1. **State the problem:** We have a patio represented as a rectangle with length $l$ and width $w$. The patio is enclosed on three sides by a fence, and the fourth side is the house. We want to analyze the constraints on $l$ and $w$ given by the fencing and area requirements. 2. **Identify the fencing constraint:** Since the patio is enclosed on three sides, the fencing length is the sum of one length side and two width sides. This gives the fencing constraint: $$l + 2w \leq 60$$ 3. **Identify the area constraint:** The area of the patio is given by: $$lw \leq 400$$ 4. **Explain the inequalities:** - The fencing length $l + 2w$ must be less than or equal to 60 feet. - The area $lw$ must be less than or equal to 400 square feet. 5. **Summary of constraints:** $$\begin{cases} l + 2w \leq 60 \\ lw \leq 400 \end{cases}$$ 6. **Interpretation:** These inequalities define the feasible region for the patio dimensions $l$ and $w$. 7. **Desmos graph function:** The fencing constraint can be rearranged to express $l$ in terms of $w$: $$l = 60 - 2w$$ The area constraint can be expressed as: $$l = \frac{400}{w}$$ These two functions can be graphed to find the feasible region. **Final answer:** The patio dimensions must satisfy: $$l + 2w \leq 60$$ $$lw \leq 400$$