query: Prove that $|x+y| \leq |x|+|y|$
positive_passages.docid: 22237
positive_passages.text: Proof of triangle inequality
negative_passages.docid: 88328
negative_passages.text: Proof of $1+x\leq e^x$ for all x?
negative_passages.docid: 129331
negative_passages.text: Prove that $|x^p - y^p| \le p|x-y|(x^{p-1} + y^{p-1})$ provided that $1 \le p \lt \infty$ and $x, y \ge 0$
negative_passages.docid: 33543
negative_passages.text: is "y1 < x < y2" an inequality or inequalities?
negative_passages.docid: 80009
negative_passages.text: Prove that $| |a|-|b| | \le |a-b|$
negative_passages.docid: 68986
negative_passages.text: Use the triangle inequality to show that $|a|+|b| \leq |a+b|+|a-b|$
negative_passages.docid: 42516
negative_passages.text: If every $x\in X$ is uniquely $x=y+z$ then $\|z\|+\|y\|\leq C\|x\|$
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: 107173
negative_passages.text: If $a\lt{b}$ and $c\le{d}$, prove that $a+c\lt b+d$
negative_passages.docid: 40930
negative_passages.text: Prove that $ex \leq e^x$ for all $x \in \mathbb{R}$
negative_passages.docid: 122115
negative_passages.text: Proving that $1+x^py^{1-p} \le (1+x)^p(1+y)^{1-p}$
negative_passages.docid: 155232
negative_passages.text: Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$
negative_passages.docid: 3109
negative_passages.text: Prove that $(p+q)^m \leq p^m+q^m$
negative_passages.docid: 5369
negative_passages.text: Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$
negative_passages.docid: 31937
negative_passages.text: How do I prove equality of x and y?
negative_passages.docid: 77026
negative_passages.text: Proof with inequalities
negative_passages.docid: 8027
negative_passages.text: How to prove $\sqrt{x + y} \le \sqrt{x} + \sqrt{y}$?