? I can show it's true for p=q=1.
But, I can't...
I don't seem to understand Clausius inequality at all. Really. It was deduced to me that the Clausius inequality is given by
dS = \frac{\delta Q_i}{T} > 0
where Q_i is the irreversible heat transferred to a system. Though I cannot find a way to prove an assertion my teacher said: through...
attached are the problems (actually i don't think i bothered with #96) I'm having trouble with.
attached is ONE of my attempts
and attached is the book's answers.
I have NO idea where to even begin with these.
I know that if \forall x \in E \subset \mathbb{R}^n we have f(x) \le g(x) then it is true that \int_E f \le \int_E g .
However, is it also true that if \forall x \in E we have f(x) < g(x) then \int_E f < \int_E g ?
Hi guys, I am reading a proof on Holder's inequality. There is a line I don't understand.
Here is the extract from Kolmogorov & Fomin, Introductory Real Analysis.
"The proof of [Minkowski's inequality] is in turn based on Holder's inequality
\sum_{k=1}^n |a_k b_k|\leq...
Homework Statement
i got stuck at the question below:-
Homework Equations
The Attempt at a Solution
I tried to solve it by simplifying it but i got stuck at:-
Please help.
Homework Statement
I'm reading the proof for the reverse triangle inequality, but I don't understand what is meant by "by symmetry"
Homework Equations
The Attempt at a Solution
(X,d) is a metric space
prove:
|d(x,y) - d(x,z)| <= d(z,y)
The triangle inequality
d(x,y) <=...
Homework Statement
Prove that if |x-x_0| < \textrm{min} \bigg ( \frac{\epsilon}{2|y_0|+1},1 \bigg ) and |y-y_0| < \frac{\epsilon}{2|x_0|+1} then xy-x_0y_0<\epsilon
Homework Equations
We can use basic algebra and the following axioms:
For any number a, one and only one of the following...
Homework Statement
For a symmetric matrix A, use the notation \lambda_{k}\left(A\right) to denote the k^{th} largest eigenvalue, thus
\lambda_{n}\left(A\righ)<=...<=\lambda_{2}\left(A\right)<=\lambda_{1}\left(A\right)
Now suppose A and A+E are nxn symmetric matrices, prove the following...
Assume:
p>1, x>0, y>0
a \geq 1 \geq b > 0
\frac{a^2}{p^2}+(1-\frac{1}{p^2})b^2 \leq 1
\frac{x^2}{a^2}+\frac{y^2}{b^2} \leq 1
Prove:
\frac{x}{p}+y\sqrt{1-\frac{1}{p^2}} \leq 1
I've been trying for 3 days and it's driving me crazy. Any ideas?
Revenue Equation: R(x)=-x^2+10x Cost Equation: C(x)= 4X+5
Average profit= profit equation, P(x)/x
therefore p(x)= R(x)-C(x)=-x^2+6x-5
(-x^2+6x-5)/x=(-1(x-5)(x-1))/x, I then found that x is positive between 1 and 5, therefore average profit is positive in that range, however, the answer...
Homework Statement
Prove that (n + 1)n - 1 < nn for n ∈ Z+. [Hint: Induction is suggested. Write out the induction statement explicitly. Make one side of the inequality look like your induction hypothesis.]
Homework Equations
The Attempt at a Solution
^ That's what I have so far. I'm good...
How can I solve, step by step, this inequality ?
The result I have is [ 1 , (-1 + sqrt33)/2 ]
but the result should be [ 0 , (-1 + sqrt33)/2 ]|x+2|-|x-1|\geq\sqrt{x^2+x+1}
thanks for ur help =)
Homework Statement
I have to approximate sin(1/2) with the taylor inequality
Homework Equations
taylors inequality |Rn(x)| ≤ M/(n+1)! | x-a|n+1
The Attempt at a Solution
Im not really sure what the significance of this is, but ill do the derivatives
f(x) = sin(x)
f'(x) = cos(x)
f''(x) =...
Homework Statement
Let\; S = \sqrt(1) + \sqrt(2) + \sqrt(3) + ... + \sqrt(10000) \;and\; I = \int_{0}^{10000} \sqrt x dx[/itex]
Show that I \leq S \leq I+100
The Attempt at a Solution
Consider\; I\;=\;\int_{0}^{10000} \sqrt x dx
I\;=\; \frac{(10000)^{3/2}}{(3/2)}...
Homework Statement
the actual problem is to show that d(x,y)=d1(x,y)/[1+d1(x,y)] expresses a distance in R^n if d1(x,y) is a distance in R^n.Based on theory I have to show that
i) d(x,y)>=0 ,
ii)d(x,y)=d(y,x) and
iii)d(x,y)<= d(x,z)+d(z,y)
i've proven the first two so basically how can i...
Homework Statement
Prove that for all numbers a, b, c, d: if 0 \leq a < b and 0 \leq c < d then ac < bd.
This is problem 5 from chapter 1 of Michael Spivak's "Calculus", 4th Edition. It is the text for my real analysis course.
I should also mention that this is not a homework problem...
While this is not technically an assignment for any particular class (that I'm aware of, at least), I think the nature of this problem makes it suitable for this forum. Please, inform me if I should direct my question elsewhere.
Find x>3 such that ln(x)<x^0.1 (hint: The number is "huge")...
Hi, I am working on a calculus of variations problem and have a general question.
Specifically, I was wondering about what kind of constraint functions are possible.
I have a constraint of the form:
f(x)x - \int_{x_0}^x f(z) dz \leq K
If I had a constraint that just depends on x or...