Final answer:
The inequality representing all possible values of the total length of rope needed for the hot air balloon is 420 ≤ x ≤ 600 inches.
Step-by-step explanation:
The total length of rope needed for the hot air balloon can be found by multiplying the number of pieces of rope by the minimum length of each piece. In this case, there are 12 separate pieces of rope and each piece must be at least 35 inches long. Therefore, the minimum total length of rope needed is 12 * 35 = 420 inches.
Similarly, the maximum total length of rope needed can be found by multiplying the number of pieces of rope by the maximum length of each piece. In this case, each piece must be no more than 50 inches long. Therefore, the maximum total length of rope needed is 12 * 50 = 600 inches.
To summarize, the inequality representing all possible values of the total length of rope x, in inches, needed for the hot air balloon is 420 ≤ x ≤ 600.