Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

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votes
3answers
125 views

Inequality with the variable in both the base and the exponent. [duplicate]

I realised that what I took as terseness of my question, actually made it look like a lazy attempt to get a homework answer. The following is the edited question, hopefully up to the standards of this ...
-2
votes
2answers
79 views

Prove using induction that $n^6 < 3^n$,for all $n > 18$

Prove using induction (or using any other elementary precalculus techniques) that $$n^6 < 3^n, \forall n \geq 19.$$ I have no idea how to do this. Writing the induction step, I get that I need to ...
-2
votes
2answers
65 views

If $1<a<b$, which of the following is larger than the other: $a\sqrt[3]{b^2}$ and $b\sqrt[3]{a^2}$? [closed]

Given that $1<a<b$, how can you determine which is the larger, out of $a\sqrt[3]{b^2}$ and $b\sqrt[3]{a^2}$? Thanks in advance.
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votes
3answers
124 views

Prove that $\displaystyle \lim_{n\to \infty} a_n\le\lim_{n\to \infty}b_n$ [closed]

Let (a$_n$) and (b$_n$) be convergent sequences. Prove the following statement: If there is a number $N$ $\in$ $\mathbb{N}$ so that a$_n$ $\le$ b$_n$ applies for all n $\ge$ $N$ follows: $\...
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votes
2answers
108 views

Proving $\frac14(a+b+c+d)\geq \sqrt[4]{abcd}$ [closed]

Can somebody help me how to prove that: $$\frac14(a+b+c+d)\geq \sqrt[4]{abcd}$$ I'm sure it's easy, but I just can't figure out how! I've tried many ways.
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votes
3answers
77 views

How to prove that $(1-x)^n<1-x^n$ [closed]

Given $x$ is a real number greater than zero and less than one, how can we prove that $(1-x)^n$ is less than $(1- x^n)$, where $n$ is a positive integer?
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votes
3answers
57 views

Just this last inequality [closed]

I can't seem to find a counterexample to disprove it: $\sqrt{k}k^c + (2\sqrt{k+1}-1)\dfrac{\sqrt{k^k}}{2^{k-1}} -\sqrt{k+1}(k+1)^c≤0$. For $k \ge 1, c>0$ both real
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votes
2answers
110 views

Work out the integer values for which $x^{2} - 20x + 96 < 0$ [closed]

Work out the integer values for which $x^{2} - 20x + 96 < 0.$ How should I approach the above question?
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votes
2answers
66 views

solve for x using inequalities

Solve $$\frac{x}{x-2} < \frac{x}{x-1}$$ I know for inequality you have to multiply by the denominator square but I'm not sure if this applies to this one since this contains two denominators.
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votes
2answers
50 views

Inequality involving summation

Can someone help me with this inequality: $\sum_{i=1}^{n}{\dfrac{1}{\sqrt i}}\leq \dfrac{2n}{\sqrt{n}}$ Thank you.
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votes
2answers
113 views

Verify if this inequality is true

If $a_1, \; a_2, \cdots , \; a_n$ are positive real numbers with product $1$, does this inequality hold, and if so how can one prove it? $$(\sum a_i)^{7} \geq n^{5} \cdot (\sum a_i^2)^2$$ Thanks ...
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votes
3answers
78 views

Does $1 + \frac{1}{x} + \frac{1}{x^2}$ have a global minimum for $0 < x \in \mathbb{R}$?

This question is related to this one. My question here is: Does the function $$f(x) = 1 + \frac{1}{x} + \frac{1}{x^2}$$ have a global minimum for $0 < x \in \mathbb{R}$? Thank you!
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votes
2answers
235 views

Simple ceiling function problem [closed]

Prove that $\lceil4n/3\rceil\le 4\lceil n/3\rceil$ for all integers $n$. Try to generalize this result to something where something other than 4 and 3 are used.
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votes
3answers
64 views

Calculate the minimum value of $\frac{a}{\sqrt{a + 2b}} + \frac{b}{\sqrt{b + 2a}}$ where $a, b > 0$ and $\sqrt{a + 2b} = 2 + \sqrt{\frac{b}{3}}$.

