Inequality Definition and 1000 Threads

see the textbook problem below; see my working to solution below; i generally examine the neighbourhood of the critical values in trying to determine the correct inequality. My question is "is there a different approach other than checking the neighbourhood of the critical values"? In other...

My prof. calls it the triangle inequality. However the wikipedia page with the same this name shows a special case of it, which is ##|x+y|\leq|x|+|y|##, and my prof. calls it the triangle inequality 2. I wonder what the formal name of the inequality in the picture above is. Thanks in adv.

Prove that $$(|z_1 + z_2| + |z_1 - z_2|)(|z_1| + |z_2|) >= \sqrt{2}$$

Hi, This is as part of a larger probability change of variables question, but it was this part that was giving me problems. Question: If we have ## 0 < x_1 < \infty ## and ## 0 < x_2 < \infty ## and the transformations: ## y_1 = x_1 - x_2 ## and ## y_2 = x_1 + x_2 ##, find inequalities for...

Assume that $x_1,\,x_2,\,\cdots,\,x_n\ge -1$ and $\displaystyle \sum_{i=1}^n x_i^3=0$. Prove that $\displaystyle \sum_{i=1}^n x_i\le \dfrac{n}{3}$.

Prove or disprove the following: There exists $A$ such that for all $a>0$ there exists $b>0$ such that for all $ x$: $|x-\ x_0|<b$ i mplies. $|\frac{1}{[x]}-A|<a$ where [x] is the floor value of x Gvf

Prove that $\dfrac{y^2z}{x}+y^2+z\ge\dfrac{9y^2z}{x+y^2+z}$ for all positive real numbers $x,\,y$ and $z$.

Hi, I've been looking at Bells inequality test. To see if I understand it correctly I'd like to state it in my own words. Could you please let me know of I have it right? Thanks Michael We have 4 measurements, A,B,C,D Each measurement is True or False A is 0 degrees B is 45 degrees C is 25.5...

I can solve (i), I got x = -1.6 For (ii), I did like this: $$(f^{-1} o ~g)(x)<1$$ $$g(x)<f(1)$$ But it is wrong, the correct one should be ##g(x) > f(1)##. Why? Thanks

Bell's inequality in it's original form is: |cor(a,b) - cor(a,c)| \le 1 - cor(b,c) where ##a,b## and ##c## are random variables with values ##\pm 1##, and the correlation is then simply the expectation value of their products, ##cor(a,b)=E[ab]## or as usually expressed ##\langle ab\rangle##...

I am having some trouble find the domain with this function: ##f(x)=\frac{1}{\sqrt{x^2-4x\cos(\theta)+4}}## and ##\theta\in[0,\pi]##.I know that the denominator needs to be greater than 0. So ##\sqrt{x^2-4x\cos(\theta)+4}>0##. I squared both side of the inequality, ##x^2-4x\cos(\theta)+4>0##...

Let $x$ be a real number such that $x=(x-1)^3$. Show that there exists an integer $N$ such that $-2^{1000}<x^{2021}-N<2^{-1000}$.

Problem: Find the set of all harmonic functions ##u(x,y,z)## that satisfy the following inequality in all of ##R^3## $$|u(x,y,z)|\leq A+A(x^2+y^2+z^2)$$ where ##A## is a nonzero constant. Work: I removed the absolute value bars by re-writing the expression $$-C-C(x^2+y^2+z^2)\leq u\leq...

see attached image, it asks to repesent it in x-graph constant "a" isn't conditioned. Do I need to separate it into a few cases of the constant a and represent each in one x-graph?

Correlation between polarization measurements of entangled photons at angles less than 45 are greater than classically statistically possible. No set of hidden variables can be preordained to explain the 75% correlation of photon measurements at 30 degrees and complete anticorrelation of...

First I give detail of what I think I understood so far. Suppose, there are three angles A, B, C separated by 120° angles. A can measure + (spin up in A direction, we call it A+), and - (spin down in A direction, we call it A-). Same goes for B+, B- and C+, C-. I have choose A direction to...

Summary:: To prove a conditional statement on a pair of inequalitites. Mentor note: Moved from technical forum section, so the post is missing the usual fields. I feel it should be possible to prove this but I keep getting lost in the symbolic manipulation. Theorem: If a>b implies a>c then...

Find all pairs of real numbers $(p,\,q)$ such that the inequality $|\sqrt{1-x^2}-px-q|\le \dfrac{\sqrt{2}-1}{2}$ holds for every $x\in [0,\,1]$.

Prove for all $n\in N$ $\dfrac{|a_1+...a_n|}{1+|a_1+...+a_n|}\leq\dfrac{|a_1|}{1+|a_1|}+...\dfrac{|a_n|}{1+|a_n|}$

Prove that $\cos(\sin x))+\cos(\cos x))<\dfrac{\pi}{2}$.

If $x$ and $y$ are positive real numbers, prove that $4x^4+4y^3+5x^2+y+1\ge 12xy$.

Let $a$ and $b$ be positive real numbers such that $a+b=1$. Prove that $a^ab^b+a^bb^a\le 1$.

Im confused as finding the minimum value of lambda is an important part of the proof but it isn't clear to me that the critical point is a minimum

I got this function in a function analysis and got confused on how to solve its positivity; I rewrote it as: $$\sqrt{x^{2}-2x}>x-1 \rightarrow x^2-2x>x^2-2x+1$$ And therefore concluded it must've been impossible... but I'm certainly missing something stupid, since plotting the graphs of the two...

I've been given this answer: to move from negative to positive angles, ##-\theta'=2\pi-\theta##; and to move from positive to negative angles, ##\theta'=\theta-2\pi##. But my question is if there is any way to calculate it in a sequence of inequalities' steps. If I am being cumbersome, forgive...

