Questions tagged [inequality]

Questions on proving, manipulating and applying inequalities.

19,204 questions
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Parametric exponential inequality

Find the values of $m$ s.t. $$\left(\frac{9}{25}\right)^x-m\left(\frac{3}{5} \right)^x+1>0,$$ for all $x<0$. My attempt is the following: let $y=(3/5)^x>1$ and the inequality transforms as ...
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Hermite - Hadamard inequality [on hold]

What is the geometric meaning of Hermite - Hadamard inequality? I.e. $$f(\frac{a+b}{2})\leq \frac{1}{b-a}\int_{a}^{b} f(x)dx \leq \frac{f(a)+f(b)}{2}.$$ Thank you for your help.
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How to prove this inequality for $a,b,c>0$?

How to prove the inequality for $a,b,c>0$ : $$\frac{2a-b-c}{2(b+c)^2}+\frac{2b-a-c}{2(a+c)^2}+\frac{2c-b-a}{2(b+a)^2}\geq 0$$ ?
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Inequality with absolute value function inside absolute value - $||x-2|-3|<4.$ [on hold]

Help me solving this. $||x-2|-3|<4.$ Find all values of x satisfying this.
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Show that if $\sigma = \frac {z(1 - \pi)}{x} < 0$ then $- \pi \sigma = \frac {- \pi z (1 - \pi)}{x} > (1 - \pi)$

I'm trying to understand a published economic paper and can't figure out the following steps: $\pi$ and $1-\pi$ are probabilities, z and x are two long terms that I have summarised for simplification,...
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Proving $a = b = c$ under certain conditions

For all real a, b, c, prove a = b = c if $$\frac{a^2+b^2+c^2}{3} = (\frac{a+b+c}{3})^2$$ The first idea that came to mind would be to prove this inequality by contradiction. However, I am unsure ...
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If $|z^2 + 2019| < 2019$ prove that $|z + \sqrt{2019}| > 31$

I got this from today's test. Let $z\in \mathbb{C}$. If $|z^2 + 2019| < 2019$ prove that $|z + \sqrt{2019}| > 31$ I tried triangle inequality, but doesn't work. I also tried using ...
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Why is there a Semi-Colon in this problem, and what does it mean? [on hold]

The problem that I am so concerned about, and that I don't understand at all is this- 4(x+3)>20;2 I don't understand what the Semi-Colon is for, and what it does. Please help me.
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Set of positive integers

Let $A$ be a set of positive integers with the following properties: a) If $n \in A$ then $n \leq 2018$ b) If $S$ is a subset of $A$ with $|S|=3$ then there are two elements $m,n \in S$ ...
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If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$

If $a,b,c>0$ and $ab+bc+ca=3$, prove that $\displaystyle \sum_{cyc} \frac{a}{\sqrt{a^3+5}} \leq \sqrt{6}/2$. My attempt was to use firstly AM-GM in the denominator, like $a^3+5 \geq 3a+3$ and the ...