Inequalities Definition and 329 Threads

In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities:

The notation a < b means that a is less than b.
The notation a > b means that a is greater than b.In either case, a is not equal to b. These relations are known as strict inequalities, meaning that a is strictly less than or strictly greater than b. Equivalence is excluded.
In contrast to strict inequalities, there are two types of inequality relations that are not strict:

The notation a ≤ b or a ⩽ b means that a is less than or equal to b (or, equivalently, at most b, or not greater than b).
The notation a ≥ b or a ⩾ b means that a is greater than or equal to b (or, equivalently, at least b, or not less than b).The relation "not greater than" can also be represented by a ≯ b, the symbol for "greater than" bisected by a slash, "not". The same is true for "not less than" and a ≮ b.
The notation a ≠ b means that a is not equal to b, and is sometimes considered a form of strict inequality. It does not say that one is greater than the other; it does not even require a and b to be member of an ordered set.
In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude. This implies that the lesser value can be neglected with little effect on the accuracy of an approximation (such as the case of ultrarelativistic limit in physics).

The notation a ≪ b means that a is much less than b. (In measure theory, however, this notation is used for absolute continuity, an unrelated concept.)
The notation a ≫ b means that a is much greater than b.In all of the cases above, any two symbols mirroring each other are symmetrical; a < b and b > a are equivalent, etc.

View More On Wikipedia.org
Recent contents Top users
  1. MathematicalPhysicist

    Proving Hard Inequalities for Acute Angles and Trigonometric Functions

    i need to prove the following: 1)let a,b,c be acute angles, if tg(a)tg(b)tg(c)=1 then sin(a)sin(b)sin(c)<=1/2sqrt2 2) prove that for every x,y cos(x^2)+cos(y^2)-cos(xy)<3 for the second question i tried to use the fact that (x^2+y^2)/2>=xy and the fact that on some intervals the function cos...
  2. MathematicalPhysicist

    Proving Inequalities with Induction: Examples and Strategies

    i need to prove the following: 1)(1+1/n)^n<3 for every n>=3. 2) (x^n+y^n)/2>=((x+y)/2)^n for every n natural and every x,y>=0. 3) |a+1/a|>=2 for every a different than 0. for the first i thought to use induction and to use the fact of increasing sequence (1+1/n)^n or of the decreasing...
  3. S

    What are the rules for solving inequalities involving logs?

    Check this out: 1 < 2 \Rightarrow \frac{1}{4} < \frac{1}{2} \Rightarrow (\frac{1}{2})^2 < \frac{1}{2} \Rightarrow \log(\frac{1}{2})^2 < \log(\frac{1}{2}) \Rightarrow 2\cdot\log(\frac{1}{2}) < \log(\frac{1}{2}) \Rightarrow 2 < 1 What happened? What did I do wrong?
  4. S

    Write inequalities to describe the region.

    I'm having some trouble figuring out the inequality that would satisfy this region: The solid rectangular box in the first octant bounded by the planes x=1, y=2, and z=3. Is it x >1, y >2 and z > 3? I can't think of anything else, really, and there's no answer in the back of the book...
  5. C

    About violation of Bell's inequalities

    About violation of Bell ineqalities (and the Loch Ness monster) Assuming no loophole experiments can be done... even then I am not sure of what this would mean beyond the fact that QM's prediction on correlation work fine (and where is the surprise there?). I know that assuming some...
  6. M

    Quadratic Inequalities: Solving x² - x < 0

    Solve the Inequality: x² - x < 0 Express the solution set as intervals or union of intervals. Use the result √a² = |a| as appropriate. What is the procedure/explanation for the answer to this question? The answer is (0,1). THat is: (0,1) is the solution set. Please help.
  7. A

    Find solution to this system of inequalities such that x+y is minimal

    I propsed myself a problem but I'm having some trouble solving it. I've narrowed it down to this but i need some help on this final part: I have this 2 inequations that must be obeyed: (17/15)x + y > 50000 (1) (2/25)x + (3/10)y > 5000 (2)...
  8. M