Prove: $A\leq B\wedge B\leq A\Rightarrow A=B$

The solution from my book: From $$\frac{3}{x+2}<\frac{13}{12}<\frac{3}{x} \tag1$$ It follows that ##13x<36<13(x+2)## x<3, i.e. x = 1 or 2. By checking, x=1 is not the solution and x = 2 satisfies the equation. However, how does the author deduce (1)?

The answer to the above problem is baffling, despite its straightforward nature. I will post the answer later, but here is my solution first. Solution : (1) ##2x+1 > x## : In this case, we have ##2x - x > -1 \Rightarrow \boxed{x > - 1}## (2) ##2x+1 < -x## : In this case, we have ##2x+x < -1...

prove the following inequality: $\dfrac{|a+b|}{1+|a+b|}$ $\leq \dfrac{|a|}{1+|a|}$ +$\dfrac{|b|}{1+|b|}$

Prove that for positive reals $a,\,b,\,c,\,d$, $\sqrt{ab}+\sqrt{cd}\le \sqrt{(a+d)(b+c)}$.

Summary:: x Hey, I'm learning calculus and had to prove the following Lemma which is used to prove AM-GM inequality, I had tried to prove it on my own and it is quite different from what is written in my lecture notes. I have a feeling that my proof of the Lemma is incorrect, but I just don't...

Prove that $\sqrt[n]{1+\dfrac{\sqrt[n]{n}}{n}}+\sqrt[n]{1-\dfrac{\sqrt[n]{n}}{n}}<2$ for any positive integer $n>1$.

Prove that $\sqrt[3]{\dfrac{2}{1}}+\sqrt[3]{\dfrac{3}{2}}+\cdots+\sqrt[3]{\dfrac{996}{995}}-\dfrac{1989}{2}<\dfrac{1}{3}+\dfrac{1}{6}+\cdots+\dfrac{1}{8961}$.

Given positive real numbers $a,\,b,\,c$ and $d$ that satisfy the following inequalities: $a \le 1 \\a+4b \le 17\\a+4b+16c \le273\\a+4b+16c+64d \le4369$ Find the minimum value of $\dfrac{1}{d}+\dfrac{2}{4c+d}+\dfrac{3}{16b+4c+d}+\dfrac{4}{64a+16b+4c+d}$.

Note that if we prove problem 4, the proof for problem 5 follows directly. We use properties of logarithms to combine the right hand side of ln into a single logarithm. Then we raise both side of the inequality to a power of e. Which leads us to the desired inequality. But, when I try to be...

Let $a,\,b$ and $c$ be the side lengths of a triangle. Prove that $\dfrac{a}{\sqrt[3]{4b^3+4c^3}}+\dfrac{c}{\sqrt[3]{4a^3+4b^3}}+\dfrac{a}{\sqrt[3]{4b^3+4c^3}}<2$.

Show that for all $0<x<\dfrac{\pi}{2}$, the following inequality holds: $\left(1+\dfrac{1}{\sin x}\right)\left(1+\dfrac{1}{\cos x}\right)\ge 5\left[1+x^4\left(\dfrac{\pi}{2}-x\right)^4\right]$

I am reading from Courant's book. He gave an example of the continuity of ##f(x)=5x+3## by finding ##\delta=\epsilon/5##. He then said that ##|x-x_0|## does not exceed ##|y-y_0|/5##, but I don't see how he came up with this inequality. I know that ##|x-x_0|<\epsilon/5##, and that...

Prove that for every $x\in (0,\,1)$ the following inequality holds: $\displaystyle \int_0^1 \sqrt{1+(\cos y)^2} dy>\sqrt{x^2+(\sin x)^2}$

Given the real numbers $a,\,b,\,c$ and $d$, prove that $(1+ab)^2+(1+cd)^2+a^2c^2+b^2d^2\ge 1$

Hello all this is my first post. I am not an engineer, but I wish I was. I have been enjoying watching Dan Gelbart!s YouTube Chanel. His air bearing has me wondering about the relationship and apparent conflict inside ball bearings. How does the ball travel around the shorter distance...

Hello everyone, I've been struggling quite a bit with this problem, since I'm not sure how to approach it correctly. The inequality form reminds me of the equation of a circle (x^2 + y^2 = r^2), but I have no idea how to be sure about it. Would it help just to simplify the inequality in terms...

In a recent https://mathhelpboards.com/threads/inequality-challenge.27634/#post-121156, anemone asked for a proof that $1-x + x^4 - x^9 + x^{16} - x^{25} + x^{36} > 0$. When I graphed that function, I noticed that in fact it is never less than $\frac12$. If you add more terms to the series, this...

Prove $x+x^9+x^{25}<1+x^4+x^{16}+x^{36}$ for $x>0$.

Prove that $2^{2\sqrt{3}}>10$.

hey so, this is an algebra assignment that we had to do and i really didn't understand the course material that well, but i managed to do the very first steps. anyways i was hoping you guys could help me finish the rest of this table. https://ufile.io/jowwrfj3 or you can see the file attached

Prove that $3^n\ge(n+3)^3$ for any natural number $n\ge6$.

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)...

I'm trying to solve the inequality: $$ \int \limits_0^1 e^{-x^2} \leq \int \limits_1^2 e^{x^2} dx $$I know that $\int \limits_0^1 e^{-x^2} \leq 1$, but don't see how to take it from there. Any ideas?

For concreteness, let me consider real numbers. If ##A > B## and ##B \sim C##, does it mean that ##A \sim C##? If instead ##B = C##, then obviously that wouldn't imply ##A = C##.