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):
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x
+
y
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≤
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x
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+
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y
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,
{\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where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in R1, and the triangle inequality expresses a relationship between absolute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.
Show that if $a,\,b$ and $c$ are the lengths of the sides of a right triangle with hypotenuse $c$, then
\frac{(c − a)(c − b)}{(c + a)(c + b)}\le 17 − 12\sqrt{2}
Mass of a book of type X = 40g;
Mass of a book of type Y=80g;
The total mass of n books of type X and one book of type Y is less than 200g;
i. Write down an inequality containing the variable n only.
ii. Solve the above inequality for n and write down the maximum possible value for n
So how...
The Rearrangement Inequality states that for two sequences ##{a_i}## and ##{b_i}##, the sum ##S_n = \sum_{i=1}^n a_ib_i## is maximized if ##a_i## and ##b_i## are similarly arranged. That is, big numbers are paired with big numbers and small numbers are paired with small numbers.
The question...
Mod note: Moved from a technical math section, so missing the homework template.
This is for an Intro to Analysis course. It's been a very long time since I've taken a math course, so I do not remember much of anything.
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Here is the problem:For the inequality below, find all values...
Homework Statement
By choosing the correct vector b in the Schwarz inequality, prove that (a1 + ... + an)^2 =< n(a1^2 + ... +an^2)
Homework Equations
Schwarz inequality
The Attempt at a Solution
since the answer key says that a1 = a2 = ... = an, i tried plugging in values, but i am not...
Homework Statement
Find coefficients a,b>0 such that a||x||∞≤||x||≤b||x||∞.Homework EquationsThe Attempt at a Solution
No idea how to get started. Help will be appreciated.
Question 1:
(a) Show that the complex number i is a root of the equation
x^4 - 5x^3 + 7x^2 - 5x + 6 = 0
(b) Find the other roots of this equation
Work:
Well, I thought about factoring the equation into (x^2 + ...) (x^2+...) but I couldn't do it. Is there a method for that? Anyways the reason I...
hi! i need help for this inequality
1. ##a\in\mathbb{N}*~and~ \frac{a}{a+1}<\frac{a+1}{a+2}<\frac{a+2}{a+3}##
show that : ##\frac{1}{2}*\frac{4}{5}*...*\frac{2005}{2006}*\frac{2008}{2009}<\frac{1}{12}##
Here i have stoped. Please tell me if is corect what i have done so far and how to continue ...
Homework Statement
Let a,b and c be lengths of sides in a triangle, show that
√(a+b-c)+√(a-b+c)+√(-a+b+c)≤√a+√b+√c
The Attempt at a Solution
With Ravi-transformation the expressions can be written as
√(2x)+√(2y)+√(2z)≤√(x+y)+√(y+z)+√(x+z).
Im stuck with this inequality. Can´t find a way to...
Let $a,\,b,\,c$ and $d$ be non-negative real numbers such that $a + b + c + d = 2$.
Prove that $ab(a^2+ b^2 + c^2) + bc(b^2+ c^2+ d^2) + cd(c^2+ d^2+ a^2) + da(d^2+ a^2+ b^2) ≤ 2$.
Currently revising for my A-Level maths (UK), there is unfortunately no key in the book;
Given the triangle with sides a,b,c respectively and the area S, show that ab+bc+ca => 4*sqrt(3)*S
I have tried using the Ravi transformation without luck, any takers?
A few weeks ago I created a discussion titled "How does Bell's inequalities rule out realism."
Essentially my question was pertaining to how does removing realism retain locality and not violet Bell's inequality.
Someone answered with the this,
I'm not really happy with this explanation, but...
I have this inequality:
$$ \frac{n^3}{n^5 + 4n + 1} \le \frac{1}{n^2}$$
for all $n \ge 1$
I get that
$$ \frac{1}{n^5 + 4n + 1} \le \frac{1}{n^2}$$
but how do I guarantee that when $n^3$ is in the numerator, this inequality holds? Is this for any numerator greater than 1? Also, why must $n$...
