A triangle has two sides of length 4 and 20. What compound inequality describes the possible lengths for the third side, x?

The possible lengths for the third side of a triangle, given two sides, can be found using the Triangle Inequality Theorem. This theorem states that the length of any side of a triangle 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.

Given a triangle with sides of length 4 and 20, we can set up the following inequalities to find the possible lengths for the third side, x:

1. x < 4 + 20
2. x > 20 - 4

Solving these inequalities gives:

1. x < 24
2. x > 16

So, the possible lengths for the third side of the triangle are between 16 and 24 (not inclusive). This can be written as the compound inequality 16 < x < 24.