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Algebra / Linear inequalities in one or two variables Difficulty: Hard

The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. If a triangle has side lengths of $6$ and $12$ , which inequality represents the possible lengths, $x$ , of the third side of the triangle?

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Explanation

Choice C is correct. It’s given that a triangle has side lengths of $6$ and $12$ , and $x$ represents the length of the third side of the triangle. It’s also given that the triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the length of the third side. Therefore, the inequalities $6 plus x greater than 12$ , $6 plus 12 greater than x$ , and $12 plus x greater than 6$ represent all possible values of $x$ . Subtracting $6$ from both sides of the inequality $6 plus x greater than 12$ yields $x greater than 12 minus 6$ , or $x greater than 6$ . Adding $6$ and $12$ in the inequality $6 plus 12 greater than x$ yields $18 greater than x$ , or $x less than 18$ . Subtracting $12$ from both sides of the inequality $12 plus x greater than 6$ yields $x greater than 6 minus 12$ , or $x greater than negative 6$ . Since all x-values that satisfy the inequality $x greater than 6$ also satisfy the inequality $x greater than negative 6$ , it follows that the inequalities $x greater than 6$ and $x less than 18$ represent the possible values of $x$ . Therefore, the inequality $6 less than x less than 18$ represents the possible lengths, $x$ , of the third side of the triangle.

Choice A is incorrect. This inequality gives the upper bound for $x$ but does not include its lower bound.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.