    1st year calc. trig, and inequalities

    Hi! Im stuck on 2 assignment questions and I was hoping to get help on whut I am doing wrong. Its 1st year Calculus 1) It says Given a right angled triangle prove that 1/1+cot^2 X=sin^2 X so I know cot=1/tan so 1/tan= 1/(opp/adj) therefore cot=1/(opp/adj) so 1/cot become (this is where...
  9. P

    Tough Olympiad-like Inequalities question

    a, b, c, and d are all positive real numbers. Given that a + b + c + d = 12 abcd = 27 + ab +ac +ad + bc + bd + cd Determine a, b, c, and d. --- The solution says that using AM - GM on the second equation gives abcd (is greater than or equal to) 27 + 6*sqrt of (abcd) From...
  10. P

    Proving Inequalities with n > 2: A Challenge

    Dear all, I want to prove that the following inequalities are true. I hope you can give some hints. Thanks a lot! Define c_{\beta}=\sum_{j=1}^n \sigma_j^{\frac{2\beta}{\beta+1}}\sum_{1\leq i<k \leq n}\Big( \sigma_k^{\frac{2}{3(\beta+1)}} + \sigma_i^{\frac{2}{3(\beta+1)}} \Big)^3 ...
  11. R

    Proof of Inequalities by Induction.

    Okay, so we are covering proof by induction, and i need some ones help on it covering inequalities. (a) (2^n) ≤ n! , n≥4 Base Step: sub in n=1 and yes, it works! Inductinve step: assume (2^n) ≤ n! and show (2^(k+1)) ≤ (k+1)! ,K≥4 holds. (2^(k+1)) ≤ (k+1)! (2)(2^k) ≤ (K!)(K+1) So...
  12. W

    Proving Inequalities for Numbers x and y: Graphs & Algebraic Methods

    The numbers x and y satisfy 0 < x \leq a^2, 0 < y \leq a^2, xy \geq a^2 where a \geq 1. By sketching suitable graphs or otherwise, show that x + y \geq 2a and x \leq a^{2}y \leq a^{4}x --- I don't know what to sketch (tried x \leq 1, y \leq 1, xy \leq 1), so I tried algebraic methods...
  13. L

    How Do You Solve Quadratic Inequalities with Positive Roots and Ratios?

    1. (a) If the roots of the equation 2(x)^2 + kx + 100 = 0 are positive, find the possible range of k. (b) If, in addition, one root is twice the other, find the roots and the value of k. I have tried (a), but incorrect: discriminate > 0 k^2 - (4)(2)(100) > 0 k^2...
  14. H

    Inequalities with the Unknown Denominator.

    I am having trouble solving these type of problems for some reason. I can't get to the answer. If anyone could care to explain to me it would be greatly appreciated. Example 1: 1/x < 1/4 Example 2: 1/x-3 > 2
  15. J

    Solving 2 inequalities with imaginary numbers?

    I have 2 equations, imaginary ones, and 2 unknowns...trying to solve for them..but the answer i got, works with one, but not the other: i*Z1 - i*Z2 = -2 - i Z1 + 3i*Z2 = 4 + 7i where i is the imaginary number, and Z1 and Z2 are the 2 unknowns the answer i got: Z1 : 1.33333 +...
  16. B

    Can Quadratics Help Solve Inequalities?

    I need help on solving inequalities? Someone please help me. I 'm currently taking Pre-calculus. :smile:
  17. T

    Calculators How can I solve inequalities with a TI-89?

