Inequality Definition and 1000 Threads

Homework Statement Using the principle of mathematical induction, prove that for all n>=10, 2^n>n^3 Homework Equations 2^(n+1) = 2(2^n) (n+1)^3 = n^3 + 3n^2 + 3n +1 The Attempt at a Solution i) (Base case) Statement is true for n=10 ii)(inductive step) Suppose 2^n > n^3 for some integer >=...

The Cauchy-Schwartz inequality (\sum_{i=1}^n x_i^2)(\sum_{i=1}^n y_i^2) - (\sum_{i=1}^n x_iy_i)^2 \geq 0 holds with equality (or is as "small" as possible) if there exists an a \gt 0 such that x_i=ay_i for all i=1,...,n . But when is the inequality as "large" as possible? That is, can we...

I need help determining the interval notation of the inequality below: -9<1/x<=1

Solve the following inequality. Represent your answer graphically: ## |z-1| + |z-5| < 4 ## Homework Equations ## z = a + bi \\ |x+y| \leq |x| + |y| ## Triangle inequality The Attempt at a Solution ## |z-1| + |z-5| < 4 \\ \\ x = z-1 \ \ , \ \ y = z-5 \\ \\ |z-1+z-5| \leq |z-1| + |z-5| \\...

Let ${y}_{n}$ be a arbitrary sequence in X metric space and ${y}_{m+1}$ convergent to ${x}^{*}$ in X...İn this case by using triangle inequality can we say that ${y}_{n}\to {x}^{*}$

Can somebody help me please, I've tried solving this for hours but I still couldn't get it. Given that a, b, c, d are positive integers and a+b=c+d. Prove that if a∗b < c∗d, then a∗log(a)+b∗log(b) > c∗log(c)+d∗log(d) How do I do it?

Can someone explain to be in detail what is quadratic inequality? It's rather confusing. Thank you

Given $x,\,y,\,z$ are positive real numbers. Prove that $\dfrac{xy}{x^2+xy+y^2}-\dfrac{1}{9}+\dfrac{yz}{y^2+yz+z^2}-\dfrac{1}{9}+\dfrac{zx}{z^2+zx+x^2}-\dfrac{1}{9}\le \dfrac{2\sqrt{xy+yz+zx}}{3\sqrt{x^2+y^2+z^2}}$

Prove $\sqrt[3]{1−12\sqrt[3]{65^2} + 48\sqrt[3]{65}} -\sqrt[3]{63}\gt \sqrt[3]{1−48\sqrt[3]{63} + 36\sqrt[3]{147}} -4 $

Hello all, I want to prove the following inequality. sin(x)<x for all x>0. Now I figured that I put a function f(x)=x-sin(x), and show that it is increasing for all x>0. But this alone doesn't prove it. I need to show we have inequality from the start. I can't show that lim f(x) as x->0 is...

Homework Statement How to solve this kind of inequality? x²-4x+3≤(3x+5)(2x-3)Homework EquationsThe Attempt at a Solution :[/B] I'm confused. Should I factor the left side or should I FOIL the right side then equate it to zero to find the critical numbers? Help pleaasee.

Homework Statement Show that |<v|w>|^2 ≤ <v|v><w|w> for any |v>,|w> ∈ ℂ^2 Homework EquationsThe Attempt at a Solution The Cauchy-Schwartz inequality is extremely relevant for the math/physics that I am interested in. I feel like I have a very good proof here, but I am interested in a few...

1. Homework Equations Solving Polynomial Inequalities The Attempt at a Solution Then I used the property of absolute value inequality to get rid of it. But I really don't know if I'm doing the right step. Is this correct? So that I could separate them in two cases and find the...

Please help me. What is the next step to get rid of the absolute value? I tried using its property but I don't know if its correct.

Homework Statement solve 3x4+2x2-4x+6≥6x4-5x3-9x+2 Do not use technology (i.e.-graphing calculators) Homework Equations Remainder Theorem The Attempt at a Solution I set the inequality equal to zero -3x4+5x3+3x2+5x+4≥0 Checking all the Possible rational roots for a possible factors... none...

For $x,\,y,\,z>0$ and $xyz=1$, prove $\sqrt{x^2+1}+\sqrt{y^2+1}+\sqrt{z^2+1}\le\sqrt{2}(x+y+z)$.

Homework Statement Let ##f,g## be two real valued functions, defined on the segment ##[a,b]## and continuous on ##[a,b]##, such that ## 0 < g < f ##. Show there exist ##\lambda > 0 ## such that ## (1+\lambda) g \le f ## Homework Equations The Attempt at a Solution Set ##h = f/g##. Since...

