# What is Inequalities: Definition and 328 Discussions

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.

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1. ### Inequalities With negative numbers and switching x -- therefore switching the sign

My attempt: 4>19-3x Subtract 19 from both sides: -15 > -3x Divide both sides by -3: 5 > x Switch sides (change sign): x < 5 ! But Maths Genie tells me the answer is x>5 Where have I gone wrong?
2. ### Prove 0<a<1, 0<b<1, then ab<a

I would like some feedback on my proof. Can I just say that : 0< a<1 ... [1] 0<b<1 ... [2] multiplying [2] by 'a' everywhere, then I get 0<ab<a And, we prove that ab<a?
3. ### Solving an Asymmetrical Inequalities Problem: Seeking Light

I'm obsessed with this problem and need some light, I've tried solving it using inequalities but there seems to be an asymmetry that's hard to deal with.
4. ### Is it Possible for x^2 to Exceed 900?

We can also write a general inequality ##x^2>a## where a is a number. If ##x^2= a##, then ##x= \pm \sqrt a## which means ##x= \sqrt a## or ##-\sqrt a##. But in this case i don’t think it will be ##x > \pm \sqrt a## because if we take ##0## , it’s greater than a negative but in the original...
5. ### Quadratic inequalities with absolute values

I was given a problem to solve that goes like this ##\frac{3}{|x+3|-1}\geq |x+2|## . I got the correct solution for all possible cases and here they are; for ##|x+3|\geq0## and ##|x+2|\geq## i got ##x\epsilon <-2, -2\sqrt{3} ]## and for ##|x+3|\leq0## , ##|x+2|\leq0## I got ##x\epsilon [-5...
6. ### B Bell's Inequalities and the issue of non-locality

QM is compatible with Bell's inequalities clearly shows that no deterministic program can give similar results for entangled particles. In other words, there is no deterministic algorithm that mimics QM that the particles already follow before the measurement. So far, so good. But of course...
7. ### B What Are Basic Questions About Understanding Inequalities?

Hello. I have a GCSE level (or below) question about inequalities. I got the following from BBC Bitesize https://www.bbc.co.uk/bitesize/guides/z9vkqhv/revision/1 "Inequalities are the relationships between two expressions which are not equal to one another. The symbols used for inequalities are...
8. ### Can this system of inequalities be solved for x?

Summary: Can these two equations be solved for x like a system of linear inequalities, and how? ##x- 2y \le 54## ##x + y \ge 93## We start with ##x- 2y \le 54## ##x + y \ge 93## Multiplying the second equation by 2, we have ##2x + 2y \ge 184##. We cannot seem to cancel the y out with the...
9. ### An inequality involving ##x## on both sides: ##\sqrt{x+2}\ge x##

Problem statement : Let me copy and paste the problem as it appears in the text on the right.Attempt (myself) : By looking at ##\large{\sqrt{x+2}\ge x}##, from my Relevant Equations above, we have the following : 1. Outcome ##\mathbf{x \ge 0}##, since square roots are always positive. 2...
10. ### Solving Inequalities: Need Help with Part (a)?

I’m not sure of how begin solving for the inequality. How do I begin solving part (a)? Could I have a hint? Thank you!
11. ### Need help in manipulating rational absolute value inequalities

How does one manipulate rational absolute inequalities? For example, I want to transform the absolute value inequality ##|x-3|<1## to ##\frac{|x+3|}{5x^2}<A \ ##, for some number ##\text{A}##, to find an upper and lower bound on the latter term using the constraint in the first term, and not...
12. ### I Finding a bound for a fraction

Hi, I have given the following, which I would like to show that this estimation is correct, where ##|\theta| \leq \frac{\pi}{^2}## and ##M \geq 1##: $$\frac{1}{M^2}\frac{\sin^2(M\theta)}{\sin^2(\theta)} \geq \frac{4}{\pi^2}$$ I would approach an estimation of the denominator via ##\sin(x)...
13. ### A The experiment for testing the CHSH inequalities

