Inequality Definition and 1000 Threads

  1. N

    Trignometric inequality problem

    Homework Statement Given that -pi < x < pi, solve the following inequality in radians root(2) - 2sin(x-(pi/3)) < 0 The Attempt at a Solution root(2) - 2sin(x-(pi/3)) < 0 - 2sin(x-(pi/3)) < -root(2) sin(x-(pi/3)) > (root(2)) -pi<x<-pi -pi - (pi/3) < x - pi/3 < pi...
  2. U

    Proving Inequality: Vector Method for Cos2A+Cos2B+Cos2C in Triangle ABC

    Homework Statement In a triangle ABC, prove by vector method cos2A+cos2B+cos2C≥ -3/2. Homework Equations The Attempt at a Solution I can change the LHS of the inequality to the form (cos2A i + cos2B j + cos2C k).(i+j+k)
  3. E

    Is There a Non-Calculus Method to Prove this Thermodynamic Inequality?

    Prove that $$ \frac{y^x-1}{xy^{x-1}(y-1)}<1$$ where x,y \in ℝ, x>1 and y>1. I was able to prove it using calculus, but am wondering if there was another way of doing so, like exploiting some inequality-theorems which involves real numbers. I'll be glad if anyone can show me a way and...
  4. M

    Is a<-24 the Correct Solution to the Inequality Problem?

    Homework Statement ##\frac{a}{4}>\frac{a}{2}+6## The Attempt at a Solution ##\frac{2a}{2}>\frac{4a+48}{2}## ##a>2a+24## So do I just plug random numbers in and see what I get? I realized right away that it has to be a negative number so I stuck in -30 and got ##-30>-60+24## Well that's...
  5. O

    Prove Inequality: |1-K+x|/|1+y| < 1

    Homework Statement Given: |x-y| < K x+y > K - 2 0 < K < 1 Prove: \frac{|1-K+x|}{|1+y|} < 1 The Attempt at a Solution I have tried using the fact that |x-y| < K \Rightarrow -K < x-y < K \Rightarrow y-K < x < y+K to write \frac{1-K+x}{1+y} < \frac{1+y}{1+y} = 1 But I can't figure out...
  6. N

    How Can Norm Integration Address Inequality?

    I am struggling with this question. I need a different perspective. Any recommendation is appreciated. Please click on the attached Thumbnail.
  7. S

    Proving an inequality using maximum modulus

    Homework Statement Let f be an analytic function on the disc |z|<1 and satisfies |f(z)|≤M if |z|<1. Show that |f(z)| \le M \left| \frac{z-a}{1-a'z} \right| when |z|<1 where a' is the complex conjugate of a Homework Equations This section uses maximum modulus principle, but I really don't...
  8. S

    MHB Trig. inequality: Strictly algebraic

    Forum, do you have any idea how to solve the trigonometric inequality \cos (x) < \sin (x) strictly algebraically? The conventional(?) approach is to first solve \cos(x) = \sin(x) and then draw the graphs for each function in order to find the correct interval. However, I would love to know if...
  9. Seydlitz

    Proving Bernoulli's Inequality

    Homework Statement Prove Bernoulli's Inequality: if ##h>-1## (1+h)^n \geq 1+hn Homework Equations Binomial Theorem (a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^{k} The Attempt at a Solution If ##h=0## (1+0)^n=1 1=1 If ##h>0## This (1+h)^n \geq 1+hn Implies...
  10. anemone

    MHB Inequality Challenge: Prove 1/44 > 1/1999

    Show that \frac{1}{44}>\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\left(\frac{5}{6}\right)\cdots\left( \frac{1997}{1998}\right)>\frac{1}{1999}
  11. morrobay

    Is Conservation of Momentum Overlooked in Bells Inequality Violations

    In the EPR scenario the correlation results are explained with the conservation laws of classical mechanics as applied to spin. The Bell type inequalities are derived on expected spin values. But the violations of these inequalities are then explained with QM: That simultaneous knowledge of...
  12. Saitama

    Proving Cosine Inequality with Triangles

    Homework Statement If ##A+B+C=\pi##, prove that ##\cos A+\cos B+\cos C \leq 3/2##. Homework Equations The Attempt at a Solution I don't really know how to start. ##A+B=\pi-C##. Taking cos on both sides doesn't seem of much help. I need a few hints to start with.
  13. Albert1

    MHB Can This Product Inequality Be Proven for Positive x and Natural n?

