We are given the following information about the production of a single product on two lines, A and B:

• Line A requires 1 hour of labor per item.
• Line B requires 3 hours of labor per item.
• The company has 800 hours of labor available each week.

Part (a): Writing the inequality that expresses the labor information.

Let:

• $x$ be the number of items produced on line A.
• $y$ be the number of items produced on line B.

The total labor used is $1 \cdot x$ hours for line A and $3 \cdot y$ hours for line B. Since the total labor must not exceed 800 hours, the inequality is:

$x + 3y \leq 800$

Part (b): Writing the inequality that expresses the cost information.

We are also told that:

• The cost per item on line A is $5.
• The cost per item on line B is $4.
• The company can spend up to $2600 on operating costs per week.

The total cost is $5 \cdot x$ for items produced on line A and $4 \cdot y$ for items produced on line B. Since the total cost must not exceed $2600, the inequality is:

$5x + 4y \leq 2600$

Summary:

(a) Labor inequality: $x + 3y \leq 800$

(b) Cost inequality: $5x + 4y \leq 2600$

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Would you like me to help solve this system of inequalities, or do you have any further questions on how these inequalities were formed?

Here are 5 related questions that might be useful:

1. How do we graph the inequalities to visualize the solution region?
2. Can the company produce an equal number of items on both lines? What constraints apply?
3. What happens if the company needs to prioritize one line over the other?
4. How can we interpret the system in terms of optimization (e.g., maximizing production)?
5. Can the company increase labor hours or costs without violating any constraints?

Tip: When dealing with systems of inequalities, plotting them on a graph can help identify feasible solutions and clarify the relationships between the constraints.