query: Prove that $(x^2-y^2)(1/y-1/x) \geq 0$ for $x,y > 0$
positive_passages.docid: 157733
positive_passages.text: Are positive real numbers $x,y$ allowed to be taken out during this proof?
negative_passages.docid: 45918
negative_passages.text: Inequality: $x^2+y^2+xy\ge 0$
negative_passages.docid: 28822
negative_passages.text: $|x| - |y| \leq |x-y|?$
negative_passages.docid: 15237
negative_passages.text: Prove an inequality using complex analysis
negative_passages.docid: 62239
negative_passages.text: For a polynomial $p(z)$, prove there exist $R>0$, such that if $|z|=R$, then $|p(z)|\geq |a_n|R^n/2$
negative_passages.docid: 77130
negative_passages.text: How to prove this inequality $\left|x\sin{\frac{1}{x}}-y\sin{\frac{1}{y}}\right|<2\sqrt{|x-y|}$?
negative_passages.docid: 145152
negative_passages.text: How to prove that $\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}$ holds for any $(x,y,z)\in[1,2]^3$
negative_passages.docid: 102019
negative_passages.text: How to solve inequality $1-\frac{1}{x}\geq 0$ correctly?
negative_passages.docid: 137614
negative_passages.text: $\left|\frac{x}{|x|}-\frac{y}{|y|}\right|\leq |x-y|$, for $|x|, |y|\geq 1$?
negative_passages.docid: 98921
negative_passages.text: Does $y(y+1) \leq (x+1)^2$ imply $y(y-1) \leq x^2$?
negative_passages.docid: 147785
negative_passages.text: Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$
negative_passages.docid: 130327
negative_passages.text: Prove that $a^2 > b^2$ ; if $a > b \geq 0$.
negative_passages.docid: 7775
negative_passages.text: Show that $|x-y|\leq |x|+|y|$.
negative_passages.docid: 103030
negative_passages.text: Proving $\frac{1}{\sqrt{x}}\ge \frac{2}{x+1}$ for $x> 0$
negative_passages.docid: 12837
negative_passages.text: Inequality with condition $x^2+y^2+z^2=1$.
negative_passages.docid: 118613
negative_passages.text: Proving an inequality: $|1-e^{i\theta}|\le|\theta|$
negative_passages.docid: 33332
negative_passages.text: A classic exponential inequality: $x^y+y^x>1$