Let's break down and solve the problem using an inequality.

Problem Summary:

• The length of the rectangle is 4 yards less than 5 times its width.
• The perimeter of the rectangle must be between 52 and 112 yards.
• We are tasked with finding the range of the width.

Step 1: Define Variables

Let:

• $w$ be the width of the rectangle (in yards).
• The length, $l$, is given as $l = 5w - 4$ (since it's 4 yards less than 5 times the width).

Step 2: Perimeter Formula

The perimeter $P$ of a rectangle is given by: $P = 2(l + w)$ Substitute $l = 5w - 4$ into the formula: $P = 2((5w - 4) + w) = 2(6w - 4) = 12w - 8$

Step 3: Set Up the Inequality

The perimeter must be between 52 and 112 yards, so we set up the inequality: $52 \leq 12w - 8 \leq 112$

Step 4: Solve the Inequality

Solve for $w$:

1. Add 8 to all parts of the inequality: $52 + 8 \leq 12w \leq 112 + 8$ $60 \leq 12w \leq 120$

2. Divide the entire inequality by 12: $\frac{60}{12} \leq w \leq \frac{120}{12}$ $5 \leq w \leq 10$

Solution:

The width of the rectangle must be between 5 yards and 10 yards.

Would you like a detailed explanation of any step?
Here are five related questions to further expand on this:

1. How would the range change if the perimeter was different?
2. What happens to the length when the width changes within this range?
3. How would you solve this if the relationship between the length and width was different?
4. Can you derive the inequality if the perimeter formula was unknown?
5. What other properties of the rectangle can you explore with the given range?

Tip: When solving inequalities, always remember to adjust all parts of the inequality simultaneously to maintain balance!