## Geometry: Common Core (15th Edition)

$I$
Using the Triangle Inequality Theorem, we need to see if the sum of each of the combinations of two sides is greater than the other side. Let's set $x$ as the length of the third side. Now, let's look at the possible combinations of sides to see if the lengths of two sides is greater than the length of a third side: 1st combination: $x + 7 > 11$ Solve for $x$ by subtracting $7$ from each side of the inequality: $x > 4$ 2nd combination: $x + 11 > 7$ Subtract $11$ from each side of the equation: $x > -4$ 3rd combination: $7 + 11 > x$ Switch the inequality around and add to simplify: $x < 18$ $x$ cannot be a negative number because it is a length, so let's eliminate the inequality that includes a negative number. We know $x$ has to be greater than $4$ and $x$ has to be less than $18$, so we have $x$ in the following range: $4 < x < 18$ Let's look at the answer options to see which one cannot be the length of the third side of the triangle. The lengths $13$, $7$, and $5$ lie within this interval; therefore, we are not looking for options $F$, $G$, and $H$. Option $I$ is correct because we are looking for the length that is impossible for this triangle.