# What is Inequality: Definition and 1000 Discussions

In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If x, y, and z are the lengths of the sides of the triangle, with no side being greater than z, then the triangle inequality states that

z

x
+
y
,

{\displaystyle z\leq x+y,}
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):

x

+

y

x

+

y

,

{\displaystyle \|\mathbf {x} +\mathbf {y} \|\leq \|\mathbf {x} \|+\|\mathbf {y} \|,}
where the length z of the third side has been replaced by the vector sum x + y. When x and y are real numbers, they can be viewed as vectors in R1, and the triangle inequality expresses a relationship between absolute values.
In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either R2 or R3. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.
In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints.The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces (p ≥ 1), and inner product spaces.

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34. ### Solving for ##x## for a given inequality

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35. ### An inequality involving ##x## on both sides: ##\sqrt{x+2}\ge x##

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36. ### Determine the set of values of ##n## that satisfy the inequality

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I have a two-variable inequality and wish to make a graph of the regions in which it is satisfied. Is any such took available online for free? A great many free online graphing calculators are available, but I expect the great majority won't do what I want. Specifically I want to find the...
38. ### I Mean value theorem - prove inequality

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39. ### Prove this inequality about the weights of code words (Coding Theory)

I'm trying to prove the following: ##wt(x+y) \leq wt(x) + wt(y)##, where "wt(x)" is referring to the weight of a specific code word. Proof: For two code words ##x, y \in F^{n}_2##, we have the inequalities ##0 \leq wt(x)## and ##0 \leq wt(y)##. Adding these together, we have ##0 \leq wt(x) +...
40. ### MHB Troubleshooting Inequality: Can't Seem to Solve the Last One

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41. ### Prove by induction the inequality

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42. ### Solve the inequality problem that involves modulus

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43. ### I Demonstration of inequality between 2 variance expressions

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44. ### I Help with rewriting a compound inequality

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45. ### MHB Inequality in Triangle: Prove $|x^2+y^2+z^2-2(xy+yz+xz)|\le \frac{1}{27}$

Let $x, y, z$ be length of the side of a triangle such that $\sqrt{x} + \sqrt{y} + \sqrt{z} = 1.$ Prove $|x^{2} + y^{2} + z^{2} - 2\left( xy+yz+xz\right)| \le \frac{1}{27}$.
46. ### MHB How to prove the normal distribution tail inequality for large x ?

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47. ### Mathematics Behind Global Currency as a Wealth Inequality Regulation System

I've no political agenda. This is strictly a math question about optimization. Those addressing it should do so strictly from a mathematical perspective. Thank you! If the world simply had one global currency, like Bitcoin, and everyone had an account, then the sum total of all accounts...
48. ### Find the inequality that satisfies this quadratic problem

see the textbook problem below; see my working to solution below; i generally examine the neighbourhood of the critical values in trying to determine the correct inequality. My question is "is there a different approach other than checking the neighbourhood of the critical values"? In other...
49. ### What is the name of this inequality?

My prof. calls it the triangle inequality. However the wikipedia page with the same this name shows a special case of it, which is ##|x+y|\leq|x|+|y|##, and my prof. calls it the triangle inequality 2. I wonder what the formal name of the inequality in the picture above is. Thanks in adv.
50. ### I Prove Complex Inequality: $(|z_1 + z_2| + |z_1 - z_2|)(|z_1| + |z_2|)>=\sqrt{2}$

Prove that $$(|z_1 + z_2| + |z_1 - z_2|)(|z_1| + |z_2|) >= \sqrt{2}$$