query: Triangle Inequality Proof: Show that:$||z_1|-|z_2||\leq |z_1+z_2|$
positive_passages.docid: 155423
positive_passages.text: lower triangle inequality for complex numbers
negative_passages.docid: 134764
negative_passages.text: Triangle Inequality question with fractions
negative_passages.docid: 77026
negative_passages.text: Proof with inequalities
negative_passages.docid: 135365
negative_passages.text: If $x, y, z$ are the side lengths of a triangle, prove that $x^2 + y^2 + z^2 < 2(xy + yz + xz)$
negative_passages.docid: 91662
negative_passages.text: Proof for triangle inequality for vectors
negative_passages.docid: 17264
negative_passages.text: Given $a,b,c$ are the sides of a triangle. Prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}<2$
negative_passages.docid: 93599
negative_passages.text: General Proof for the triangle inequality
negative_passages.docid: 39015
negative_passages.text: Prove the triangle inequality on the sphere $S^2$ in $\mathbb{R}^3$.
negative_passages.docid: 58969
negative_passages.text: A proof for the inequality $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2} $ for all $t \geq 2$
negative_passages.docid: 126557
negative_passages.text: Proving the inequality $\sum_{\text{cyc}} \frac{a}{a^2+b^3+c^3}\le\frac1{5abc}$
negative_passages.docid: 152311
negative_passages.text: Proving expression$\geq 2$, where variables are $x,y,z$
negative_passages.docid: 63958
negative_passages.text: how to prove this inequality $(ab+bc+ac)^2 ≥ 3abc(a+b+c)$
negative_passages.docid: 111903
negative_passages.text: An inequality for sides of a triangle
negative_passages.docid: 89170
negative_passages.text: How prove this inequality $\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y}\ge x^2+y^2+z^2$
negative_passages.docid: 23264
negative_passages.text: Inequality $ \vert \sqrt{a}-\sqrt{b} \vert \leq \sqrt{ \vert a -b \vert } $
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: 73987
negative_passages.text: Prove the inequality $({1+\frac{a}b})^n$ + $(1+\frac{b}a)^n$ $\geq$ $2^{n+1}$