SAT Question 5bf5136d: Linear Inequalities answer and explanation with Desmos solution

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 $6+x>12$, $6+12>x$, and $12+x>6$ represent all possible values of $x$. Subtracting $6$ from both sides of the inequality $6+x>12$ yields $x>12-6$, or $x>6$. Adding $6$ and $12$ in the inequality $6+12>x$ yields $18>x$, or $x<18$. Subtracting $12$ from both sides of the inequality $12+x>6$ yields $x>6-12$, or $x>-6$. Since all x-values that satisfy the inequality $x>6$ also satisfy the inequality $x>-6$, it follows that the inequalities $x>6$ and $x<18$ represent the possible values of $x$. Therefore, the inequality $6<x<18$ represents the possible lengths, $x$, of the third side of the triangle.