Can the numbers 24, 32, and 40 be the lengths of three sides of a triangle? why or why not

Yes it can.

Because the 3 lengths satisfy the triangle inequality theorem.

The triangle inequality theorem states that the third side of a triangle must be greater than the positive difference of the given two sides and must be less than the sum of the given two sides.

For example, given 24 and 32.

The third side, x > (32 - 24)      x >8  and

x < (24 + 32)        x < 56

Therefore              8 < x < 56.    So x must be greater than 8 and less than 56.

So the third side, x, is between 8 and 56.  The third side can not be 8 and it can not be 56, but must fall in between 8 and 56.

And the third side 40 satisfies the inequality    8 < x < 56.

40 falls between 8 and 56.

Therefore 24, 32 and 40 can be the the three sides of a triangle.