Given that $a$ and $b$ are positives such that $\sqrt{a + 2b} = 2 + \sqrt{\dfrac{b}{3}}$, calculate the minimum value of $$\large \frac{a}{\sqrt{a + 2b}} + \frac{b}{\sqrt{b + 2a}}$$ I have provided ...
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votes
2answers
50 views

Help me proving the inequality [closed]

If $a,b,c$ are sides of a triangle then prove that, $$\left(\frac{a+b}{c}\right)^3+\left(\frac{b+c}{a}\right)^3+\left(\frac{c+a}{b}\right)^3 < 24$$
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votes
2answers
85 views

Prove $1/(x^5+y^2+z^2)+1/(x^2+y^5+z^2)+1/(x^2+y^2+z^5) \leq 3/(x^2+y^2+z^2)$ when $xyz \geq 1$ ($x,y,z$ are positive real numbers) [closed]

Prove $$ \frac{1}{x^5+y^2+z^2} + \frac{1}{x^2+y^5+z^2} + \frac{1}{x^2+y^2+z^5} \leq \frac{3}{x^2+y^2+z^2} ,$$ when $xyz \geq 1$ ($x,y,z$ are positive real numbers). I need this for lemma but I don'...
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votes
3answers
45 views

Solving $-1\le \frac{2}{x}$, I came up with 2 opposite solutions

Now let me reclassify my problem: I was solving some inequality until I stopped at this step $-1\le\frac{2}{x}$ Why did I stop? Because if I do this next step $-x\le 2$ and then multiply both sides ...
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votes
4answers
51 views

Proof of inequality in $\mathbb{R^+}$

I have this statement: You have this fraction: $\frac{p}{q}$, where: $q > p$ and $q,p \in \mathbb{R^+}$ Proof if $\frac{p}{q} < \frac{p+1}{q+2}$ Basically my development was: $i)$ ...
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votes
2answers
106 views

Minimum value of $ f(x,y,z)=\left(x+\frac{1}{y}\right)^2+\left(y+\frac{1}{z}\right)^2+\left(z+\frac{1}{x}\right)^2. $ [duplicate]

If $x>0$, $y>0$, $z>0$ and $x+y+z=6$ then find the minimum value of $$ f(x,y,z)=\left(x+\frac{1}{y}\right)^2+\left(y+\frac{1}{z}\right)^2+\left(z+\frac{1}{x}\right)^2. $$ Thanks in ...
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votes
2answers
196 views

How to prove that $\left(1+\frac 1 n\right)^n < \left(1+\frac 1 {n+1}\right)^{n+1}$? [duplicate]

How would one prove the following:$$\left(1+\frac 1 n\right)^n < \left(1+\frac 1 {n+1}\right)^{n+1}$$ This is taken from the book challenge and thrill of precollege mathematics.
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votes
4answers
74 views

How do i prove this inequality related to quadratic equations?

For any real positive value of $x$, show that $3-x\not>\frac{7}{x+2}$.(Using properties of quadratic equation) I have not posted any attempt because i have no idea where to start from. I have ...
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votes
2answers
115 views

If $xy+xz+yz=3$ so $\sum\limits_{cyc}\left(x^2y+x^2z+2\sqrt{xyz(x^3+3x)}\right)\geq2xyz\sum\limits_{cyc}(x^2+2)$

Prove that for any set of three positive real $x, y, z$ such that $xy+yz+zx=3$ $x^2(y+z)+y^2(x+z)+z^2(x+y)+2\sqrt {xyz}\left(\sqrt{x^3+3x}+\sqrt{y^3+3x}+\sqrt{z^3+3x}\right)\ge$ $\ge 2xyz(x^2+y^2+z^2+...
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votes
1answer
46 views

How to find the smallest integer $k > 0$ such that the following inequality holds

I need to find the smallest integer $k>0$ such that the following inequality holds: $$(1-\frac{1}{365})^k\le\frac{1}{2}$$ The answer is supposedly greater than $200$. How can I find $k$?
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votes
1answer
81 views

If $a+b=1$ where $a,b\in\Bbb R^+$ then prove that $a^ab^b+b^aa^b\le1$

If $a+b=1$ such that $a$ and $b$ are positive real numbers, then prove $$a^a b^b+b^a a^b\le1$$ I tried applying Arithmetic and Geometric mean but it doesn't work. I also tried to equate the given ...
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votes
1answer
49 views

Neighbourhood set in Graph theory [closed]

Let $G$ be any connected graph with $\Delta(G)$ be maximum degree. If $D \subseteq V(G)$ then how can we say that $\left | \bigcup \limits_{v \in D} N(v) \right | \leq |D| \Delta(G)$.
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votes
3answers
100 views

Prove that, if $a>c$ and $b>d$, thus $ab>cd$ [closed]

I would like to ask you a question: how could I prove that, if $a>c$ and $b>d$, thus $ab>cd$? Thank you for help. P.s. I forgot to tell you that $a>0, b>0, c>0, d>0.$
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votes
3answers
531 views

Why is $\pi^2$ so close to $10$?

