Inequalities Definition and 329 Threads

The Coopersmith inequolity: T=T_c, H\rightarrow 0^+ I'm confused by few things. What means H\rightarrow 0^+? And what difference will be if H\rightarrow 0^-? And what means T=T_c if we can't measure T_c in experiments? Then there is relation M \sim H^{\frac{1}{\delta}} That means if I...

Homework Statement a. Prove: If a≠b≠c are real numbers, then a2+b2+c2>ab+bc+ca b. Prove: If a>0, b>0and a≠b, then a/b+b/a>2 Homework Equations (real numbers)2>0 The Attempt at a Solution a. (a+b+c)2>0 a2+b2+c2>-2(ab+bc+ca) Try to prove -2(ab+bc+ca) > ab+bc+ca but not true, -2.4...

Use Binomial Theorem and appropriate inequalities to prove! Homework Statement Use Binomial Theorem and appropriate inequalities to prove 0<(1+1/n)^n<3 Homework Equations The Attempt at a Solution So I started by.. \sum ^{n}_{k=0} (n!/(n-k)! k!) a^{n-k}b^{k} = n!/(n-k)!k! (1)^{n-k}...

Hi everybody, I'd love to pick your brains about a problem in (seemingly?) linear algebra I've run into, trying to find the most efficient algorithm for solving a set of linear inequalities involving absolute magnitudes. During my research I've run into a problem that involves solving a...

Using the inequalities: \sin x \leq x \leq \tan x valid in a zero range, prove that: \displaystyle\lim_{x \to{0}}{\frac{x}{\sin x}}= 1 Thank you!

When we multiply or divide by a negative number a inequality of the type ≤, the symbol will become ≥, or >? -2x≥-4y, will become x ≤ 2y, or x < 2y?

Hi everyone. I am having a problem trying to understand the solutions of a homework problem that I had. Really need some help! Basically, I am trying to establish an inequality on kT using a given set of inequalities to work with. we have L \leq Q \leq H L \leq Q < Q+R_{1} \leq H and L...

Have you ever see any books discussing these problems? I don't know the name of these topic.

This is part of a larger problem and I'm trying to solve 2h+1≤n≤2h+1 for h. If the equation had two equals signs or one inequality I think I could do it but I'm not sure how to proceed with both. In other words, I don't know how to manipulate an expression of the form (expr1)≤(expr2)≤(expr3)...

Hello all! :smile: In my quest to re-teach myself the basics of mathematics in a more rigorous fashion, I have found out that inequalities and absolute values are a weak point if mine. So I am working to address that. I am getting much better at it (with help from PF), but I have recently...

Hello everyone, I'm posting here since I'm only having trouble with an intermediate step in proving that \sqrt{x} \text{ is uniformly continuous on } [0, \infty] . By definition, |x - x_0| < ε^2 \Longleftrightarrow -ε^2 < x - x_0 < ε^2 \Longleftrightarrow -ε^2 + x_0 < x < ε^2 + x_0 1...

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 Use a sketch graph to show the region defined by y<2 and y>x The Attempt at a Solution y<2 is easy as its anything below y=2... but I am totally stuck on y>x.. How can I know what y is if i don't know what x is?? thanks for any help

Homework Statement I just want to show that given x<0, \frac{x-1}{x-2} <1. The Attempt at a Solution I don't know why I am having trouble with this! I feel like this is so easy! So if x<0, then we know x-1<-1, x-2<-2 . So \frac{-1}{2}<\frac{1}{x-2} and...

How do you see if the following inequality holds true for (-2,0)? (-x/4)*(x+2)>1 For that matter how do you test inequalities for a given interval in general? Certainly there must be a way other than to check all values of (-x/4)*(x+2) in (-2,0) and see if they are greater than 1?

Homework Statement 4. Give a c > 0 and an integer n0 ≥ 1 such that, for all n ≥ n0. b. 16n log (n²) ≤ cn² The answer (from the sheet) is c = 32 Homework Equations ..The Attempt at a Solution When I attempt to solve such an equation I start at n=1, then go to n=2. but that way I get the...

Homework Statement (a) Show that for x,y\in \mathbb{R}^N: (i) d_{\infty} (x,y) \leq d_1 (x,y) \leq N d_{\infty} (x,y) ; (ii) d_{\infty} (x,y) \leq d_2 (x,y) \leq \sqrt{N} d_{\infty} (x,y). (b) Find constants A and B such that A d_1 (x,y) \leq d_2 (x,y) \leq B d_1 (x,y). Homework...

