M31-20

If a question asks which of the options MUST be true, an option to be considered true, MUST be true for ANY value.

If a question asks which of the options COULD be true, an option to be considered true, should be true for at least one value.

Here are must or could be true questions to practice: search.php?search_id=tag&tag_id=193

I think this is a high-quality question and I agree with explanation. The "Could" part is the real trap i failed to consider option 3 and eliminated it when it was greater than the sum of the other 2 sides when x=10. Great question

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

I agree with logic and that's the same I used to solve. But II doesn't make sense. If you substitute x with 1 then 7x + 7 = 14 and 6x+9 =15 hence it wouldn't fit the inequality.

How can option II. be correct?

The question asks for the lengths that could potentially be the length of the third side of the triangle, not the lengths that must be true for all values of x. Therefore, even if the inequality does not hold for certain values of x, as long as there exists at least one positive integer value of x for which the inequality does hold, the option is considered valid.

For example, while II does not hold true for x=1 or x=2, it does hold true for x=3. When x=3, the two given sides of the triangle are 3x+2 = 3(3)+2 = 11 and 4x+5 = 4(3)+5 = 17. For II, the third side would be 6x+9 = 6(3)+9 = 27. Since 27 is greater than 17-11=6 and less than 17+11=28, it "could" be the length of the third side of the triangle, making it a valid option.

Thus, the official solution is correct, and the answer is indeed E, I, II, and III.