In order to help return the struggling furniture business to profitability using linear programming, we need to translate the given statements into equations or inequalities:
1. **Time Constraints:**
- Dressers take 5 hours to build, and tables take 3 hours to build.
- Let \( x \) be the number of dressers and \( y \) be the number of tables.
- The total number of hours must be less than or equal to 400:
\( 5x + 3y \leq 400 \)
2. **Profit Calculation:**
- Dressers sell for $325 in profit, and tables sell for $125 in profit.
- The total profit is given by \( 325x + 125y \).
3. **Cost Constraints:**
- Dressers cost $100 and tables cost $75 for materials manufacture.
- The total materials cost must not be more than $9,750:
\( 100x + 75y \leq 9750 \)
By setting up and solving these equations and inequalities simultaneously, the business can optimize the production of dressers and tables to maximize profit while staying within the constraints of time and material costs. This process will involve graphing these constraints on a graph to identify the feasible region and find the optimal solution for the business to achieve maximum profitability.