I have
$$-1 \le \cos\left({2x}\right) \le 1 $$
If everything is squared, it goes to
$$0 \le \cos^2\left({2x}\right) \le 1 $$
and I'm not sure how $(-1)^2$ turns into $0$
I understand Bell's inequality, and I can see how removing locality can produce the observed statistical correlations. However something that I often read is that eradicating realism can also generate the correlation observed in entanglement. I don't see how a particle not having definite...
Given that ##v_0## and ##v_f## are positive variables related by the equation
where ##g## and ##\alpha## are positive constants.
Can you show that ##v_f<v_0## for all positive values of ##v_0## using a non-graphical method?
Physically, ##v_0## and ##v_f## are the initial and final speeds (at...
In the derivation of triangle inequality |(x,y)| \leq ||x|| ||y|| one use some ##z=x-ty## where ##t## is real number. And then from ##(z,z) \geq 0## one gets quadratic inequality
||x||^2+||y||^2t^2-2tRe(x,y) \geq 0
And from here they said that discriminant of quadratic equation
D=4(Re(x,y))^2-4...
Let $a,\,b$ and $c$ be real numbers such that $a+b+c=1$, prove that
\frac{1}{3^{a+1}}+\frac{1}{3^{b+1}}+\frac{1}{3^{c+1}}\ge \left(\frac{a}{3^a}+\frac{b}{3^b}+\frac{c}{3^c}\right).
Homework Statement
part c
Homework Equations
The Attempt at a Solution
Jm+2=m+2-1/m+2 Jm=m+1/m+2 Jm
hence Jm+2<Jm
should i expend Jm+2 Jm+1 Jm to the term J0 then compare them?
why the inequality is <= but not <?
should i use M.I to proof it??[/B]
Homework Statement
If a,b,c are positive real numbers such that ##{loga}/(b-c) = {logb}/(c-a)={logc}/(a-b)## then prove that
(a) ##a^{b+c} + b^{c+a} + c^{a+b} >= 3##
(b) ##a^a + b^b + c^c >=3##
Homework Equations
A.M ##>=## G.M
The Attempt at a Solution
Using the above inequation, I am able...
Let the reals $a, b, c∈(1,\,∞)$ with $a + b + c = 9$.
Prove the following inequality holds:
$\sqrt{(\log_3a^b +\log_3a^c)}+\sqrt{(\log_3b^c +\log_3b^a)}+\sqrt{(\log_3c^a +\log_3c^b)}\le 3\sqrt{6}$.
Homework Statement
Solution set of the inequality (cot-1(x))2 -(5 cot-1(x)) +6 >0 is?
Homework EquationsThe Attempt at a Solution
Subs cot-1(x)=y
We get a quadratic inequality in y.
y2-5y+6>0
(y-2)(y-3)>0
Using the wavy curve method, the solution set is...
I am trying to solve this inequality without using a factor table.
The problem
$$ \frac{x+4}{x-1} > 0 $$
The attempt at a solution
As I can see ##x \neq 1##. I want to muliply both sides of the expression with x-1 to get rid of it, from the fraction. But before that, I have to consider two...
trying to soak in this paper.
https://arxiv.org/abs/gr-qc/0205035
The following statement is found early on:
"The violation of Bell’s inequality proves that any realistic interpretation
of quantum theory needs a preferred frame."
Whether anyone agrees or disagrees I'd appreciate a sketch of...
I have attached two images from my textbook one of which is a diagram and the other a paragraph with which I am having problems. The last sentence mentions that due to violation of 2nd law we cannot convert all the heat to work in this thermodynamic cycle. However what is preventing the carnot...
Hello! (Wave)
Suppose that $k$ football matches are being done and a bet consists of the prediction of the result of each match, where the result can be 1 if the first group wins, 2 if the second group wins, or 0 if we have tie. So a bet is an element of $\{0,1,2 \}^k$. I want to show that...
Prove, with no knowledge of the decimal value of $\pi$ should be assumed or used that 1\lt \int_{3}^{5} \frac{1}{\sqrt{-x^2+8x-12}}\,dx \lt \frac{2\sqrt{3}}{3}.
Hi
I have a double integral
$\iint(f(x) g(x,y)dxdy$ over $x\in[a,y]$ , $y\in[a,b]$
and I wish to bound the integral in terms of integral f times integral g.
I suppose there must exist a form of holder inequality to do that ?
many thanks
Sarrah