Homework Statement Prove by induction that: (Please see attachment) Homework Equations The Attempt at a Solution Can someone please confirm if I have worked the question out correctly. Many thanks.

1. Let m, n, p, q \in Z If 0 < m < n and 0< p \leq q, then mp < nq 2. Propositions/axioms I can use that relate to inequalities 2.4 Let m,n,p \in Z. If m < n and n < p, then m < p 2.5 For each n \in N there exists an m \in N such that m > n 2.6 Let m,n \in Z. If m \leq n \leq m then...

Hello, I am struggling with solving trigonometric inequalities. For example, solve: cos(\frac{\pi t}{3}) < \frac{1}{2}, 0<t<50 I wonder if one of these solutions is true: 1/ \frac{\pi}{3} + k2\pi < \frac{\pi t}{3} < \frac{5\pi}{3} + k2\pi, k \in Z 2/ \frac{\pi}{3} + 6k < \frac{\pi...

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 Find the dual of -d \leq Ax-b \leq d x \geq 0; c \cdot x = min where A is mxn matrix and x,d,b \in \mathbb{R}^n Homework Equations dual of canonical is of the form maximize b \cdot y A^{T}y \leq where y \in \mathbb{R}^m The Attempt at a Solution I tried...

Hello folks :smile: I always thought that inequalities could be treated exactly like equations but somehow I seem to be loosing information or something. For example, if I wish to find all values of x for which the following is true: 1/x + 1/(1-x) > 0 I would 'solve' it as follows 1...

Hi all, Given... a + b > p b > q Is there no way to place any limits on a in terms of p and q only? I know that one is allowed to add inequalities together but not subtract, but is there any other tricks one can play to solve this? Thanks, Natski

Homework Statement d) Show that \left|x-y\right| \leq \left|x\right|+\left|y\right| e) Show that \left|x\right|-\left|y\right| \leq \left|x-y\right| The Attempt at a Solution For item d) I've tried some approaches but none was promising. For item e), I tried squaring...

Inequalities- tricky question! hiii, i was wondering if anybody knew how to help me with this one tricky homework question. i can do most of the inequaliies I've come across, but how do you solve an inequality if you can't factor it? the question is: (x^2-4x+7)/(x^2+x-6) i know that the...

Homework Statement PROBLEM: Match the inequalities with the corresponding statements. INEQUALITIES: 1) |a-5|< 1/3 2) |a- 1/3|< 5 STATEMENTS: a) The distance from a to 5 is less than 1/3 b) a is less than 5 units from 1/3 The Attempt at a...

Homework Statement DIRECTIONS: Express the intercal in terms of an inequality involving absolute value. PROBLEM: (-4, 4) MY STEPS: 1: (-4, 4) 2: -4<x<4 3: |x|< 4 MY ANSWER Is that correct? Is step 3 correct? The only reason that I included that part is becuase it says...

do we have any proof of entanglement other than Bells Inequalities?bell's inequalities says that: - no physical theory of local hidden variables can reproduce all of the predictions of quantum mechanics or in other words - the correlations in/during Quantum Entanglement (QE) are stronger...

Polynomial Inequalities - Finding the solution set?? Homework Statement Solve the Inequality 2x^3 >-8x^2 Homework Equations The Attempt at a Solution Ok I am able to solve this by first figuring out the zeroes, and then testing with regions, So my answer is x=0 and x =...

Hey guys, Just having a bit of trouble with inequalities. Homework Statement Sketch all complex numbers 'z' which satisfy the given condition: |z + i + 1| \leq |z - i| Homework Equations --- The Attempt at a Solution z + i + 1\leq z - i z + 2i + 1\leq z 2i + 1\leq...

Homework Statement Given: a>=b>=c>=0, d>=e>=f>=0, a>=d a+b>=d+e a+b+c=d+e+f a,b,c,d,e,f belong to Real numbers Prove that d, e, f can be expressed as linear combinations of a, b and c in such way: d=(c1+c2)*a+(c3+c4)*b+(c5+c6)*c e = (c1+c6)*a+(c2+c4)*b+(c3+c5)*c...

Got a load of Logarithmic Inequalities questions. Solved almost all of them but got stuck in a question. here's the question:- log3 |3-4x| > 2 Please help.

