To determine the range of possible lengths for the third side [tex]\( x \)[/tex] of a triangle with given sides of 200 units and 300 units, we can use the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This leads to three inequalities:
1. [tex]\( 200 + 300 > x \)[/tex]
2. [tex]\( 200 + x > 300 \)[/tex]
3. [tex]\( 300 + x > 200 \)[/tex]
Let's examine and simplify each inequality:
1. [tex]\( 200 + 300 > x \)[/tex]
[tex]\[ 500 > x \][/tex]
[tex]\[ x < 500 \][/tex]
2. [tex]\( 200 + x > 300 \)[/tex]
[tex]\[ x > 300 - 200 \][/tex]
[tex]\[ x > 100 \][/tex]
3. [tex]\( 300 + x > 200 \)[/tex]
[tex]\[ x > 200 - 300 \][/tex]
[tex]\[ x > -100 \][/tex]
This inequality [tex]\( x > -100 \)[/tex] is always true for any positive value of [tex]\( x \)[/tex], so it does not affect the final range in this context.
Combining the relevant inequalities [tex]\( x < 500 \)[/tex] and [tex]\( x > 100 \)[/tex], we get the compound inequality:
[tex]\[ 100 < x < 500 \][/tex]
Therefore, the range of possible lengths for the third side [tex]\( x \)[/tex] is:
[tex]\[ 100 < x < 500 \][/tex]
