To determine the possible range for the third side of a triangle (denoted as [tex]\( x \)[/tex]) given the lengths of the other two sides, we can use the triangle inequality theorem. This theorem provides us with two critical inequalities that must be satisfied for any valid triangle:
1. The sum of any two sides of a triangle must be greater than the third side.
2. The difference between any two sides of a triangle must be less than the third side.
Given the lengths of two sides, [tex]\( a = 198 \)[/tex] and [tex]\( b = 222 \)[/tex], we can apply these inequalities as follows:
### Step-by-Step Solution:
1. Determine the lower bound for the third side:
- According to the triangle inequality theorem, the third side [tex]\( x \)[/tex] must be greater than the absolute difference of the other two sides.
- Calculate the absolute difference:
[tex]\[
|a - b| = |198 - 222| = 222 - 198 = 24
\][/tex]
- Therefore, [tex]\( x \)[/tex] must be greater than 24.
[tex]\[
x > 24
\][/tex]
2. Determine the upper bound for the third side:
- According to the theorem, the third side [tex]\( x \)[/tex] must be less than the sum of the other two sides.
- Calculate the sum:
[tex]\[
a + b = 198 + 222 = 420
\][/tex]
- Therefore, [tex]\( x \)[/tex] must be less than 420.
[tex]\[
x < 420
\][/tex]
### Conclusion:
Putting it all together, the range of values for the third side [tex]\( x \)[/tex] must satisfy:
[tex]\[
24 < x < 420
\][/tex]
Thus, in the anser choices presented in the problem, the number that belongs in the green box is:
[tex]\[
\boxed{24}
\][/tex]