Homework Statement I was trying to show that 1) |a+b|≤|a|+|b| 2) |a+b|≥|a|-|b| and find out how they were true when a,b>0, a,b<0, and a>0,b<0 Homework Equations 1) |a+b|≤|a|+|b| 2) |a+b|≥|a|-|b| The Attempt at a Solution For |a+b|≤|a|+|b| a,b>0 I got that |a+b|=a+b...

Hi, Prove that the inequality 3x2 + 13 < 12x has no real solution Is it because: 3x2 - 12x + 13 < 0 And, using the quadratic equation we have to square root a negative number, meaning, the answer will be always greater than 0, not smaller? Thanks, Peter G.

So, I have this book that doesn't explain why you have to first find the domain, consolidate logs, solve the function as a rational inequality, find the key numbers, then find which numbers of the key numbers are actually in the domain according to the inequality, and finally write out the...

(x+2)/(x+4) greater or equal to 1. I got two different answers here. X is greater than 4. Or a interval notation (-Infinite, 4) - which doesn't make sense but wouldn't the correct answer just be X is greater than 4? which would mean (4, infinite)? Homework Equations...

I've been playing around with some proofs and find myself relearning how I do my mathmatical thinking. Just a general question regarding how to handle something like this. 0<a<b So a and b must be positive a/2 <b I just divided one side by two instead of "dividing through" like you...

so I have 1/(n-2). I have that n>max(epsilon+2,1). I need to get 1/(n-2) < epsilon. I know that 1/(n-2)<1/(epsilon+2-2)=1/epsilon. but 1/epsilon is not always less than epsilon. can you see any errors?

Hello my question is why do we set some equations and inequalities to 0, for example quadratic equations/inequalities. I know that they should be 0, but why. How did people come with this when they invented it.

If x<=b does this mean max(x)=b? is x<=b equivalent to the interval (-infinity, b]?

Homework Statement Hey, just wondering how I might go about doing this problem, as I really have very little idea... Prove the following inequality: \frac{1}{e}\leq\frac{1}{4\pi^{2}}\int_{R}e^{cos(x-y)}dxdy\leqe (hopefully this reads "one over e is less than or equal to one over four pi...

Please help me to confirm, weather the following step is correct |\gamma| \leq \cos (\beta) \arccos (|\gamma|) \leq \beta does taking the arccos() on both sides of equation changes the relational operator??

Homework Statement Prove that -1< x < 0 implies |x^2 - 2x +1| < 1.25|x-1| The Attempt at a Solution Attempt at 1st question: |(x-1)(x^2 + x -1)| < 1.25|x-1| |(x^2 + x -1)| < 1.25 -1.25 < (x^2 + x -1) < 1.25 -0.25 < x^2 + x < 2.25 -0.5 < (x + 0.5)^2 < 2.25 ** this leads to 0 < (x...

Hi, Could you clarify the relationship between proofs that use ≤ and those that use <? For example, if it's already proven that "abs(b) ≤ a if and only if -a≤ b≤a" can we say this implies that "abs(b) < a if and only if -a< b<a"? It seems that since the first statement holds for all abs(b)...

I need to graph/find numbers for S∩T where S is x^2+y^2 <=100 and T is x+y<=14. I know I can find them simply by choosing/picking them, but are there any other solution ? I thought maybe doing x^2+y^2 <=100 + x+y<=14 = x^2+y^2 + x+y<=14 +100 = x^2+y^2 + x+y<=114 = x^2+y^2 <=...

New J. Phys. 12 123007, 2010. Violation of Leggett inequalities in orbital angular momentum subspaces. We report an experimental test of Leggett's non-local hidden variable theory in an orbital angular momentum (OAM) state space of light. We show that the correlations we observe are in...

Homework Statement I have a function h(t) = 30t - 5t^2. I need to find the interval for which h is > 25. Homework Equations The Attempt at a Solution h(t) = 30t - 5t2 - 25 > 0 -5(t2 - 6t + 5) > 0 iff t2 - 6t + 5 > 0 Then the answer is t > 5 and t < 1. But it is actually...

Ive got a few questions id like checking please but i start with the one I got no clue about :S 1)Slove the equation x(x-2)=2-x So i asumme i slove it to zero, x^2-x-2=0 B)Use the solution to part A and the illustrated grapg to write down the solutions of I) X(x-2)<2-x II)X(x-2)>0...

Hi I'm doing the first chapter of Spivak's Calculus and just a little concerned about a particular thing he does in the chapter. He is talking about the trichotomy axiom and that if a > b then a - b, this can be understood as expressing (a - b) > 0 and then the axiom can be interpreted...