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The problem you uploaded describes a situation where a furniture manufacturing company produces tables and chairs, each requiring different amounts of labor for assembly and finishing. We need to set up a system of linear inequalities to model the constraints on labor hours for assembly and finishing.

Here's the relevant information:

- Each table requires:
  - 10 labor-hours for assembly
  - 3 labor-hours for finishing

- Each chair requires:
  - 9 labor-hours for assembly
  - 1 labor-hour for finishing

- Maximum labor-hours available:
  - 400 labor-hours for assembly
  - 140 labor-hours for finishing

We will use the variables:
- \( x \): the number of tables produced in a day
- \( y \): the number of chairs produced in a day

### Assembly constraint:
- Each table takes 10 labor-hours and each chair takes 9 labor-hours for assembly.
- The total available labor-hours for assembly is 400.
- The inequality for assembly is:
  \[
  10x + 9y \leq 400
  \]

### Finishing constraint:
- Each table takes 3 labor-hours and each chair takes 1 labor-hour for finishing.
- The total available labor-hours for finishing is 140.
- The inequality for finishing is:
  \[
  3x + y \leq 140
  \]

### Non-negativity constraints:
Since the number of tables and chairs cannot be negative, we also have:
\[
x \geq 0 \quad \text{and} \quad y \geq 0
\]

### System of inequalities:
\[
\begin{cases}
10x + 9y \leq 400 & \text{(assembly constraint)} \\
3x + y \leq 140 & \text{(finishing constraint)} \\
x \geq 0 & \text{(non-negativity)} \\
y \geq 0 & \text{(non-negativity)}
\end{cases}
\]

Would you like more details on how to graph these inequalities or solve this system? Here are some related questions:

1. How do we solve this system of inequalities graphically?
2. What is the feasible region in terms of the number of tables and chairs?
3. How can we determine the optimal number of tables and chairs to maximize production?
4. Can we use a method like linear programming to optimize production?
5. How would the constraints change if labor hours were different?

**Tip:** When dealing with systems of inequalities, it helps to plot the constraints on a graph to visually identify the feasible region.

A furniture manufacturing company manufactures dining-room tables and chairs. A table requires 10 labor-hours for assembly and 3 labor-hours for finishing. A chair requires 9 labor-hours for assembly and 1 labor-hour for finishing. The maximum labor hours available per day for assembly and finishing are 400 and 140, respectively. Write a system of linear inequalities that represents this situation. Use the following variables: Use x for the number of tables and y for the number of chairs manufactured in a day.

Explore how to set up and solve a system of linear inequalities for a furniture manufacturing problem. This problem includes assembly and finishing labor constraints, represented by the inequalities 10x + 9y ≤ 400 and 3x + y ≤ 140. Suitable for high school algebra students.

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