3
votes
4answers
231 views

inequality method of solution

Im looking for an efficent method of solving the following inequality: $$\left(\frac{x-3}{x+1}\right)^2-7 \left|\frac{x-3}{x+1}\right|+ 10 <0$$ I've tried first determining when the absolute value ...
-2
votes
2answers
109 views

How to show $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$?

I was trying to solve $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$ and I keep getting a partial answer of $x>4$ though answer key suggests a more expanded ...
0
votes
6answers
62 views

solving the inequalty

are there any ways to solve :$ x^4 -6x^3 +28x^2 -64x +96 >0$ ?
0
votes
1answer
34 views

$f(x)=sec(x)$ inequality inconsistency\trouble

I'm currently attempting to find the range of $f(x)=\sec(x)$ by considering $\cos(x)$ in the intervals of $0<\cos(x)\leqslant 1$ and $-1\leqslant \cos(x)<0$ (as $\sec(x)$ is undefined for ...
0
votes
0answers
25 views

calculate the sup of the max of 3 functions

Let a function be the variable, how the calculate the following expression? $$\inf_{c(t) \in C[-1,0]} \max \{ \max_{-1 \leq t \leq 0} |c(t)| , \max_{0 \leq t \leq 1} | \int_{0}^{t} c(v-1) +1 dv +c ...
0
votes
2answers
49 views

What does | mean in this exercise? And how do I solve it?

I was doing a practice exam for my SATs and I stumbled across this problem in the inequality section of the Algebra part. And I don't know what that symbol means and how to solve the problem with that ...
2
votes
0answers
36 views

Find Max : $P=\frac{1}{x^2+1}+\frac{4}{y^2+4}+\frac{3z}{9+z^2}$

Let $x,y,z>0$ and satisfying $3xy+yz+2zx=6$ Find Maximum of this expression: $P=\frac{1}{x^2+1}+\frac{4}{y^2+4}+\frac{3z}{9+z^2}$
1
vote
1answer
37 views

Bessel's inequality for expected value

Let $X_1, X_2,\ldots$ be independent random variables with expected value $\mathbb{E}[X_i]=0$ and variance $V[X_i]=1$. Let $Y$ be another random variable, such that $\mathbb{E}[Y^2] < \infty$. I ...
2
votes
2answers
53 views

Inequality in complex numbers

Prove that for all $z\in \mathbb{C}$ $$\frac{\Vert z+i\Vert z\Vert \Vert}{\Vert z+1\Vert}\leq \frac{2\Vert z \Vert}{\Vert z \Vert +1}$$
-2
votes
1answer
81 views

Prove that exactly one number is negative.

For $j\in\{1,2,3\}$ let $x_j,y_j\in\Bbb{R}$ be nonzero and let $v_j=x_j+y_j$. Suppose that following holds: $$x_1x_2x_3=-y_1y_2y_3\qquad\text{ and }\qquad x_1^2+x_2^2+x_3^2=y_1^2+y_2^2+y_3^2,$$ and ...
1
vote
5answers
47 views

Algebra - solve given inequality

I am having problems understanding how to solve: $ x^2 - x - 1 > 0 $. Any help would be much appreciated.
0
votes
2answers
20 views

Proving an inequality involving radicals

I need some help proving the follwoing inequality: Knowing that $a \ge 1$, prove that: $$ \frac {1}{\sqrt{a}}\gt 2\sqrt{a+1}-2\sqrt{a}$$ I need to prove this in an algebric fashion. I was thinking ...
0
votes
1answer
39 views

A proof regarding connected components of a graph

Been struggling with this home-work question for some days now. Will appreciate an explanation. Let $c(G)$ denote the amount of connected components in a graph $G$. a. prove that $\forall e\in E: ...
3
votes
1answer
84 views

solve the inequality: $\displaystyle \ln|x^2 -3x+2|+\frac{2x-3}{x-2} \geq 0$

How can I solve the following inequality in the set of real numbers: $\displaystyle \ln|x^2 -3x+2|+\frac{2x-3}{x-2} \geq 0$ Thanks in advace!
1
vote
1answer
29 views

Help understanding the solutions and answers to this inequalities question

Find the set of solutions that satisfy $\dfrac{3}{x+3}>\dfrac{x-4}{x}$ So I began by multiplying out to get $3x>x^2-x-12$ $x^2-4x-12=0$, which gives $x=-2$ and $x=6$. So these two x-values ...
0
votes
1answer
24 views

Interpolation based on $n$ uniformly distributed points

We are given $n+1$ uniformly distributed points in the segment $[0,1]$: $x_i=\frac{i}{n}$, $i=0,1,...,n$ and a function $f(x)=e^{-x}$ $P(x)$ is the interpolation polynomial of $f(x)$ where ...
1
vote
1answer
46 views

Binomial Number

Please give me feedback on my answer to this question. Question: $\binom{2\cdot2^{1000000}}{2^{1000000}}>2^{2\cdot2^{1000000}}$ . Answer: True. Let $2^{1000000}=x.$ Then ...
10
votes
5answers
229 views

Proof for $\sin(x) > x - \frac{x^3}{3!}$

They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried: ...
1
vote
1answer
54 views