Noam Elkies explained why $\pi^2=9.8696...$ is so close to $10$ using an inequality on Euler's solution to Basel problem $$\frac{\pi^2}{6}=\sum_{k=0}^{\infty} \frac{1}{\left(k+1\right)^2}$$ to form ...
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votes
3answers
133 views

Why $\sin x + \cos x$ is non-decreasing in $0 \le x \le \pi/4$? [closed]

How can I prove that $\sin x + \cos x$ is non-decreasing in $0 \le x \le \pi/4$ ?
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votes
3answers
140 views

Prove the inequality by induction: $3^n > n^3$ for $n\ge4$ [duplicate]

Prove the inequality by induction: $3^n > n^3\ $ for $\ n \geq 4$ Edit: 1) Base case: $n=4$, $3^4>4^3, 81>64$ 2) Assume true for n=k: so $3^k>k^3$ 3) Consider $(k+1)^3$, $(k+1)^3 = k^...
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votes
3answers
64 views

Prove a simple inequality (triangle inequality on a norm) [closed]

Let $a_1,b_1,a_2,b_2\geq 0$. Prove that $\left(|a_1+a_2|^2+|b_1+b_2|^2\right)^{1/2}\leq\left(a_1^2+b_1^2\right)^{1/2}+\left(a_2^2+b_2^2\right)^{1/2}$
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votes
2answers
169 views

$x_1+x_2+\cdots+x_n\leq M$: Cardinality of Solution Set is $C(M+n, n)$

Show that the number of solutions in nonnegative integers of the inequality $$x_1+x_2+\cdots+x_n\leq M,$$ where $M$ is a nonnegative integer, is $C(M+n, n)$.
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votes
1answer
144 views

Lower bound for $\ln x$ using Lagrange's mean value theorem or Rolle's theorem

I have to prove this inequality. $$ \ln x>\frac{2(x-1)}{x+1} \hspace{15pt}, \hspace{15pt}\text{where}\hspace{5pt}x>1 $$ using either Lagrange's mean value theorem or Rolle's theorem. Can ...
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votes
2answers
74 views

Sum of numerators divided by sum of denominators $\leq$ the maximum fraction [duplicate]

Let $\tfrac{a_1}{b_1},\dots,\tfrac{a_n}{b_n}$ where $a_i,b_i>0$. How can one prove that $$\frac{\sum_i a_i}{\sum_i b_i}\leq \max_j \tfrac{a_j}{b_j}$$?
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votes
3answers
52 views

How to prove $|e^{i\theta} - 1| \leq |\theta|$?

Prove $|e^{i\theta} - 1| \leq |\theta|$, for all real numbers $\theta$.
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votes
2answers
54 views

Finding maximum value of squared sum

Given k positive integers such that $x_1+x_2+...+x_k=n$, find the maximum possible value of $x_1^2+x_2^2+...+x_k^2$. I know we can find its minimum value using Cauchy-Schwarz inequality, but is ...
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votes
4answers
61 views

Can We Use $\log(1+x/y) \leq x/y$? [duplicate]

I want to know that can we have the following inequality for $x>0$, $y>0$? $$\log(1+x/y) \leq x/y$$
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votes
3answers
81 views

Find the values that $(x-y)(x+y)$ can take. [closed]

$$-3 \le x \lt 2 $$ $$1 \lt y \lt 3 $$ Find the values that $(x-y)(x+y)$ can take. I know that $(x-y)(x+y)$ is given by $$x^2-y^2$$ So, we have to square the both inequality but I couldn't do ...
-2
votes
2answers
44 views

Prove |a|<2|a-b| if 2|b|<|a| [closed]