    Given: x>0,\, n\in\mathbb{N} Prove: (1+x)\times\left(1+x^2 \right)\times\left(1+x^3 \right)\times\cdots\times\left(1+x^n \right)\geq\left(1+x^{\large{\frac{n+1}{2}}} \right)^n
  14. Seydlitz

    How does the Triangle Inequality apply in this situation?

    I'm beginning to read Spivak's Calculus 3ed, and everything is smooth until I reach page 12. My question is marked, between line 2 and 3. Why there's such sign change suddenly? In fact I tried with simple line 4 case and it's not in fact equal. I'm assuming that a and b is valid for all...
  15. Albert1

    MHB Prove Inequality: $x^4,y^4,z^4 \geq 48(y-1)^2(z-1)^2(x-1)^2$

    x>1,y>1 and z>1 prove :$\dfrac {x^4}{(y-1)^2}+\dfrac {y^4}{(z-1)^2}+\dfrac {z^4}{(x-1)^2}\geq 48$
  16. A

    Is the Integral Inequality Possible to Prove for Certain Parameters?

    I want to know that is it possible to show that $$ \int_{0}^{T}\Bigr(a(t )\Bigr)^{\frac{p+1}{2p}}dt\leq C\Bigr(\int_{0}^{T}a(t)dt\Bigr)^{\frac{p+1}{2p}} $$ for some ##C>0## where ##a(t)>0## and integrable on ##(0,T)## and ##p\in(\frac{1}{2},1)##. It is worth noting that this range for ##p##...
  17. A

    Integral Inequality for Measurable Functions

    For what class of functions we have: $$ \int_{\Omega} [f(x)]^m dx \leq C\Bigr ( \int_{\Omega} f(x)dx\Bigr)^{m}, $$ where ##\Omega## is open bounded and ##f## is measurable on ##\Omega## and ##C,m>0##.
  18. anemone

    MHB Find x in [0,2π] to Solve Inequality

    Find all x in the interval [0, 2\pi] which satisfies 2\cos(x) \le \left|\sqrt{1+\sin (2x)}-\sqrt{1-\sin (2x)} \right|\le \sqrt{2}
  19. morrobay

    Bells' Inequality Spin Violations

    When entangled photons are generated from a cascade of a Calciums' 6s level this inequality : n[y+z-] + n[x-y+] ≥ n[x-z-] is derived for what is equivalent to spin in photons. When the detectors at A and B are parallel the perfect anti correlations are due to conservation laws of angular...
  20. Saitama

    Find Integer Values of a for Inequality Problem

    Homework Statement Let ##x^2+y^2+xy+1 \geq a(x+y)## for all ##x,y \in R##. Find the possible integer(s) in the range of ##a##. Homework Equations The Attempt at a Solution I can rewrite this into ##(x+y)^2-xy+1 \geq a(x+y) \Rightarrow (x+y)(x+y-a)+1-xy \geq 0## but I don't think...
  21. Saitama

    What Went Wrong with Solving a Simple Inequality?

    Homework Statement 3\sqrt{x}-\sqrt{x+3}>1 Homework Equations The Attempt at a Solution As obvious from the given inequality, x must be greater than zero. Rearranging and squaring both the sides, 9x>1+x+3+2\sqrt{x+3} \Rightarrow 4x-2>\sqrt{x+3} Squaring again, 16x^2+4-16x>x+3...
  22. MarkFL

    MHB Can you prove this trigonometric inequality?

    Show that : \left( {\sin x + a\cos x} \right)\left( {\sin x + b\cos x} \right) \leq 1 + \left( \frac{a + b}{2} \right)^2
  23. A

    Can You Solve the Inequality x^2 + x < 0?

    Homework Statement x^2+x<0 [
  24. Fernando Revilla

    MHB Solve Simple Inequality: x < 0 or x > 2

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  25. Albert1

    MHB Prove A < B with Log Inequality $\pi\approx3.1416$

    $\pi\approx3.1416$ $A=\dfrac{1}{log_5 19}+\dfrac{2}{log_3 19}+\dfrac{3}{log_2 19}$ $B=\dfrac{1}{log_2\pi}+\dfrac{1}{log_3\pi}$ edit :$B=\dfrac{1}{log_2\pi}+\dfrac{1}{log_{\color{red}5}\pi}$ $Prove: \,\, A < B$
  26. S

    MHB Proof of Inequality by Induction

    need help on this Show by induction that n^3 <= 3^n for all natural numbers n.
  27. Saitama

