The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the

Question

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?

A.

$x less than 18$

B.

$x greater than 18$

C.

$6 less than x less than 18$

D.

$x less than 6$ or $x greater than 18$

Accepted Answer

Choice C is correct. It&rsquo;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&rsquo;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 + x &#62; 12 , 6 + 12 &#62; x , and 12 + x &#62; 6 represent all possible values of x . Subtracting 6 from both sides of the inequality 6 + x &#62; 12 yields x &#62; 12 - 6 , or x &#62; 6 . Adding 6 and 12 in the inequality 6 + 12 &#62; x yields 18 &#62; x , or x &#60; 18 . Subtracting 12 from both sides of the inequality 12 + x &#62; 6 yields x &#62; 6 - 12 , or x &#62; - 6 . Since all x -values that satisfy the inequality x &#62; 6 also satisfy the inequality x &#62; - 6 , it follows that the inequalities x &#62; 6 and x &#60; 18 represent the possible values of x . Therefore, the inequality 6 &#60; x &#60; 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.

Question 176e3a