# inequalities

1. ### Inequalities #4

1. Prove that n!^{2}\geq n^n \; , n\in \mathbb{N}. 2. Prove that (n+1/n)^n \geq n^2 +n .
2. ### Inequalities for idontknow, and for everyone else.

1. If a, b, c are sides of the triangle, prove for any $x \in \mathbb{R}$ $$b^2x^2+(b^2+c^2-a^2)x+c^2>0$$ 2. $$(a_1b_1+a_2b_2+a_3b_3)^2 \le ({a_1}^2+{a_2}^2+{a_3}^2)({b_1}^2+{b_2}^2+{b_3}^2)$$ 3. Prove, if $a>0, \; b>0$, $$\dfrac{a+b}{1+a+b} < \dfrac{a}{1+a} + \dfrac{b}{1+b}$$ 4. If...
3. ### Inequalities #3

Prove that \: \displaystyle (1+e^{-1})\cdot (1+e^{-2}) \cdot ... \cdot (1+e^{-n}) < \displaystyle (e^{-1^2 }+e^{-2^2 }+e^{-3^2}+...+e^{-n^2 } )^{\displaystyle -1} \; ,n >1 . e-euler constant . Method required !
4. ### Limes superior and inferior inequalities with functions and its derivatives

Hi, I have got some troubles with Lemma 2. from N.H. Duu, On The Existence of Bounded Solutions for Lotka- Volterra Equations, Acta Mathematica Vietnamica 25(2) (2000), 145-159. Can anybody help me understand the last part of this proof? Exactly, how this transformation using Cauchy theorem...
5. ### Union or intersection in inequalities?

I was solving this inequality $\left\lvert\frac{x+2}{x+1}\right\rvert>1$ and got \begin{aligned}x\in(-\frac{3}{2},-1)\quad\lor\quad x\in(-1,+\infty)\end{aligned} Apparently the solution should be the union of the two, not the intersection. How do I know which to do when? Both rules and...
6. ### Inequalities important problem

Good Morning, I've recently been working on a problem, which I haven' been able to solve: "Find the biggest positive integer with the following property: each digit except for the first and the last one must be smaller than the average mean of the two immediately adjacent digits. The...
7. ### Problem with inequalities

Good Morning, I've recently been working on a problem, which I haven't been able to solve: "Find the biggest positive integer with the following property: each digit except for the first and the last one must be smaller than the arithmetic mean of the two immediately adjacent digits. The...
8. ### Exercises about tricky inequalities (real analysis)

Where can I find some tricky inequalities? (Like this one: $$\sqrt{(\log_9 |x| -1)\log_3(x^2)}>(1/2) - 2\log_9 |x|$$) (I don't know whether it is the correct section of the forum to post this; sorry if it's not.)
9. ### Is the square operation in inequalities?

5 > -5 5^2 > (-5)^2 Does one reverse the signs at this point? (Since one of the sides is multiplied by a negative?) Either way, you arrive at the contradiction: 25>25 Am I doing something wrong here, or is it illegal to square both sides of an inequality all together?
10. ### Solve Inequalities

2(x^2-2)>7x Expand out the brackets: 2x^2-4>7x Tidy it up a little: 2x^2-7x>4 But where do I go from here? The real question is, how can I go about removing the x^2, I can never quite remember the rules. Many thanks.
11. ### Integration inequalities

Trying to prove the following: Let $g(x),l(x),h(x)\geq0$ for all $x\in[a,b]$ with the inequality being strict for at least some $x_1,x_2\in[a,b]$, and let $g$'$(x)$,$l$'$(x)>0$, then $\int_{a}^{b}h(x)g(x)l(x)dx\int_{a}^{b}h(x)dx-\int_{a}^{b}h(x)g(x)dx\int_{a}^{b}h(x)l(x)dx>0$
12. ### Inequality help

Assume a,b,c,d are positive and a>b as well as c<d Can we add the the two the following way? a>b --- (1) d>c --- (2) so a+d>b+c. My teacher said it is wrong, but i am not able to find a counterexample. Appreciate any help.
13. ### Inequalities

Determine all real numbers r such that the inequality a^3 + ab + b^3>= a^2 +b^2 applies to all pairs of real numbers a, b that are greater or equal to r
14. ### Linear inequalities

X is a whole number.if three quarters of x is subtracted from 1,the result is always greater than 0.

17. ### Inequalities

Hi! I need help for these exercises:
18. ### how to selecte 2 inequalities between three inequalities for solving linear programmi

hi i saw a question in a book for liner programming in solving the question i faced three inequalities as follow 1- 3x+2y<= 5000 2- 1x+.5y<= 1600 3- 4x+5y<= 9600 in book they select 1 and 3 inequalities for solving simultaneous equation, i cant understand the logic for this. anyone...
19. ### Absolute value inequalities

| x + 1 | > | 2x - 4 | For which real x values is the absolute value of x+1 greater than the absolute value of 2x-4? Should I split it into the three cases below? -1 > x 2 > x > -1 x > 2 I wrote x = 2, 3 and 4 But I don't seem to remember why it was this answer and I lost the papers where I...
20. ### Given a>c and b>c, can I write {a,b}>c?

Hi, is there a shorter way of writing that a>c and b>c? Can I, for example, write {a,b}>c? Thomas