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where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in R1, and the triangle inequality expresses a relationship between absolute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.
Homework Statement
Find the solution of the inequality ## \sqrt{5-2sin(x)}\geq6sin(x)-1 ##
Answer: ## [\frac{\pi(12n-7)}{6} ,\frac{\pi(12n+1)}{6}]~~; n \in Z##Homework Equations
None.
The Attempt at a Solution
There are two cases possible;
Case-1: ##6sin(x)-1\geq0##
or...
Homework Statement
Homework EquationsThe Attempt at a Solution
i just straight up applied am gm
##\frac {x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} \geq 3(\sqrt[3]{\frac{1}{(y+z)(x+z)(x+y)}})
##
so the denominator is which i had to maximise
##
x^2(y+z) + y^2(x+z) +z^2(x + y) + 2\\...
1)Prove without using AM-GM :\frac{ab}{c}+\frac{ac}{b}+\frac{bc}{a}\geq a+b+c...... a,b,c >02) Prove without using contradiction :
a\leq b\wedge b\leq a\Longrightarrow a=b
I'm pretty sure that the following is true, but I don't see an immediate compelling proof, so I'm going to throw it out as a challenge:
Let A,A', B, B' be four real numbers, each in the range [0,1]. Show that:
AB + AB' + A'B \leq A' B' + A + B
(or show a counter-example, if it's not true)...
Hello.
I am trying to prove that the uncertainty in energy for a normalized state limits the speed at which the state can become orthogonal to itself.
The problem is number 2 on https://ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/assignments/MIT8_05F13_ps6.pdf
Having issues...
Using high school mathematics prove the following inequality:
\sqrt{a_{1}^2+...+a_{n}^2}\leq\sqrt{(a_{1}-b_{1})^2+...+(a_{n}-b_{n})^2}+\sqrt{b_{1}^2+...+b_{n}^2}
Hello all,
Jensen's inequality says that for some random x,
f(E[x])≤E[f(x)]
if f(x) is convex.
Is there any generality that might help specify under what circumstances this inequality is...equal?
Thanks
Homework Statement
F(x,y,z)=4x i - 2y^2 j +z^2 k
S is the cylinder x^2+y^2<=4, The plane 0<=z<=6-x-y
Find the flux of F
Homework Equations
The Attempt at a Solution
What is the difference after if I change the equation to inequality?
For example :
x^2+y^2<=4, z=0
x^2+y^2<=4 , z=6-x-y...
I began reading Mehran Kardar's Statistical Physics of Particles and about halfway through the first chapter, there was a discussion on the second law of thermodynamics. He makes no mention of the old tenet that 'the total entropy in the universe must always increase' (I'll refer to this as the...
Homework Statement
Hello!
The task is to express the exact answer in interval notation, restricting your attention to -2π ≤ x ≤ 2π.
Homework Equations
The given inequality:
cos(x) ≤ 5/3
The Attempt at a Solution
I have only one doubt here, and I don't see my mistake.
I see that if cos(x)...
How do I find the range of [(4 - 4x^2)/(x^2 + 1)^2] > 0 algebraically?
Do I set the numerator to 0 and solve for x?
Do I set the denominator to 0 and solve for x?
Hi.
I wonder if following thought experiment (which is most probably impossible to be put into practice) could have any implications concerning interpretations of QM.
Consider five parties A, B, C, D and E, lined up in that order and with no relevant relative motion. No pair of them have ever...
Hi, I am reading a paper and in this equation is given.
I don't quite follow how they end up with the last (Uw-UL)<a-B. If I do it myself I get the inequality sign wrong.
Any help?
Thx
In a proof, I encountered the following expressions:
\[\sum_{cyc}\frac{\cos^2 A}{\sin B \sin C}\geq \sum_{cyc}\frac{\cos B \cos C}{\sin B \sin C}=\sum_{cyc}\cot B \cot C =1\]
My question is concerned with the validity of the inequality.
The inequality is based on the use of the Rearrangement...
Homework Statement
Consider a real valued function f which satisfies the equation f (x+y) = f (x) . f (y) for all real numbers x and y. Prove:
f ((x + y) / 2) ≤ 1/2 (f(x) + f(y))
Homework Equations
Not sure
The Attempt at a Solution
Please give me a hint to start solving this question. I...