How do you solve x for the below inequality? ##\frac{a}{x^2} < -b## My attempt is: ##\frac{a}{x^2} + b < 0## ##\frac{a + bx^2}{x^2} < 0##

Homework Statement [/B] As attached Homework EquationsThe Attempt at a Solution [/B] The answer is stated as option A. However, my solution is -6≤x≤3; I can seems to find an option that fits the solution.

5/(x-1) - (2x)/(x+1) - 1 < 0 How does one solve this inequality?

(3x - 5)/(x - 5) > 4 How does one complete this problem?

So, i need to proof the triangle inequality ( d(x,y)<=d(x,z)+d(z,y) ) for the distance below But I'm stuck at In those fractions i need Xk-Zk and Zk-Yk in the denominators, not Xk-Yk and Xk-Yk. Thanks in advance

Prove $\dfrac{1}{x^4}+\dfrac{1}{4x^3y} + \dfrac{1}{6x^2y^2}+ \dfrac{1}{4xy^3}+ \dfrac{1}{y^4} ≥ \dfrac{128}{3(x+y)^4}$, given $x,\,y$ are positive real numbers.

I am facing some doubts trying to understand the illustration my textbook has adopted for the development of the Clausius inequality for thermodynamic cycles.I have attached an image of the content from my textbook. As one could see the author has assumed a closed system connected to a thermal...

Prove that $9\gt \sqrt{a-1}+\sqrt{19-3a}+\sqrt{2a+9}$ for all real $a$.

A great new experiment is reported closing simultaneously the loopholes of detection (fair sampling assumption) and distance (locality assumption): Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km B. Hensen, H. Bernien, A.E. Dréau, A...

Let $a_1,\,a_2,\,\cdots,\,a_{12}$ be positive numbers. Show that at least of the following must be true: $\dfrac{a_1}{a_2}+\dfrac{a_3}{a_4}+\dfrac{a_5}{a_6}+\dfrac{a_7}{a_8}+\dfrac{a_9}{a_{10}}\ge 5$, $\dfrac{a_{11}}{a_{12}}+\dfrac{a_2}{a_1}+\dfrac{a_4}{a_3}+\dfrac{a_6}{a_5}\ge 4$, or...

Homework Statement \ln \frac{(x+1)}{(x-1)} \geq 0 Homework Equations \ln \frac {a}{b} = \ln a - \ln b \ln \frac {(x+1)}{(x-1)} = \ln (x+1) - \ln (x-1) The Attempt at a Solution \ln (x+1) \geq \ln (x-1) e^{\ln (x+1)} \geq e^{\ln (x-1)} (x+1) \geq (x-1) According to Wolfram, the...

Hello all, I have this formula ##\left[2\sqrt{Q\left(\sqrt{2\eta}\right)}\right]^N## where Q is the Q Gaussian function which can be upper bounded by the Chernoff bound ##Q\left(\sqrt{2\eta}\right)\leq exp\left(-\eta\right)##, and thus the original formula can be upper bounded as...

https://www.osapublishing.org/optica/fulltext.cfm?uri=optica-2-7-611&id=321243 "In our experimental test, we used light whose statistical behavior (field second-order statistics) is indistinguishable from classical, viz., the light from a broadband laser diode operating below threshold. Our...

Prove $\dfrac{(x+y)^2}{2}+\dfrac{x+y}{4}\ge x\sqrt{y}+y\sqrt{x}$.

Prove $35\sqrt{55}+55\sqrt{77}+77\sqrt{35}+35\sqrt{77}+55\sqrt{35}+77\sqrt{55}\gt 2310$.

How is the generalized triangle inequality in b-metric spaces ? I find something...But I wonder your opinion...Thank you for your attention... Especially if you write for n,m>0 m>n $d({x}_{n},{x}_{m})$$\le$..... I will be happy...

Let $X$ be a non-empty set and let $s\ge1$ be a given real number. A function $d:$ X $\times$ X$\to$ ${R}^{+}$ , is called a b-metric provided that, for all x,y,z $\in$ X, 1) d(x,y)=0 iff x=y, 2)d(x,y)=d(y,x), 3)d(x,z)$\le$s[d(x,y)+d(y,z)]. A pair (X,d) is called b-metric space. İt is clear...

