To determine the range for the possible lengths of the third side [tex]\( x \)[/tex] of a triangle with the given side lengths of 200 units and 300 units, we will use the triangle inequality theorem. The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Let's denote the sides of the triangle as [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( x \)[/tex] (the unknown side length).
1. Lower Bound: The length of the third side must be greater than the absolute difference of the lengths of the other two sides [tex]\( \left| a - b \right| \)[/tex]:
[tex]\[
x > \left| 200 - 300 \right| = 100
\][/tex]
2. Upper Bound: The length of the third side must be less than the sum of the lengths of the other two sides [tex]\( a + b \)[/tex]:
[tex]\[
x < 200 + 300 = 500
\][/tex]
Combining these inequalities, we get the compound inequality:
[tex]\[
100 < x < 500
\][/tex]
Thus, the range of the possible lengths for the third side [tex]\( x \)[/tex] is:
[tex]\[
\boxed{100 < x < 500}
\][/tex]
