Question

Two sides of a triangle measure 9 and 19 which inequality shows all the possible lengths of the third side x

Answer 1

In a triangle, the lengths of any two sides must be greater than the length of the third side. Using the sides 9 and 19, we can establish inequalities for the third side \( x \).

1. The sum of the lengths of any two sides must be greater than the length of the third side:
- \( 9 + 19 > x \) (which simplifies to \( x < 28 \))

2. The sum of the lengths of the other side and the third side must be greater than the length of the first side:
- \( 9 + x > 19 \) (which simplifies to \( x > 10 \))

3. The sum of the lengths of the remaining side and the third side must be greater than the length of the first side:
- \( 19 + x > 9 \) (this inequality is always true for positive values of \( x \))

Combining the valid inequalities gives us:
\[
10 < x < 28
\]

Thus, the inequality that shows all possible lengths of the third side \( x \) is:
\[
10 < x < 28
\]