Triangle Inequality

The triangle inequality states that the sum of any two sides of a non-degenerate triangle must equal more than the third side. Thus a triangle with side lengths $3, 4$ , and $7$ is not possible since $3+4 \not > 7$ .

This property sets a boundary on the possible lengths of an unknown side of a given triangle. For example, if we know that a triangle has side lenghts of $4, 8,$ , and $x$ , we know that $4 < x < 12$ since if $x \leq 4$ it will be too short to satisfy the triangle inequality ( $4 + x \not > 8$ ), and if $x \geq 12$ , then it will be too long, since $4 + 8 = 12 \not > x$ .

Easy Math Editor

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`bold` or `__bold__`	bold
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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$