Number of possible values for fourth side of quadrilateral

Question

This math question is from KVPY SA 2009 exam :

The sides of a quadrilateral are all positive integers and three of them are 5, 10, 20. How many values are possible for fourth side?

Options: $\quad$ A. 29 $\quad$ B. 31 $\quad$ C. 32 $\quad$ D. 34

D option is the correct answer.

But when I tried to solve this by the method in picture (shared below) , I got 27 as answer with the values ranging from 7 to 33. Please correct me where I am wrong.

How I solved the problem.

Answer 1

You are assuming that $y$ must be an integer, but this is not stated in the problem. It is necessary and sufficient that the longest side be greater than the sum of the other three sides. If the longest side is $20$, we have $20<x+15$ so that $6\leq x\leq20$. If $x$ is the longest side, we have $20\leq x<35$, so $6\leq x\leq 34$ and A is the correct answer.

Answer 2

Arrange $a = 20, b = 10, c = 5$ for maximum value through sum and min value through difference $(\ge 0$).

$a + b + c = 35$ which is the longest line segment using them.

$a - b - c = 5$ which is the smallest line segment using them.

So $6 \le x \le 34$ should form a quadrilateral.