I have already proved the triangle inequality $\vert a+b \vert \le \vert a \vert + \vert b \vert$ I also proved that $\vert a \vert - \vert b \vert \le \vert a-b \vert $ and that $\vert \vert ...
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votes
3answers
63 views

Prove If $\lim_{x\to x_0 }f(x) > C$ then $f(x) > C$ in a deleted neighborhood of $x_0$ and… [closed]

Prove: If $$\lim_{x\to x_0 }f(x) > C$$ then $f(x) > C$ in a deleted neighborhood of $x_0$. Similarly, if $$\lim_{x\to x_0 }f(x) < C$$ then $f(x) < C$ in a deleted neighborhood of $x_0$.
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votes
3answers
185 views

How to solve the inequality $x^3 + 2x^2 - 2x - 1 < 0$? [closed]

What will be the solution to the following inequality? $$x^3 + 2x^2 - 2x - 1 < 0$$ I can't solve this using the table of signs method. Please include an explanation as to how it was solved.
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votes
3answers
63 views

Solving Quadratic Inequality

I can't solve this math problem: $$\frac{x}{2x-8} > 3 $$ To find the values of x. I have been trying for over an hour and now my head hurts! Here is what I have done so far: I tried: $$ (2x-...
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votes
3answers
52 views

A Problem Involving an Inequality [closed]

How to prove that $\frac1{a^2} + \frac1{b^2} + \frac1{c^2} \geq \frac1{ab} + \frac1{bc} + \frac1{ac}$ Assume that given symbolic terms are REAL and POSITIVE
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votes
1answer
64 views

Solve the inequality $\sqrt{4x+1}+\sqrt{x+1}<x+1$

Can you show the steps for solving this inequality: $$\sqrt{4x+1}+\sqrt{x+1}<x+1$$ Condition: $x \geq -1/4$ and $x\geq -1$. I'm stuck here: $$2\cdot \sqrt{(4x+1)(x+1)}<x^2-3x-1$$
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votes
1answer
137 views

Not so easy inequality: $(x+1)(y+1)(z+1)\ge8$ [closed]

Let $x$, $y$ and $z$ be three positive numbers such that $x+y+z=xy+xz+yz$ prove that $(x+1)(y+1)(z+1)\geq8$
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votes
1answer
561 views

Lower and Upper bounds for Ratio of Sum of two Sequences of positive numbers [closed]

Given two sequences of positive numbers $a_1, a_2, \dots a_k$ and $b_1, b_2, \dots b_k$. Prove that $$ \min_{i} \frac{a_i}{b_i} \leq \frac{\sum_{i}a_i}{\sum_{i}b_i} \leq \max_{i} \frac{a_i}{b_i} $$ ...
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votes
3answers
107 views

Inequalities with quadratics [closed]

$$\frac{12}{x^2 + 2x} < \frac{3}{x^2 + 4x + 4}$$ I am confused. Can someone help me? Update : you can see my work in the comments. i figured out the answer but the answers other people gave was ...
-2
votes
2answers
100 views

For nonnegative numbers $x + y + z = \pi$, prove that $1 \le \cos x+\cos y+\cos z \le \dfrac 3 2 $

Let $x,y,z$ be nonnegative real numbers and $x+y+z=\pi$. Prove the inequality $$1 \le \cos x+\cos y+\cos z \le \frac 3 2.$$ I tried to put $z=\pi-x-y$ and then calculate the extremas of two ...
-2
votes
1answer
30 views

problem from inequalities chapter [closed]

prove that if $(1+a_1)(1+a_2)...(1+a_n)=2^n$, then $a_1a_2a_3...a_n\le1$
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votes
2answers
162 views

How to show $\frac12\cdot\frac34\cdot\frac56\cdots\frac{99}{100}<\frac{1}{12}$? [closed]

How can I show that $$\frac12\cdot\frac34\cdot\frac56\cdots\frac{99}{100}<\frac{1}{12}?$$
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votes
1answer
35 views

simple inequality in $R ^ 2$ [closed]

Given that, $D\{(a, b), (c, d)\}$ is the usual distance between $(a, b)$ and $(c, d)$ in $R ^ 2$, $0 < k$ and $(x, y) \in R ^ 2$ Is the following true? $D\{(x, y), (-k, 0)\} - D\{(x, y), (k, 0)...