Prove homogeneous inequality

Prove that for all $a,b,c>0$ we have $$\frac{ab}{(a+b)^2}+\frac{bc}{(b+c)^2}+\frac{ac}{(a+c)^2}\leq\frac{1}{4}+\frac{4abc}{(a+b)(b+c)(c+a)}.$$ Please help me prove this homogeneous inequality.
2
votes
2answers
54 views

Given two series $ x_n $ and $ y_n $

Let $ x_n = \sum_{k=n}^{\ 2n-1} \frac{1}{k} $ , $ y_n = \sum_{k=n+1}^{\ 2n} \frac{1}{k} $ b) Show that $ y_n \leq \ln2 \leq x_n $ for all $n$
2
votes
1answer
26 views

probability theory proof of exponential chebyshev inequality

This is a question about my homework. I am not sure about what is exponential Chebyshev inequality, also how do I get rid of the absolute value and prove it directly by PDF? As well as the ...
1
vote
1answer
54 views

Why does this equation work?

let $ P(x) := \sum_{p \leq x} Log [p]$, then we have $P(2^{k+1}) = \sum_{i=0}^k ( P(2^{i+1}) - P(2^i)) < 2 \cdot Log[2] \cdot (1 + 2 + 4 +... + 2^k) \leq 4 \cdot Log[2] \cdot 2^k$. Why does ...
0
votes
2answers
69 views

Question on transformations in the complex plane

In the image (part (b)), Since $z < |3|$ before the transformation, does that simply imply that the region to be shaded after the transformation is definitely the inside of the circle and not it's ...
7
votes
2answers
196 views

Smallest K for which $ |\sin^2x - \sin^2y | \le K|x - y|$ holds [closed]

What is the smallest positive number K for which the following inequality holds $\forall$ $x$ and $y$? $$ |\sin^2x - \sin^2y | \le K|x - y|$$
0
votes
3answers
46 views

Solve $6\sin x \cos 2x\ge 0$

How can I solve the following inequality: $${6\sin x\cos 2x\ge 0}$$ Can you give me an explicit explanation of how this exercise can be understood. I have no problems with trigonometric equations, but ...
3
votes
1answer
106 views

Inequality problem about sides of a triangle and the semiperimeter

Let $a,b,c$ the sides of a triangle and $s$ be the semi perimeter. Then show that $$ a^2+b^2+c^2 > \frac{36}{35}(a^2+\frac{abc}{s}) $$ I tried it doing in many ways using some ...
0
votes
2answers
30 views

Simple inequation equivalence with n as exponent

$5^{n+1 }-4 \cdot 5^n \geq 4 \cdot 3^n -3^{n+1} \iff 5^n \geq 3^n$ where $n$ is a positive integer How is this possible? I can't find a connection. I put spaces to distinguish fractions.
0
votes
2answers
95 views

Prove: $\frac{1}{2}\cdot\frac{2}{3}\cdots\frac{2n-1}{2n}<\frac{1}{\sqrt{2n+1}}$

prove the inequality if you can: $\frac{1}{2}\cdot\frac{2}{3}\cdots\frac{2n-1}{2n}<\frac{1}{\sqrt{2n+1}}$ Thanks.
0
votes
3answers
46 views

Prove the inequality $\frac{a}{b}<\frac{a+k}{b+k},(a<b, a,b,k>0)$

Help me please to prove the follow inequality: $\frac{a}{b}<\frac{a+k}{b+k},(a<b, a,b,k>0)$ thanky very much
0
votes
1answer
33 views

Establish a relation between p and q

For positive real numbers $a_1, a_2, ... ,a_{100}$, let $$p = \sum_{i=1}^{100} a_i $$ and $$q = \sum_{1\le i \lt j \le 100} a_ia_j \space .$$ Then establish an inequality(or equality) among $p$ and ...
3
votes
2answers
60 views

An inequality for some series

Consider real positive numbers $t_1,t_2,\cdots, t_n$ for some $n\in\Bbb N$, with $\sum_{i=1}^nt_i^2=n$, such that if $0<t_i<1$ then $$\frac{t_i}{\sin\left(\frac{t_i\pi}{1+t_i}\right)}<1$$ ...
3
votes
1answer
83 views

Prove that inequality holds for all real number from $[0,1]$

There are given real numbers $a_1,a_2, ... , a_n \in [0,1]$ Prove that $\displaystyle \sum_{1\le i\le n} a_i \le 1+ \sum_{1\le i} \sum_{<j\le n} a_ia_j$ I have problem here because I can't find ...
2
votes
4answers
66 views

Proving for all n that $\sum_{i=0}^n \frac1{2^{i}} < 2$

Proving for all n $\in \mathbb N$, $$\sum_{i=0}^n \frac1{2^{i}} < 2$$ Hint. First prove that the left hand side can be expressed in closed form, i.e. without using the summation operator. This is ...
0
votes
2answers
85 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 ...
6
votes
4answers
90 views

A question about inequality ${(n+1)\over e^n}^n<n!$

How to prove the inequality $${(n+1)\over e^n}^n<n!$$ I have tried mathematical induction, but it doesn't work! Are there other methods to solve it?
0
votes
1answer
48 views

