0
votes
2answers
67 views

How to solve this inequality?

I have the following problem in an assignment and have been struggling to do it. $2 + 2x - x^2 \geq 2 \sqrt{1+2x}$ I have tried solving for $x$ but have not been able to do so. Any hints to solve ...
1
vote
1answer
30 views

Finding the minimum value of a 6th degree polynomial algebraically

Is it possible to answer this question using methods of basic algebra? Find the least value of the expression $a^6 + a^4 - a^3 - a + 1$ for real value of $a$. This question is from the 2013 Philippine ...
4
votes
1answer
82 views

A tough inequality problem with condition $a+b+c+abc=4$

If, $a+b+c+abc=4$, with $a,b,c$ being positive reals, then prove or disprove the following inequality: $$\frac{a}{\sqrt{b+c}}+\frac{b}{\sqrt{a+c}}+\frac{c}{\sqrt{a+b}}\geq\frac{a+b+c}{\sqrt2}$$ I ...
0
votes
1answer
34 views

Solve a system of inequalities

$$\begin{cases} \log_{2}^{2}(-\log_{2}x) + \log_{2}\log_{2}^{2}x \leq 3 & \\-4 |x^2-1|-3\geq \frac{1}{x^2-1}& \end{cases}$$ What I've tried: Make substitution $t=x^2-1$ and solve second ...
5
votes
7answers
115 views

Find value range of $2^x+2^y$

Assume $x,y \in \Bbb{R}$ satisfy $$4^x+4^y = 2^{x+1} + 2^{y+1}$$, Find the value range of $$2^x+2^y$$ I know $x=y=1$ is a solution of $4^x+4^y = 2^{x+1} + 2^{y+1}$ , but I can't go further more. I ...
6
votes
3answers
166 views

An Inequality Problem with not nice conditions

How to show that $\dfrac{a^3}{a^2+b^2} + \dfrac{b^3}{b^2+c^2} + \dfrac{c^3}{c^2+a^2} \ge \dfrac32$, where $a^2+b^2+c^2=3$, and $a,b,c > 0$ ?
0
votes
1answer
19 views

Proving elementary inequalities with equations

Assume $b > 0,\ d > 0$. Assume: $$ \frac{a}{b} < \frac{c}{d} $$. Prove that: $$ \frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d} $$. I would like to find an intuitive way to solve ...
5
votes
2answers
56 views

How to show $a+b+ad\geq c+d+bc$ given $a\geq c$ and $a+b\geq c+d$?

Let $0\leq a,b,c,d\leq 1$ and $a\geq c$ and $a+b\geq c+d$. Show that $a+b+ad\geq c+d+bc.$ Of course we have $a+b\geq c+d$, but how to relate $ad$ and $bc$?
0
votes
4answers
122 views

Does $xy\geq x+y$?

I just see the GM-AM inequality. But I would like to compare $xy$ with $x+y$ for any $(x, y)\in\mathbb{R}^2$. It looks like $xy>x+y$ since the first one is multiplication and the second one is ...
2
votes
3answers
54 views

How to prove this inequation?

$$ 1+\frac{2}{3n-2}\leqslant \sqrt[n]{3}\leqslant 1+\frac{2}{n}, n\in \mathbb{Z}^{+} $$ How to prove this inequation?
3
votes
3answers
46 views

Inequality Exercise in Apostol's Calculus I

Let p and n denote positive integers. Show that: $$n^{p} \lt \frac{(n+1)^{p+1} - n^{p+1}}{p+1} < (n+1)^{p}$$ Attempt at Solution Using the identity $b^{p+1}-a^{p+1} = ...
1
vote
3answers
89 views

if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$

This is supposed to be an application of AM-GM inequality. if $abc=1$, then the following holds true: $a^2+b^2+c^2\ge a+b+c$ First of all, $a^2+b^2+c^2\ge 3$ by a direct application of ...
6
votes
1answer
71 views

Show that $0 < \frac{y-x}{1+xy} < 1$.

