To determine the possible length of the third side, [tex]\( x \)[/tex], in a triangle with side lengths of 200 units and 300 units, we use the triangle inequality theorem. This theorem states that the length of any side of a triangle must be greater than the absolute value of the difference of the other two sides and less than the sum of the other two sides.
First, let's find the lower bound for [tex]\( x \)[/tex]:
[tex]\[ x > |200 - 300| \][/tex]
[tex]\[ x > 100 \][/tex]
Next, let's find the upper bound for [tex]\( x \)[/tex]:
[tex]\[ x < 200 + 300 \][/tex]
[tex]\[ x < 500 \][/tex]
Combining these inequalities, we get the compound inequality that describes the possible lengths for the third side [tex]\( x \)[/tex]:
[tex]\[ 100 < x < 500 \][/tex]
So, the range of possible lengths for the third side is:
[tex]\[ 100 < x < 500 \][/tex]
