A triangle has side lengths of 200 units and 300 units. Write a compound inequality for the range of the possible lengths for the third side, x.

Answer 1

To determine the range of the possible lengths for the third side of a triangle given side lengths of 200 units and 300 units, we can use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we need to establish three inequalities based on this rule:

1. \(200 + 300 > x\)
2. \(200 + x > 300\)
3. \(300 + x > 200\)

Let's solve each inequality separately:

1. \(200 + 300 > x\)
\[
500 > x \implies x < 500
\]

2. \(200 + x > 300\)
\[
x > 100
\]

3. \(300 + x > 200\)
\[
x > -100
\]

Since side lengths must be positive, \( x > -100 \) is always true and does not influence our range.

Combining the relevant inequalities from (1) and (2), we get a compound inequality:

\[
100 < x < 500
\]

Therefore, the range of the possible lengths for the third side \( x \) is \( \boxed{100 < x < 500} \).