I want to understand how an experiment is carried out to test violations of the CHSH inequalities. I have read Wikipedia and one popular book on quantum mechanics. The Wikipedia article is too short and incomprehensible to me; in addition, the description from Wikipedia and from the book is...
14. ### I A popular explanation of Bell's inequalities

In one book on quantum mechanics, I found a very simple analogy to Bell's inequalities, which explains their essence without delving into the details of the physical experiment. I liked this analogy, but it seems the authors got it a little confused, so I ask for clarification if it is correct...
15. ### A Bell inequalities demonstration

Hello, In this thesis https://tel.archives-ouvertes.fr/tel-01743877/document at "1.2.2 Bell inequalities" page 7-8 it's define a correlation function : C(x y) = P(+ + |x y) + P(− − |x y) − P(+ − |x y) − P(− + |x y), with −1 ≤ C(x y) ≤ 1. How do one get to this relationship −1 ≤ C(x y) ≤ 1 ...
16. ### B Inequalities in the real numbers

How do we get ##\epsilon(2p+\epsilon)<\epsilon(2p+1)<2-p^2## from ##0<\epsilon<1## and ##\epsilon<\dfrac{2-p^2}{2p+1}##? Answer: As we have ##\epsilon<1##, we've got ##2p+\epsilon<2p+1##; therefore, ## \epsilon(2p+\epsilon)<\epsilon(2p+1) ##; -as we have ##\epsilon<\dfrac{2-p^2}{2p+1}##, we...
17. ### Inequalities Since ε is arbitrarily small, do the inequalities hold?

#### If ##b \leq x_n \leq c## for all but a finite number of n, show that ##b \leq \operatorname{lim inf}_{n \to \infty} x_n## and ##\operatorname{lim sup}_{n \to \infty} x_n \leq c_n## (Buck, Advanced Calculus, Section 1.6, Exercise 24) Let ##\beta =\operatorname{lim inf}_{n \to \infty} x_n##...
18. ### MHB How to Graph a System of Inequalities for Investment Allocation?

A person plans to invest up to 20,000 dollars in two different interest-bearing accounts. Each account must contain at least 5,000 dollars. The amount in one account is to be at least twice the amount in the other account. Write and graph a system of inequalities that describes the various...

20. ### Modulus Inequalities — Help Needed Please

1. I think the question is asking where is the graph of |3x-2| below the graph of 1/x. To sketch the graph of y= |3x-2| draw the line of y=3x-2 and reflect the section with negative y-values in the x-axis. Alternatively, I could set 3x-2 ≥0, meaning |3x-2|=3x-2 so draw the line of y=3x-2. Then...
21. ### Rational inequalities homework help please

My attempt so far: I put all the terms to become smaller than zero: so ##x<-4## becomes ##x-4<0## ##-1\leq x\leq 3## becomes ##-1-x\leq 0## and ##x-3 \leq 0## ##x>6## becomes ##x-6>0## which is the same as ##-x+6<0## (i think)... I am now stuck on making it a rational inequality... anyone...
22. ### Hey I have a question about inequalities

I am honestly not sure where to start. Please give me a hint on how to solve this problem. Thanks!
23. ### B Thought Experiment: Exploring Violations of Bell Inequalities

I have been thinking about the Violation of bell inequalities , trying to justify how non locality can be determined from violation of bell tests. I have been through Dr. Chinese page which has partially convinced me that there can be no hidden variables , but I need to understand what...
24. ### Proving rational surd inequalities

my attempt, i am not good in this kind of questions ...i need guidance.
25. ### MHB Solve System of Inequalities: 2x+3y-4, 3x-4y+5

Solve the following system of imequalities: 2x+3y-4>0 3x-4y+5>0
26. ### I Maximum violation of Bell's Inequalities?

Can someone please tell me(and in simple terms-like in percentages), what the maximal violations of Bell's inequality has been recorded at in actual experiments and in an ideal scenario? Thank you.
27. ### Having problems in multiplying the inequalities

Let’s say we are given two inequalities $$x \lt y \\ a \lt b$$ then we can write (we can even prove it using logarithms) $$ax \lt by$$ given that every number is positive. In this article (Fact 4.1, point (ii) ) it is given that if ##x, y## are positive numbers and...
28. ### Algebra Book about irrational inequalities and other....