    Solving Inequality Problem: Find a for 3 on 0-2 Interval

    Homework Statement Find all numbers ##a## for each of which the least value of the quadratic trinomial ##4x^2-4ax+a^2-2a+2## on the interval ##0\leq x \leq 2## is equal to 3.Homework Equations The Attempt at a Solution I don't really know what should be the best way to start with this type of...
  28. R

    MHB Lagrange's Identity and Cauhchy-Schwarz Inequality for complex numbers

    I guess the best way to start this is by admitting that my conceptual understanding of the Cauchy-Schwarz Inequality and the Lagrange Identity, as the title suggests, is not as deep as it could be. I'm working through Marsden's 3e "Basic Complex Analysis" and it contains a proof of the Cauchy...
  29. mathworker

    MHB Proving Inequality in Mathematics: Vacation Edition

    i found this problem interesting in stack exchange unfortunately i will participate in discussion for 4 days(vacation) inequality - Prove $\sum_{i=1}^{n}\frac{a_{i}}{a_{i+1}}\ge\sum_{i=1}^{n}\frac{1-a_{i+1}}{1-a_{i}}$ if $a_{i}>0$ and $a_{1}+a_{2}+\cdots+a_{n}=1$ - Mathematics Stack Exchange...
  30. mathworker

    MHB Prove Inequality IMO-2012: a2a3⋯an=1

    IMO-2012: let a_2,a_3,...,a_n be positive real numbers that satisfy a2.a3...a​n=1 .Prove that, (a_2+1)^2.(a_3+1)^3...(a_n+1)^n>n^n hint:
  31. I like Serena

    MHB AM-GM inequality for sum of 3 square roots

    Let $a,b,c$ be positive real numbers with sum $3$. Prove that $√a+√b+√c≥ab+bc+ca$.
  32. Saitama

    Prove: Inequality of Sums of Square Roots of Positive Reals

    Homework Statement Let ##a,b,c## be positive real numbers with sum 3. Prove that \sqrt{a}+\sqrt{b}+\sqrt{c} \geq ab+bc+ca Homework Equations AM-GM inequalityThe Attempt at a Solution I don't really know how to start with. We are given ##a+b+c=3##. Also, ##2(ab+bc+ca)=(a+b+c)^2-(a^2+b^2+c^2)##...
  33. U

    Prove this inequality via graph

    Homework Statement 2x>3sinx-xcosx, 0<x<∏/2 Homework Equations The Attempt at a Solution One possible way is to draw the graph of the functions and compare but plotting a graph manually is not easy in this case. I want some other methods.
  34. X

    Inverse function of inequality function

    Homework Statement Find inverse of each. 1. y<x+1 2. y=2x/(x-2) Homework Equations Switch y and x? The Attempt at a Solution For 1. I switched y and x, so x<y+1. Do I have to switch the sign also? For 2. I switched y and x, so x=2y/(y-2). But I have to express the inverse...
  35. T

    Application of the Schwarz Inequality

    Here's yet another assigned problem that I'm having difficulty with. I think I'm close to the end but am "nervous" (for lack of a better word) about whether or not I have used summation notation properly throughout the problem. Here it is: "Use the Schwarz inequality to establish that (...
  36. G

    Quick Chebychev Inequality Question

    Hello all, I am currently working through a proof in my Real Analysis book, by Royden/Fitzpatrick and I'm confused on a part. if f is a measurable function on E, f is integrable over E, and A is a measurable subset of E with measure less than δ, then ∫|f| < ε...
  37. pratikaman

    How Many Integers Meet the Condition {√n - √(23×24)}² < 1?

    How many integers satisfy {√n-√(23×24)}^2<1 I was able to solved this by trial and error method , but i want to know systematic step-wise solution.
  38. V

    Solve Spivak Inequality for x: 0<x<1

    Question: Find all numbers x for which \frac{1}{x}+\frac{1}{1-x}>0. Solution: If \frac{1}{x}+\frac{1}{1-x}>0, then \frac{1-x}{x(1-x)}+\frac{x}{x(1-x)}>0; hence \frac{1}{x(1-x)}>0. Now we note that \frac{1}{x(1-x)} \rightarrow ∞ as x \rightarrow 0 and \frac{1}{x(1-x)} \rightarrow 0 as x...
  39. B

    How to use the triangle inequality to solve a proof involving absolute values?