Solve the following inequality…

Can you please verify if I've done this exercise correctly, and if you have a better solution, please, show it to me. Thank you! (The exercise is in the left top corner.)
1
vote
1answer
28 views

Prove triangle inequality of vector norm

I am trying to show that $||x+y||_p \leq ||x||_p + ||y||_p$ where $p$ is an integer larger than 1, but not infinity (I proved those cases already), and $||x||_p = (\sum_{i=1}^n |x_i|^p)^{\frac{1}{p}}$ ...
0
votes
4answers
77 views

Elementary proofs of inequalities

I was just introduced into elementary proofs of inequalities, my text's explanation however feels incomplete. I did further research on the subject, my question is thus: Prove: If $0 < a < b$, ...
0
votes
1answer
52 views

Caratheodory's theorem and outer measure

I'm trying to show that $$\lambda(A)=\lambda(A\cap E)+\lambda(A\cap E^c)$$ where $\lambda$ is an outer measure, $A\subset \mathbb{R}$, $E \subset \mathbb{R}$, and $E$ is an elementary set; that is, ...
2
votes
1answer
71 views

To prove $2^{n(n+1)} >(n+1)^{(n+1)} \prod\limits_{j=1}^n \left(\dfrac {j}{n+1-j}\right)^j , \forall n\in \mathbb N $ \ { $1$ }

How do we prove that $2^{n(n+1)} >(n+1)^{(n+1)} \prod_{j=1}^n \Bigg(\dfrac {j}{n+1-j}\Bigg)^j , \forall n\in \mathbb N$ \ {$1$} ?
2
votes
3answers
151 views

Finding the values of $A,B,C,D,E,F,G,H,J$

Given that the letters $A,B,C,D,E,F,G,H,J$ represents a distinct number from $1$ to $9$ each and $$\frac{A}{J}\left((B+C)^{D-E} - F^{GH}\right) = 10$$ $$C = B + 1$$ $$H = G + 3$$ find (edit: ...
1
vote
4answers
185 views

Induction proof of $n^{(n+1) }> n(n+1)^{(n-1)}$

The question statement from my homework booklet goes: Prove by mathematical induction that $n^{n+1} > n(n+1)^{n-1}$ is true for all integers $n \geq 2$. I've managed to come up with this ...
1
vote
2answers
92 views

Positivness of the sum of $\frac{\sin(2k-1)x}{2k-1}$.

For $n\in \mathbb{N}$, $x\in (0,\pi)$. Prove that : $$f_n(x)=\sum_{k=1}^n \frac{\sin [(2k-1)x]}{2k-1} \geq 0.$$ I've tried to do it by differentiation : I Calculate $f_n'(x)$ (sum of ...
2
votes
2answers
154 views

Solve inequality: $\frac{2x}{x^2-2x+5} + \frac{3x}{x^2+2x+5} \leq \frac{7}{8}$

Rational method to solve $\frac{2x}{x^2-2x+5} + \frac{3x}{x^2+2x+5} \leq \frac{7}{8}$ inequality? I tried to lead fractions to a common denominator, but I think that this way is wrong, because I had ...
3
votes
12answers
380 views

How to prove this inequality $ x + \frac{1}{x} \geq 2 $

I was asked to prove that: $$x + \frac{1}{x}\geqslant 2$$ for all values of $ x > 0 $ I tried substituting random numbers into $x$ and I did get the answer greater than $2$. But I have a ...
0
votes
3answers
100 views

Solving $x^2 - 16 > 0$ for $x$

This may be a very simple question, but I can't figure out the correct reason behind it. If $x^2 - 16 >0$, which of the following must be true? a. $4 < x$ b. $-4 > x > 4$ ...
3
votes
2answers
111 views

Showing that $\sum_{i=1}^n \frac{1}{i} \geq \log{n}$

I have been trying to prove this by induction on $n\in \mathbb{N}$, but this approach seemed to get me nowhere. I have a suspicion it might be necessary to express $\log{n}$ as $\int_1^n 1/x\text{ ...
3
votes
1answer
146 views

Proving the AM-GM Inequality with Lagrange Multipliers

Exercise: Let $x_1,x_2,...,x_n$ be real positive numbers. Prove the arithmetic-geometric mean inequality, $(x_1x_2...x_n)^{1/n}\le (x_1+x_2+...+x_n)/n$. Hint: Consider the function ...
0
votes
1answer
44 views

Inequality in non-decreasing sequence

Let $a, b$ be two sequences of real numbers such that $a_1 \le a_2 \le \dots \le a_n$ and $b_1 \le b_2 \le \dots \le b_n$. Prove (or disprove) that ...
2
votes
1answer
40 views

Prove that $P[X>\epsilon] \leq M(t)/e^{\epsilon t}$

Prove that $P[X>\epsilon] \leq \dfrac{M(t)}{e^{\epsilon t}}$ Looks like Markov's inequality, it's very easy to derive for $t>0$ $P[X>\epsilon] =P[Xt>\epsilon t]=[e^{Xt}>e^{\epsilon ...