Let $E$ a finite set of real, with at least 5 elements. I remember that my teacher proved that there exist two of its elements $x<y$, such that $0 < \dfrac{y-x}{1+xy} < 1$. Unfortunately I ...
1
vote
1answer
57 views

prove that $\sqrt{4-a^2}+\sqrt{4-b^2}+\sqrt{4-c^2}+(\sqrt{3}-1)(|a-1|+|b-1|+|c-1|)\ge 3\sqrt{3}$ if $a+b+c=3$

$a,b,c\in[0,2]$ observation by triangle inequality $|a|+|b|\ge |a+b| $ $|a-1|+|b-1|+|c-1|\ge |a+b+c-3|$ but $a+b+c=3$ hence ...
1
vote
3answers
80 views

Express $|a+b|-|b|$ without absolute value signs

I am having trouble understanding what cases I need to evaluate. So far I've checked $a = b = 0$ and that results in the expression being equal to $0$. I've checked $0 \le b < a$ which results in ...
1
vote
3answers
50 views

Problem about an Inequality

I need a hint to solve this inequality If $x_i >0$ for all $i$ then $(x_1 x_2 ... x_n)^{1/n}$ $<$ $(x_1+...x_n)\over n$ I tried a little by induction over n, but i dont go anywhere with that
0
votes
2answers
35 views

How to use $t(29/\sqrt{2})<0$ where $t(x)=x^2-41x+420$ to prove that $41/29<\sqrt{2}<42/29$??

So I was investigating different ways to approximate $\sqrt{2}$. Here's my latest: $$Let:t(x)=x^2-41x+420$$ then the roots of $t(x)$ are $20$ and $21$. I showed that then $t(x)=(x-20)(x-21)$ and ...
2
votes
5answers
61 views

How can I prove that $40/29<\sqrt{2}<42/29$ given $20<29/\sqrt{2}<21$

How can I prove that $40/29<\sqrt{2}<42/29$ given $20<29/\sqrt{2}<21$? I did lot of approaches I obtained in one that: $$\dfrac{29}{42}<\sqrt{2}<\dfrac{29}{10}$$ but couldn't do ...
0
votes
0answers
44 views

An inequality with hyperbolic function and exponential function

I was encountering with a conjecture about the following statement: Let $Y>0$ and $X \neq 0$ it follows that , $$\cosh(KX) >e^{K^2Y}$$ for all $0<K<\frac{|X|}{2Y}$ it seems to me I need ...
-2
votes
2answers
34 views

Find the range of values of $x$ for the inequality $x^2-4x-1>0$ [closed]

Find the range of values of $x$ for the inequality given. $x^2-4x-1>0$
0
votes
1answer
34 views

How to tell if the roots to a quadratic equation is always positive using the quadratic formula?

Suppose $k>0$ and $(k+1)^2>8k$. Let: $\alpha,\beta = \frac{(k+1)/2 \pm \sqrt{(k+1)^2/4-2k}}{2}$ The solution I have says that $\alpha$ and $\beta$ will always be positive. I don't know how ...
1
vote
4answers
61 views

Find the range of values of $x$ which satisfies the inequality.

Find the range of values of $x$ which satisfies the inequality $(2x+1)(3x-1)<14$. I have done more similar sums and I know how to solve it. I tried this one too but my answer doesn't matches the ...
0
votes
1answer
46 views

Inequality challenge

I was studying inequations when I encountered this problem here. How can I find a region of values for m where this inequation is true? $$-3<\frac{x^2+mx-2}{x^2-x+1}>2$$ Thanks
1
vote
3answers
36 views

Minimal answer possible.

Are there a finite or infinite set of solutions that would satisfy this equation? If there are a finite set of solutions, what would it be? $$(2x+2y+z)/125\leq9.5$$ where $x$, $y$ and $z$ $\leq$ 250. ...
0
votes
2answers
27 views

Help interpreting question about a system of equations

I'm currently working through Beckenbach and Bellman's book "An Introduction to Inequalities." One of the questions has me a little stumped, as I'm not really sure what they're asking for. For fixed ...
0
votes
1answer
28 views

Inequality between the coeficients of a quartic equation

Given the inequlity $$ ax^4-bx-c\geq 0, \quad \forall x\in \mathbb R $$ where $a, b$ and $c$ are real positive constants. Is it possible to conclude some inequality between the coefficients like ...
-1
votes
2answers
70 views

Prove! $\sqrt{(a+c)(b+d)} \geq \sqrt{ab+cd}$

Let $a,b,c,d \in (0,\infty)$. Show that $$\sqrt{(a+c)(b+d)} \geq \sqrt{ab+cd}.$$ When does the equality hold?
7
votes
7answers
269 views

$211!$ or $106^{211}$:Which is greater?