Homework Statement
For a,b,c,d >1, Show that (a+1)(b+1)(c+1)(d+1) < 8(abcd+1)
Homework Equations
How to show this?
The Attempt at a Solution
I could show for two variables, (a+1)(b+1)<2(ab+1). Tried C-S, AM-GM inequalities in different form and variable transformations. But still no result...
Everything I've seen about Bell's inequality has had the setup of 120 degree angles between the axis of measurements. The experiment then proves that the basic hidden variable theory can't be true. But the actual measurement has always been told to me as a 0.5 correlation. 50% of the time the...
Homework Statement
"Given: ##a,b,c∈ℤ##,
Prove: If ##2a+3b≥12m+1##, then ##a≥3m+1## or ##b≥2m+1##."
Homework Equations
##P:a≥3m+1##
##Q:b≥2m+1##
##R:2a+3b≥12m+1##
The Attempt at a Solution
Goal: ##~(P∨Q)≅(~P)∧(~Q)⇒~R##
Assume that ##a<3m+1## and ##b<2m+1##. Then...
Hi all,
https://en.m.wikipedia.org/wiki/Bonse's_inequality
It seems to me that the inequality can be true for higher powers (if not any given higher power), for an appropriately higher (lower) bound for "n".
Any thoughts, proofs, counter proofs your insights are appreciated.
In particular, I...
I believe this is probably a high level undergraduate question, but i could easily be underestimating it and it's actually quite a bit higher than that.
I'm reading the Prime number theorem wikipedia page and I'm in part 4 under Proof sketch where sometime down they give in inequality:
x is a...
Homework Statement
Prove that for any naturam number n > 1 :
\frac{1}{n+1} + \frac{1}{n+2} + \frac{1}{n+3} + ... + \frac{1}{2n} > \frac{13}{24}
Homework Equations
Not sure
The Attempt at a Solution
\frac{1}{n+1} + \frac{1}{n+2} + \frac{1}{n+3} + ... + \frac{1}{2n} > \frac{1}{2n} +...
Bell inequality in page 171 of
https://www.scientificamerican.com/media/pdf/197911_0158.pdf
is
##n[A^+B^+] \le n[A^+C^+]+n[B^+C^+]##
In page 174 we can see that this causes linear dependency according to angle. How to derive this?
Let us suppose that angle between ##A^+## and ##B^+## is 30°...
Homework Statement
If a,b,c,d,e>1
then prove that
a^2/(c-1)+b^2/(d-1)+c^2/(e-1)+d^2/(a-1)+e^2/(b-1)=>20
The Attempt at a Solution
Given a,b,c,d,e are roots of a polynomial equation of a degree 5 then
x^2/(x-1)+x^2/(x-1)+x^2/(x-1)+x^2/(x-1)+x^2/(x-1)=>20
5 x^2/(x-1)=>20
x^2/(x-1)=>4
x^2=>4x-4...
Homework Statement Homework EquationsThe Attempt at a Solution
Hi
How do I go about showing ##0 \leq \frac{2x}{\pi} \leq sin x ##?
for ## 0 \leq x \leq \pi /2 ##
I am completely stuck where to start.
Many thanks.
(I see it is a step in the proof of Jordan's lemma, but I'm not interested in...
Homework Statement
please see attached, I am stuck on the second inequality.
Homework Equations
attached
The Attempt at a Solution
I have no idea where the ##2/\pi## has come from, I'm guessing it is a bound on ##sin \theta ## for ##\theta## between ##\pi/4## and ##0## ?
I know ##sin...
Prove the inequality:
\[\left | \cos x \right |+ \left | \cos 2x \right |+\left | \cos 2^2x \right |+...+ \left | \cos 2^nx \right |\geq \frac{n}{2\sqrt{2}}\]
- for any real x and any natural number, n.
Assume f and g are two continuous functions in (a, b).
If at the start of the segment I've shown f>g by taking the lim where x ---> a+ and the f ' > g ' for every x in (a,b )
can i say that f >g for all x in (a,b )? is there a theorem for that? that looks intuitively right.