$a,b,c \in N$,prove : $\dfrac{a^3}{a^2+ab+b^2}+\dfrac{b^3}{b^2+bc+c^2}+\dfrac{c^3}{c^2+ca+a^2}\geq\dfrac{a+b+c}{3}$

The problem I want to solve the following inequality: $$ x+\frac{1}{x}<1 $$ The attempt ## x+\frac{1}{x}<1 \\ x+\frac{1}{x}-1<0 \\ \frac{x^2}{x}+\frac{1}{x}-\frac{x}{x}<0 \\ \frac{x^2-x+1}{x}<0 ## ## x \neq 0 ## I tried to factor the numerator to examine the polynomial with a character table...

Suppose $k>0$. Show that $\dfrac{2015}{(k+1)(k+4030)}<\dfrac{1}{k+1}-\dfrac{1}{k+2}+\dfrac{1}{k+3}-\dfrac{1}{k+4}+\cdots+\dfrac{1}{k+4029}-\dfrac{1}{k+4030}$.

Thought I knew this, but am confused by the following example: Show $ |z^3 - 5iz + 4| \ge 8 $ The example goes on: $ |z^3 - 5iz + 4| \ge ||z^3 - 5iz| - |4|| $, using the reverse triangle inequality It's probably right, but I don't get why the +4 can just be made into a -4 ?

What is a square of a number? A^2=A*A. If A=B squaring both sides will give A^2=B^2. How I think about squaring is we multiply both sides of A=B by A(we could also do this for B) we get A*A=B*A but A=B so this will result in A*A=B*B. But if we do this for an inequality, A>B, multiplying both...

Homework Statement A small smooth sphere of mass 3 kg moving on a smooth horizontal plane with speed 8 ms-1 collides directly with a sphere of mass 12 kg which is at rest. Given that the spheres move in opposite directions after the collision, obtain the inequality satisfied by e. Homework...

Hello, I'm having a small problem with Schwarz inequality, |u⋅v|≤||u||||v|| the statement is true if and only if cosΘ≤1 !, I'm familiar with this result but how could it be more than 1? what is so special in higher dimensions that it gave the ability for cosine to be more than 1...

given : $a>b>c>0$ prove : $a^{2a}\times b^{2b}\times c^{2c}>a^{b+c}\times b^{c+a}\times c^{a+b}$

I'm interested in the following inequality (which may or may not be true) Theorem 1: ##( \sum_{i=1}^n \frac{a_i} {n}\ )( \sum_{i=1}^n \frac{1} {b_i}\ ) > \sum_{i=1}^n \frac{a_i} {b_i}\ ## Where ##n ≥ 2, a_1 < a_2 < ... < a_n## and ##b_1 < b_2 < ... < b_n##. My attempt at a proof: 1) When n =...

Hello! (Wave) Let $u(x,y), x^2+y^2 \leq 1$, a solution of $$u_{xx}(x,y)+2u_{yy}(x,y)+e^{u(x,y)}=0, x^2+y^2\leq 1$$ Show that $\min_{x^2+y^2 \leq 1} u(x,y)= \min_{x^2+y^2=1} u(x,y) $. We suppose that $\min_{x^2+y^2 \leq 1} u(x,y) \neq \min_{x^2+y^2=1} u(x,y) $.At the solution it is said...

I recently attended a presentation on the fundamentals of quantum mechanics which focused on the most recent experimental tests on Bells Inequality. As part of the introduction the speaker derived Bells Inequality. The speaker made it sound very straightforward and it was, the proof was a piece...

Homework Statement $$x+\frac{16}{\sqrt{x}} \geq 12$$ How do I show that only x>0 satisfies the inequality above. Homework EquationsThe Attempt at a Solution I have not made a lot of progress here. I tried the following: $$x+\frac{16}{\sqrt{x}} - 12 \geq 0$$ I tried to multiply with $$...

Real numbers $u,\,v,\,x,\,y$ satisfy the following conditions: $|u|>1$, $|v|>1$, $|x|>1$, $|y|>1$, and $u+v+x+y+uv(x+y)+xy(u+v)=0$ Prove that $\dfrac{1}{u-1}+\dfrac{1}{v-1}+\dfrac{1}{x-1}+\dfrac{1}{y-1}>0$.

Homework Statement What is the set of values of p for which p(x^2+2) < 2x^2+6x+1 for all real values of x? Homework Equations p(x^2+2) < 2x^2+6x+1 3. The Attempt at a Solution I know I need to use my knowledge of the discriminant here, but the fact that its an inequality is confusing me...

How would I solve the inequality (X-4)/X>0. I thought that inequalities were solved in the same way equations were, but when I solve that way I get X>4 which isn't the entire answer.

Proof of Cauchy-Schwarty Inequality from the Book "Quantum Mechanics Demystified" Page 133. I do not understand one key step! Most appreciated someone could help. Please see attached file.