An acute triangle has side lengths 21 cm, x cm, and 2x cm. if 21 is one of the shorter side of the triangle, what is the greatest possible length of the longest side, rounded to the nearest 10th?

The triangle inequality tells us that the sum of the two shorter sides of a triangle must be greater than the third side.

Here, 21 cm is one of the shorter sides, and x cm must be the length of the other shorter side.

So using the triangle inequality, 
21+x > 2x
or x<21 cm
To the nearest 0.1 cm, the greatest possible length of the longest side (2x) is 2*21-0.1 cm = 41.9 cm < 2x=42 cm.