Let x be the shortest side length, in inches.
Then the medium side is (x + 3) inches,
while the longest side is 5x inches long.

The perimeter is the sum  x + (x+3) + 5x, so your equation is

x + (x+3) + 5x = 24.

Simplify and solve it

7x + 3 = 24  --->  7x = 24-3  --->  7x = 21  --->  x =  = 3.

Hence, the shortest side length is 3 in.
Then the medium side is x + 3 = 3 + 3 = 6 in
and the longst side is 5x = 5*3 = 15 in.

Now check if such a triangle does exist.
The sum of the lengths of the shortest and the medium sides is 3 + 6 in = 9 in.

It is less than the length 15 in of the longest side.

Hence, the triangle (3,6,15) actually doesn't exist.

It implies that the problem HAS NO solution.

Answer.  The problem HAS NO solution.  Such a triangle doesn't exist.