query: Prove the inequality$|x|-|y|≤|x+y|$
positive_passages.docid: 3820
positive_passages.text: Prove that $||x|-|y||\le |x-y|$
negative_passages.docid: 147785
negative_passages.text: Proving inequality $x^xy^y \geq (\frac{x+y}{2})^{x+y}$
negative_passages.docid: 27570
negative_passages.text: Prove $(1-x)\ln(1-x)+(1+x)\ln(1+x)\leq 2x^2$ for $0<x<1$
negative_passages.docid: 147892
negative_passages.text: Proving if $x<y$ then $\sqrt{x} < \sqrt{y}$
negative_passages.docid: 64457
negative_passages.text: Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$
negative_passages.docid: 126165
negative_passages.text: Prove $|\arctan x - \arctan y|\leq |x-y|$
negative_passages.docid: 124968
negative_passages.text: How prove this inequality $\sum\limits_{cyc}\frac{x+y}{\sqrt{x^2+xy+y^2+yz}}\ge 2+\sqrt{\frac{xy+yz+xz}{x^2+y^2+z^2}}$
negative_passages.docid: 1305
negative_passages.text: Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$
negative_passages.docid: 8207
negative_passages.text: Can I prove this inequality algebraically?
negative_passages.docid: 138931
negative_passages.text: Show that $||x+y||\cdot||x-y||\le||x||^2+||y||^2$
negative_passages.docid: 88838
negative_passages.text: How can I prove this inequality for $n\geq 2$?
negative_passages.docid: 14677
negative_passages.text: Proving the exponential inequality: $x^y+y^x\gt1$
negative_passages.docid: 148451
negative_passages.text: Polynomial inequality proof
negative_passages.docid: 44233
negative_passages.text: How to prove this inequality $ x + \frac{1}{x} \geq 2 $
negative_passages.docid: 143072
negative_passages.text: Proving $|\sin x - \sin y| < |x - y|$
negative_passages.docid: 29198
negative_passages.text: How to prove the following inequality $|\prod_{i=1}^{i=n}a_i-\prod_{i=1}^{i=n}b_i| < n\delta$?
negative_passages.docid: 117262
negative_passages.text: Proving a logarithm inequality