Good evening, I have consulted several precalculus books, intermediate algebra but none of these lists irrational inequalities, trigonometric inequalities and more. In which book I can find them? Thank you :)
29. ### Proving inequalities for these Sequences

I need help only in section 3 I have some kind of solution but I'm not sure because it seems too short and too simple. We showed in section 1 that an> 0 per n. Given that an + 1 <0 and an + 1 = an / a1 therefore a1 <0 is warranted
30. ### MHB Possible mistake in question? (system of linear inequalities)

Hello, so there's a graph provided in the task which I'm trying to solve for a quite a while and I am really confused where the 5 to 8 line came from, because (5x+8y<5) doesn't create that sort of line. Is it possible that there's a misstake done by my teacher or am I understanding something wrong?
31. ### Finding range of a function using inequalities

My attempt : Given ##f(x)## and ##g(x)## for ## -1.6 < x < 1.6## we get ##0\leq f(x)<1.6## Thus, for ##f(g(x))## we get ## -3 \leq g(f(x)) < -1.4## Thus the required set should be the interval ##[-3, -1.4)##? My Questions : 1. What have I missed since my answer does not match the given...
32. ### MHB Proof of vector dimensions using inequalities

Hello all! I've got this problem I'm trying to do, but I'm not sure what the best way to approach it is. It's obvious that there can only be 2 dimensions, because there's only two linearly independent vectors in the span. However, what would be a good way of using the inequalities to prove...
33. ### Is (3/4)*(a^2/c) less than a with multiple inequalities?

Homework Statement I for some reason can't seem do become sure of this. There are 2 variables x and y. And two constants, a and c, which are both positive. Homework Equations x+2y ≤ (3/4)*(a^2/c) x + 2y < a The Attempt at a Solution Does this mean that: (3/4)*(a^2/c) < a ? Mons
34. ### Need help in solving this question about a rational inequality

Homework Statement Go through question number 4 Homework Equations The Attempt at a Solution See basically the question is asking us to find the range of the given function x/(x^2+x+1). So,I began solving it this way... I am stuck at this step. I asked my friend for a hint and he told me to...
35. ### I Do these Bell inequalities rely on probability concept?

This question came up in another thread. I will post again the link to Nick Herbert's proof here: https://www.physicsforums.com/threads/a-simple-proof-of-bells-theorem.417173/#post-2817138 I don't see where the probability shows up in Nick Herbert's proof. For discussion of Eberhard's proof I...
36. ### I When does an inequality indicate a maximum or minimum value?

If I write ##-1 \le\cos x \le 1##, we all clearly know what this means, that not only is ##\cos x## contained in this interval but that its max is 1 and its min is -1. However, what if I write ##-1 \le \cos n \le 1##, where ##n \in \mathbb{N}##. What does the inequality mean in this case? This...
37. ### MHB Solving Inequalities Involving |x+2| & |x2 -3ax+2a2|

1) Solve for x, in terms of a, the inequality |x2 -3ax + 2a2| < |x2 +3a - a2| where x is real . a is not 0. 2ai) By means of a sketch or otherwise, state the range of values of a for which the equation |x+2| = ax + 4 has 2 distinct real roots. 2aii)Solve the inequality |x+2| < ax + 4.
38. ### MHB R^n as a normed space .... D&K Lemma 1.1.7 .... .... some inequalities ....

I am reading "Multidimensional Real Analysis I: Differentiation by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Lemma 1,1,7 (iv) ... Duistermaat and Kolk"s Lemma 1.1.7 reads as follows...
39. ### Inner Product, Triangle and Cauchy Schwarz Inequalities

Homework Statement Homework Equations I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm...
40. ### B Irrational inequalities √f(x)>g(x) and √f(x)>g(x)

So, I know that the inequality √f(x)<g(x) is equivalent to f(x)≥0 ∧ g(x)> 0 ∧ f(x)<(g(x))^2. However, why does g(x) have to be greater and not greater or equal to zero? Is it because for some x, f(x) = g(x)=0, and then > wouldn't hold? Doesn't f(x)<(g(x))^2 make sure that f(x) will not be...
41. ### MHB Can Sum to Product Inequalities Hold for Non-Negative Reals?