A BdMO question: Let $a=211!$ and $b=106^{211}$. Show which is greater with proper logic. By matching term by term,it is pretty easy to note that $106!<106^{106}$ $106^{105}<107\cdot ...
0
votes
1answer
102 views

My proof of the inequality of arithmetic and geometric means

Doing the exercises from the Apostol's Calculus I give my proof that the geometric mean is less than or equal the arithmetic mean ($G \le M_1$). I followed the hints from the book and I think I've ...
0
votes
4answers
46 views

Where is a flaw in these logical implications?

We have a theorem: If $a \le x < a + {\frac yn}$ for $y > 0$ and all natural $n \ge 1$ then $x = a$. Suppose I derive that $a < x$ and $x < a + {\frac yn}$ for all $n \ge 1$. In other ...
0
votes
1answer
34 views

Implications using inequality signs <= and <

Suppose we have a theorem that says: If $A \le X \le B$ and $A$, $B$ both have property $p$ then $X$ has property $p$. I'm working on some problem and I derive that $A < X < B$ and $A$, $B$ ...
0
votes
1answer
80 views

How $x^2$ increases by $x+\frac{1}{x}$?

I was going through one of the topic "Introduction to Formal proof".In one example while explaining "Hypothesis" and "conclusion" got confused. The example is as follows: If $x\geq 4$ then $2^x \geq ...
3
votes
0answers
72 views

Trigonometry or inequality problem

Today, I saw this question: If $x,y,z \in [0,\frac\pi 2]$, $x+y+z=\frac{3\pi}{4}$ and $\sec^2(x)\sec^2(y)\sec^2(z)=8$, calculate $E=\tan x\tan y+\tan y\tan z+\tan z\tan x$ My first thought was ...
8
votes
1answer
243 views

Why does Group Theory not come in here?

Here is a list of questions that I find quite similar, for the one and only reason that they all show much "symmetry". Which is at the same time my problem, because I don't have a very precise notion ...
1
vote
1answer
52 views

Prove the following inequality on the unit interval: $\sqrt{\frac{1-x}{1+x}}\lt\frac{\ln(1+x)}{\arcsin(x)}\lt1$

For $0\lt x\lt1$ show $$\sqrt{\frac{1-x}{1+x}}\lt\frac{\ln(1+x)}{\arcsin(x)}\lt1$$
3
votes
4answers
56 views

Exercise about inequalities

This exercise is from Apostol's Calculus: $|1 + 3x| \le 1$ implies $x \ge -{\frac 23}$ I answered false, but the right answer is true. My understanding is — of course all $x$ are bigger ...
0
votes
1answer
35 views

Prove that $\forall \, a,b \in \mathbb{N}- \{0,1\}\,\, \wedge \,\,a<b \,\, ; \,\, a^{1/a} > b^{1/b}$

Prove that $\forall \, a,b \in \mathbb{N}- \{0,1\}\,\, \wedge \,\,a<b \,\, ; $ $$\,\, a^{1/a} > b^{1/b}$$ I need some tip to start it. Thank you.
2
votes
1answer
40 views

Solving inequation with two absoulte values

I need to solve the following inequation: $$ |x| \cdot |x-1|-1>-x\\ $$ I cant get the correct result. I tried to solve it like this: $$ |x| \cdot |x-1|-1>-x $$ I know that I can write $|x ...
5
votes
7answers
162 views

Solutions for $\frac{3}{x+1}\le\frac{2}{2x+5}$

Im in search of the solutions for: $$\frac{3}{x+1}\le\frac{2}{2x+5}$$ So first i tried to combine the two sites: $$\frac{6x + 15 - 2x + 2}{2x^2 +7x + 5}\le{0}$$ $$\frac{4x + 17}{2x^2 +7x ...
0
votes
2answers
46 views

