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 \( 4 + 8 = 12 \not > x \).

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