Final answer:
To find the maximum length of the longest side of a triangle with sides in the ratio 5:6:7 and a perimeter less than 54 cm, the inequality 7x < 21 cm reveals that this side should be less than 21 cm.
Step-by-step explanation:
To write the inequality for the lengths of the sides of a triangle with a perimeter less than 54 cm, we can first denote the lengths of the triangle's sides as 5x, 6x, and 7x, where x is a common factor. The ratios reflect the proportional relationships between the sides. Since the perimeter is the sum of all sides, we create the inequality:
5x + 6x + 7x < 54
This simplifies to:
18x < 54
Dividing both sides of the inequality by 18 gives us:
x < 3
The length of the longest side will be 7 times x, so we write the inequality for the longest side as:
7x < 21
Therefore, the length of the longest side must be less than 21 cm if the perimeter is to remain less than 54 cm.