    I've got a TI-89, with which I'm trying to solve an inequality. Not a specific one, just in general. I try to use the 'solve' command just as for equations but with an inequal-sign instead, but it doesn't work. Anyone?
  18. C

    Simple absolute value problem with inequalities

    "Simple" absolute value problem with inequalities OK...Im totally stuck and could use some help :) given...for all e>0, d>0...the following holds |x-a|<d => |f(x) - f(a)| < e where f(x) = sqrt(x) how do I find d in terms of e? Thanks in advance
  19. A

    Solving inequalities, need some confirmation

    I have these three inequalities that I am supposed to solve, I think I came up with the right answer but I'm not even 100% sure it's in the correct format. A. 6x^2 < 6+5x my work: 6x^2-5x-6 < 0 solutions are then 3/2 and -2/3 so the answer I got is: -2/3 < x < 3/2 B. x^2+8x > 0 my...
  20. L

    What Values of b Satisfy the Inequality 1296(b^3) - 324(b^2) - 1008b + 108 > 0?

    I have an inequality: 1296(b^3) - 324(b^2) - 1008b + 108 > 0. I want to know for what values of b this inequality is true. Any suggestions?
  21. C

    Proving Inequalities: Tips and Examples for Solving with Different Methods

    Hello all How would you prove the following: (a) x + \frac{1}{x} \geq 2, x > 0 (b) x + \frac{1}{x} \leq -2, x < 0 (c) |x+\frac{1}{x}| \geq 2, x\neq 0. For all of these inequalities would I simply solve for x, or would I have to use things like the triangle inequality of...
  22. P

    What Are the Correct Steps to Determine the Range of a Function?

    What do you in these cases: A function is g(x) = 2 (x-3) ^2 + 4 Find the range when 0 <= x <= 6 steps i took: 0 <= x <=6 0-3 <= x-3 <= 6-3 -3^2 <= (x-3)^2 <= 3^2 9 <= (x-3)^2 <= 9 22 <= 2 (x-3)^2 + 4 <=22 :: what did i do wrong? correct answer is: { y | 4 <= y <= 22 }...
  23. R

    Understanding Inequalities: Explaining the Concept and Significance

    hello please could someone explain to me inequalities ? I don't understand how it works Roger
  24. P

    Is There an Easier Way to Understand Polynomial Inequalities?

    We just started these at school, but I have some questions.. Inequalities on number line (x) graphs.. Using x = 0 testpoint.. Let's say I have (x+2)(x-4) > 0, (0 + 2)(0 - 4) would be -8, and -8 !>0, so it would be a disjunction, right? But if it was positive and satisfied the inequality, it...
  25. I

    Is there a simpler method to prove absolute inequalities?

    prove the folowing and state when the inequality holds... |x+y+z|<=|x|+|y|+|z| i was thinking that i consider all the possible cases, ie x is positive, y positive, z positive; then the various combinations with negative as well... is there another shorter method of doing it? help...
  26. K

    Absolute quadratic inequalities.

    A bit of a newbie question, but I was wondering how does one go about solving these? For example: (I was working on a problem posted on another thread on Homework Help) |3n-4| < 9\epsilon n^2 + 3 \epsilon Epsilon is a small positive number of course :P The tricky part is when I split...
  27. S

    Explaining EPR after Bell's inequalities

    We (three students from the Netherlands) are working on a project on Bell's inequalities. We have studied the original EPR-paper from 1935, which states that quantum mechanics may well be an incomplete theory. Reactions on this paper. Von Neumann's completeness theorem. The Kochen Specker...
  28. G

    Solve Inequalities: Find p Values for p(x^2+x) < 2x^2 + 6x +1

    Can someone help me please with inequalities, I have been attempting this question quite a few times but I still can't get the same answer as the textbook. What is the set values of p for which p(x^2+x) < 2x^2 + 6x +1 for all real values of x? Here is my best attempt: p(x^2+x) < 2x^2 +...
  29. K

    2 questions - series & inequalities

    2 questions -- series & inequalities 1. By differentiate the function 1/(1-x), or otherwise, show that inf [sum] n2/2n = 6 n=1 2) Given Holder's Inequality http://mathworld.wolfram.com/HoeldersInequalities.html(equation 4) show that (attached file)
Back
Top