inequalities

  1. idontknow

    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. tahirimanov19

    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. idontknow

    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. D

    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. B

    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. M

    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. M

    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. B

    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. A

    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. EvolvedPie

    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. P

    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. M

    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. M

    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. C

    Linear inequalities

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

    Write a System of Inequalities to Graph?

    It costs a bakery \$3 to make a cake and \$4 to make a pie. If production costs cannot exceed \$120 per day, graph the system of inequalities that represents how many cakes and pies can be made. How would you find a system of inequalities for this problem? I came up with one inequality, but I...
  16. I

    Write a System of Inequalities for this Problem..?

    A music store is selling CDs for either $10 or $15. A customer wants to spend between $30 and $60 on the CDs. Write the system of inequalities that represents this situation and graph its solution region. Could anyone point me in the right direction with this? I really have no clue where to...
  17. L

    Inequalities

    Hi! I need help for these exercises:
  18. M

    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. D

    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. G

    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