Properties of Inequalities

I am unable to understand the following statements. If someone can explain with the help of Numerical it would be great. Please see the attached image, and explain! If sides of an inequality are ...
2
votes
1answer
44 views

Proving Bernoulli's Inequality for $h<0$

I'm answering question 19 of chapter two of Spivak's Calculus and I can't seem to think of a way of doing it. I don't want to look up the answer so I thought I'd ask for a hint as to the general ...
4
votes
1answer
93 views

Prove that :$\frac{1}{a+b} +\frac{1}{b+c} +\frac{1}{c+a}\ge \frac{4}{a^2+7} +\frac{4}{b^2+7} +\frac{4}{c^2+7}$

Let $a,b,c>0$ and satisfying $a^2+b^2+c^2=3$. Prove that :$\dfrac{1}{a+b} +\dfrac{1}{b+c} +\dfrac{1}{c+a}\ge \dfrac{4}{a^2+7} +\dfrac{4}{b^2+7} +\dfrac{4}{c^2+7}$
0
votes
4answers
87 views

If $x>10^2$ then is the following statement true? $1-\frac{2}{x}+\frac{3}{x^2}>0.9$

If $x>10^2$ then is the following statement true? $$1\color{red}-\frac{2}{x}+\frac{3}{x^2}>0.9$$ I already figure out that $x>10^2$ implied: $$-2/x>-2\cdot10^{-2}$$ ...
2
votes
2answers
176 views

Solving $x\; \leq \; \sqrt{20\; -\; x}$

This is how I tried to solve it: By squaring both sides: $x^{2}\; \leq \; 20\; -\; x$ $x^{2}\; +\; x\; -\; 20\; \leq \; 0$ Thus $-5\; \leq \; x\; \leq \; 4$ However, it seems that values less ...
0
votes
2answers
41 views

How do I graph this inequality?

$$y\le x\le \sqrt{1-y^2}$$ Graph: http://www.wolframalpha.com/input/?i=y%3C%3Dx%3C%3Dsqrt%281-y%5E2%29 How would I approach this problem? I graphed $0\le x\le \sqrt{1-y^2}$ easily, but this one is ...
0
votes
1answer
38 views

inequality of sums

I think about the following inequality: $$ \sum_{i=1}^n a_ib_i \le \left(\sum_{i=1}^n a_i \right)\left(\sum_{i=1}^n b_i \right) $$ Is the inequality true for all $a_i$ and $b_i$, or just correct ...
4
votes
4answers
61 views

Proving $\frac{(n+1)^4}{4}+(n+1)^3\le\frac{(n+2)^4}{4}$ for all $n \ge 1$.

$$\frac{(n+1)^4}{4}+(n+1)^3\le\frac{(n+2)^4}{4}$$ For all $n\ge 1$. I thought that I could get rid of the denominators like this: $$(n+1)^4+4(n+1)^3\le(n+2)^4$$ Then, maybe, take $(n+1)^3$ as ...
0
votes
0answers
34 views

approximating a square root

It is given that, $$a < \sqrt{b}<a+1$$ where, a and b are both positive numbers. I need to prove the inequality $$a + \frac{b-a^2}{2a+1}<\sqrt{b}<a+\frac{b-a^2}{2a+1}+ ...
0
votes
1answer
59 views

Solve the inequality $2^{\left( x^{3}-x\right) } < 1$

$2^{\left( x^{3}-x\right) } < 1$ Let $2^{\left( x^{3}-x\right) }-1=f\left( x\right)$ To find the values for which $f(x)<0$ I let $f(x)=0$: $2^{\left( x^{3}-x\right) }-1=0$ $2^{\left( ...
0
votes
1answer
28 views

Prove that there exist two infinite sequences that simultaneously satisfies all these conditions

Prove that there exist two infinite sequences $\langle a_n\rangle_{n\geq 1}$ and $\langle b_n\rangle_{n\geq 1}$ of positive integers such that the following conditions hold simultaneously: $$1 < ...