The problem
Show that ## 1-x+ \int^x_1 \frac{\sin t}{t} \ dt < 0## for ## x > 1 ##
The attempt
I rewrite the integral as
##\int^x_1 \frac{\sin t}{t} \ dt < x-1 ##
This is about where I get. Can someone give any suggestions on how to continue from here?
Hey! :o
Let $\mathbb{K}$ be a fiels and $A\in \mathbb{K}^{p\times q}$ and $B\in \mathbb{K}^{q\times r}$.
I want to show that $\text{Rank}(AB)\leq \text{Rank}(A)$ and $\text{Rank}(AB)\leq \text{Rank}(B)$.
We have that every column of $AB$ is a linear combination of the columns of $A$, or not...
Homework Statement
Hi guys, I would just like someone to go over my method for this derivation/proof ( not sure of the right word to use here). Anyway I think this is right method, but just feel like I am missing something. Could someone please check my method. Thanks in advance.
Homework...
Homework Statement
Given : (y+2)(y-3) <= 0Homework EquationsThe Attempt at a Solution
Now, I have y-3 <= 0 or y+2 <= 0
Hence, y <= 3 or y <= -2
But how
is correct?
I think
is wrong because y <= -2.
Can someone please clarify?
Homework Statement
Prove the following facts about inequalities. [In each problem you will have to consider several
cases separately, e.g. ##a > 0## and ##a = 0##.]
(a) If ##a \leq b##, then ##a + c \leq b + c##.
(b) If ##a \geq b##, then ##a + c \geq b + c##.
(c) If ##a \leq b## and ##c \geq...
Proof: If either x or y is zero, then the inequality |x · y| ≤ | x | | y | is trivially correct because both sides are zero.
If neither x nor y is zero, then by x · y = | x | | y | cos θ,
|x · y|=| x | | y | cos θ | ≤ | x | | y |
since -1 ≤ cos θ ≤ 1
How valid is this a proof of the...
Homework Statement
Let ##a,b,c## be positive integers and consider all the quadratic equations of the form ##ax^2-bx+c=0## which have two distinct real roots in ##(0,1)##. Find the least positive integers ##a## and ##b## for which such a quadratic equation exist.
Homework EquationsThe Attempt...
\sin{(\pi x)}>\cos{(\pi \sqrt{x})}
I don't know how to solve this. I would really appreciate some help.
I tried to do something, but didn't get anything.
If I move cos to the left side, I can't apply formulas for sum. Since arguments of sin and cos have \pi , I think there is no way I can...
Homework Statement
show that 1-t^2/2 <=cos(t) <=1 for 0<=t<=1
Homework Equations
Trigonometry knowledge
The Attempt at a Solution
I don't know how to relate t with cos(t), and I also try to find out cos(1), but there is no result, so how can I start with this problem.
So I am trying to solve a simple rational inequality: ##\sqrt{x} < 2x##. Now, why can't I just square the inequality and go on my way solving what results? What precisely is the reason that I need to be careful when squaring the square root?
Hello,
if we consider the vector spaces of integrable real functions on [a,b] with the inner product defined as: \left \langle f,g \right \rangle=\int _a^bf(x)g(x)dx the Cauchy-Schwarz inequality can be written as: \left | \int_{a}^{b} f(x)g(x)dx\right | \leq \sqrt{\int_{a}^{b}f(x)^ 2dx}...
If the system of inequalities y ≥ 2x + 1 and y> x/2-1
is graphed in the xy-plane above, which
quadrant contains no solutions to the system?
A) Quadrant II
B) Quadrant III
C) Quadrant IV
D) There are solutions in all four quadrants.
I thing the answer is D . But book says that it is C. I...
The polynomial: $P(x) = 1 + a_1x +a_2x^2+...+a_{n-1}x^{n-1}+x^n$
with non-negative integer coefficients has $n$ real roots. Prove, that $P(2) \ge 3^{n}$
Homework Statement
$$
\left | \frac{z}{\left | z \right |} + \frac{w}{\left | w \right |} \right |\left ( \left | z \right | +\left | w \right | \right )\leq 2\left | z+w \right |
$$
Where z and w are complex numbers not equal to zero.
2.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$...