Given non-negative reals, $\alpha_i$, where $i = 1,2,...,n.$ Prove, that $\alpha_1+\alpha_2+...+\alpha_n \leq \frac{1}{2}$ $\Rightarrow$ $(1-\alpha_1)(1-\alpha_2)...(1-\alpha_n) \geq \frac{1}{2}.$
42. H

### I Difference between CHSH and Bell inequalities

Bell's Inequality, P(a,b)-P(a,d)+P(c,b)+P(c,d) is calculated as: S = a*b - a*d + c*b + c*d <= 2. It is valid for all values of a, b, c and d between -1 and +1 It is also valid for counts, a=a's counts/total counts of a,b,c &d. b, c,and d are derived similarly. Negative counts are not allowed...
43. ### Trigonometric inequality problem.

Homework Statement Find the solution of the inequality ## \sqrt{5-2sin(x)}\geq6sin(x)-1 ## Answer: ## [\frac{\pi(12n-7)}{6} ,\frac{\pi(12n+1)}{6}]~~; n \in Z##Homework Equations None. The Attempt at a Solution There are two cases possible; Case-1: ##6sin(x)-1\geq0## or...
44. ### MHB Solving $y>0$ given $[y]^2=y\times(y)$ and Proving Inequalities

(1) Given :$y>0$,let $y=[y]+(y)$ where we define $[y]$ the integer part of $y$ and $(y)$ the decimal part of $y$ here $0≤(y)<1$ if $[y]^2=y\times(y)$ find $y=?$ (2) $y\in R,y=[y]+(y)$ the definition is the same as (1) prove :$[y]+[y+\dfrac {1}{2}]=[2y]$ (3) if $0<y<2^{10}$ using (2)...
45. ### Are My Proofs of Triangle Inequalities Correct?

Just wondering if anyone could confirm if I've headed in the right direction with these (a) Prove the triangular inequality: |x + y| ≤ |x| + |y|. (b) Use triangular inequality to prove |x − y| ≥ ||x| − |y||. (c) Show that if |x − a| < c/2 and |y − b| < c/2 then |(x + y) − (a + b)| < c. So for...
46. ### MHB Solving system of linear inequalities

How do I solve this? Which of the following ordered pairs (x,y) satisfies the system of inequalities below? y>-3x+5 y≤ x-2 Choices: (2,1) (1,4) (4,1) (3,-6) Please. Thanks.

You can work a total of no more than 41 hours each week at your two jobs. Housecleaning pays \$6 per hour and your sales job pays \$9 per hour. You need to earn at least \$252 each week to pay your bills. Are the answers: x+y\le 41 6x+9y\ge252 How would you solve this? Thanks 48. ### MHB Buy 7 Pencils & Notebooks with \$15 - Solve Inequalities

I have \$15 to buy pencils and notebooks from a bookstore. If pencils are \$0.50 each and a notebook costs \\$1.50, then how many pencils and notebooks can I buy if I spent all of the money? I reasoned: .5x+1.5x=15 x= 7.5 So 7 notebooks and 7 pencils. Is this correct?Thanks
49. ### Show two inequalities - (context gamma function converges)

Homework Statement I'm not after another proof. I've just got a couple of inequalities I don't know how to show when following a given proof in my book. These are: Q1) ## 0\leq x \leq 1 \implies x^{t-1} e^{-x} \leq x^{t-1} ## So this is obvioulsy true, however I think I'm being dumb because...
50. ### MHB What is the Factorization and Inequality of a Lagoon's Depth Function?

Hello Everybody , First of all, I would like to apologize that this problem contains 3 parts to it (3 questions) but they all relate to each other. You must complete one part to move on to the next part. With that being said, I have 3-part problem that I could use some assistance with. 1a...