    Homework Statement Use the triangle inequality to prove that \left| s_n - s \right| < 1 \implies \left| s_n \right| < \left| s \right| +1 Homework Equations The triangle inequality states that \left| a-b \right| \leq \left| a-c \right| + \left| c-b \right| The Attempt at a Solution...
  40. U

    Proving Inequality for Linear Functions: |h(h(x))+h(h(1/x))|>2

    Homework Statement If f and g are two distinct linear functions defined on R such that they map[-1,1] onto [0,2] and h:R-{-1,0,1}→R defined by h(x)=f(x)/g(x) then show that |h(h(x))+h(h(1/x))|>2 Homework Equations The Attempt at a Solution I assume f(x) to be ax+b and g(x) to be lx+m so...
  41. L

    Proving probability inequality

    Homework Statement Prove the following a>0, X is a non-negative function Ʃ_{n\in N} P(X>an)≥\frac{1}{a}(E[X]-a) Ʃ_{n\in N} P(X>an)≤\frac{E[X]}{a} The Attempt at a Solution I know that \sum_{n\in N} P(X>an)=\sum_{k \in N} kP((k+1)a≥X>ka)=\sum_{k \in N} E[k1_{[(k+1)a,ka)}(X)]...
  42. A

    MHB Solve Probability Inequality | Finite Probabilistic Space

    Hi! I have to do this exercise: Define a finite probabilistic Space (Ω; Pr[ ]) and 2 events A,B⊆ Ω and Pr[A] ≠ Pr[B] so that we can verify that Pr[A∩B]>=9*Pr[A]*Pr[B] > 0. (1) ___________________________________________ I've been trying it but i have reached this conclusion: If Pr[A]>0...
  43. L

    Clausius inequality temperature

    in the clausius inequality is the temperature that of the system or of the surroundings? or is it temperature of the body receiving positive heat? (assuming the irreversibility is due to heat transfer with finite temperature difference) [borgnakke and sonntag-principle of entropy increase for...
  44. J

    I'm not sure what you're saying. Can you please clarify?

    I need a bit of help proving the following statement (n + 2)^n ≤ (n + 1)^n+1 where n is a positive integer. The (n+2) and (n+1) bases are making it hard for me solve this. I tried several time, I can't get the inductive step. Can someone lend me a little hand here? The base case is real...
  45. U

    How to Solve an Inequality with Greatest Integer Function and Fractional Part?

    Homework Statement x^2 \geq [x]^2 [] denotes Greatest Integer Function {} denotes Fractional Part Homework Equations The Attempt at a Solution x^2-[x]^2 \geq 0 \\ (x+[x])(x-[x]) \geq 0 \\ -[x] \leq x \leq [x] \\ Considering left inequality x \geq -[x] \\ \left\{x\right\}...
  46. B

    Is my proof of this inequality correct?

    Homework Statement Prove that |a + b| ≤ |a| + |b|. Homework Equations |a| = √a2 The Attempt at a Solution Since |a| = √a2, then |a + b| = √(a + b)2 = √(a2 + 2ab + b2) = √a2 + √b2 + √(2ab) = |a| + |b| + √(2ab). And since the square root of a negative number is not defined...
  47. phosgene

    Valid proof of Cauchy-Schwarz inequality?

    Homework Statement I was discussing the proof for the Cauchy-Schwarz inequality used in our lectures, and another student suggested an easier way of doing it. It's really, really simple. But I haven't seen it anywhere online or in textbooks, so I'm wondering if it's either wrong or is only...
  48. Albert1

    MHB Prove Inequality: $\frac{1}{2}$ Bound w/ x,y,z>0

    x>0 ,y>0 ,z>0 and xyz=1 ,prove : $ \dfrac{1}{(x+1)^2+y^2+1}+\dfrac{1}{(y+1)^2+z^2+1}+\dfrac{1}{(z+1)^2+x^2+1}\leq \dfrac{1}{2}$
  49. K

    Can var(x+y) Be Less Than or Equal to 2(var(x) + var(y))?

    Hi, I was hoping that someone might be able to please help me with this proof. Prove that var(x+y) ≤ 2(var(x) + var(y)). So far I have: var(x+y) = var(x) + var(y) + 2cov(x,y) where the cov(x,y) = E(xy) - E(x)E(y), but I'm not really sure to go from there. Any insight would be very...
  50. P

    Solving an absolute value inequality

    Homework Statement lx/(x-2)l < 5 Homework Equations The Attempt at a Solution x/(x-2) < 5 x< 5x-10 10 < 4x 5/2 < x x/(x-2) > -5 x > -5x+10 6x > 10 x > 5/3 The answer is x < 5/3 and x > 5/2 so where did I go wrong on the second one?
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