Geometry and Triangle Inequality Question

If k is an integer and 2 < k < 7, for how many different values of k is there a triangle with sides of lengths 2, 7, and k?
(A) One
(B) Two
(C) Three
(D) Four
(E) Five


Please explain your answer with cogent and pellucid explanation

A is the right answer.

K is an integer and values of K are 3,4,5,6.

Now according to the Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

It is given that the triangle two sides are 2 and 7.

Based on the triangle inequality theorem, the third side can be only 6.

In a triangle, the sum of the smaller two sides must be larger than the largest side.
For k values 3, 4, 5, and 6, the only triangle possible is 2, 7, and k = 6 because only 2 + 6 > 7.
For k values 3, 4, and 5, the sum of the smaller two sides is not larger than the third side; thus, 6 is the only possible value of k that satisfies the conditions.

Answer is (A) One