Let's analyze the problem and the system of inequalities based on the image.

Problem Summary:

A manufacturing company produces two models of entertainment centers:

• Model A requires 6 feet of molding and takes 2 hours to manufacture.
• Model B requires 12 feet of molding and takes 6 hours to manufacture.

Given constraints:

• The company has 360 feet of molding available each week.
• The company has 162 hours of labor available each week.

Inequalities:

Let:

• $x$ = units of Model A produced
• $y$ = units of Model B produced

The problem provides the following system of inequalities:

1. Molding constraint: $6x + 12y \leq 360$ This inequality represents the total molding used by both models, which must not exceed 360 feet.

2. Labor constraint: $2x + 6y \leq 162$ This inequality represents the total labor hours required to produce both models, which must not exceed 162 hours.

These inequalities need to be graphed on the coordinate plane where $x$ (units of Model A) is on the horizontal axis and $y$ (units of Model B) is on the vertical axis.

Steps for Graphing:

1. Solve each inequality for $y$ to convert them into slope-intercept form:

   • From $6x + 12y \leq 360$: $y \leq \frac{-x}{2} + 30$
   • From $2x + 6y \leq 162$: $y \leq \frac{-x}{3} + 27$

2. Graph these two lines on the coordinate plane and shade the region below each line, as the inequalities are "less than or equal to."

Would you like me to explain the graphing in more detail or proceed with plotting the points and lines for the solution set?

Related Questions:

1. How would the solution set change if the available hours were reduced to 120?
2. What happens to the inequalities if Model A starts requiring 8 feet of molding?
3. How would you interpret the region where the two inequalities overlap?
4. Can the company produce 40 units of Model A under the current constraints?
5. How can you verify if a specific production plan (e.g., 10 units of Model A and 15 units of Model B) is feasible?

Tip:

When graphing systems of inequalities, always check the boundary lines first and test a point to determine which region to shade.