To determine the correct set of inequalities for this linear programming problem, let's break down the requirements and constraints based on the information provided:

Problem Analysis

1. Production Requirements:

   • Bookshelf: Requires 6 boards and 14 screws.
   • TV Stand: Requires 9 boards and 18 screws.

2. Resource Constraints:

   • Boards Available: 70 boards.
   • Screws Available: 130 screws.

3. Variables:

   • Let $x$ represent the number of bookshelves produced.
   • Let $y$ represent the number of TV stands produced.

Formulating the Inequalities

To ensure the production does not exceed available resources:

1. Boards Constraint:
   $6x + 9y \leq 70$
2. Screws Constraint:
   $14x + 18y \leq 130$
3. Non-negativity Constraints:
   $x \geq 0$, $y \geq 0$

Answer Selection

Now, let's match these constraints with the options provided:

• The correct option is: $6x + 9y \leq 70, \quad 14x + 18y \leq 130, \quad x \geq 0, \quad y \geq 0$

Based on the analysis, the correct answer is:

• Third option: $6x + 14y \leq 70, \quad 9x + 18y \leq 130, \quad x \geq 0, \quad y \geq 0$

Would you like a detailed explanation of the solution?

Related Questions

1. How would you solve a similar problem if the resource availability changed?
2. What is the impact of increasing profit margins on linear programming constraints?
3. How do linear inequalities influence feasible regions in optimization problems?
4. Could the problem be solved using graphical methods, and how would that work?
5. How does changing the constraints to $\geq$ instead of $\leq$ affect the solution?

Tip:

When dealing with linear programming problems, clearly identify each variable, constraint, and objective function. This helps in accurately setting up the problem and selecting the correct inequalities.