To determine the range of possible values for the third side [tex]\( x \)[/tex] in a triangle with sides measuring 200 units and 300 units, we use the triangle inequality theorem. According to this theorem, in any triangle, the length of any side must be less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.
To find this range for the third side [tex]\( x \)[/tex], follow these steps:
1. Calculate the Difference of the Given Sides:
[tex]\[ \text{Difference} = |200 - 300| = 100 \][/tex]
2. Calculate the Sum of the Given Sides:
[tex]\[ \text{Sum} = 200 + 300 = 500 \][/tex]
3. Set Up the Compound Inequality:
The length of the third side [tex]\( x \)[/tex] must be greater than the difference and less than the sum of the other two sides:
[tex]\[ 100 < x < 500 \][/tex]
Thus, the compound inequality that represents the range of possible lengths for the third side [tex]\( x \)[/tex] is:
[tex]\[ 100 < x < 500 \][/tex]
This means the length of the third side must be more than 100 units and less than 500 units.
