https://file.aiquickdraw.com/mathbot/file/1728985136463l8ldecyf.jpg

The problem is asking to model a system of inequalities based on the amount of coffee available and the amount used in two types of blends.

Let's define the variables based on the problem:

- \( x \): the number of deluxe blend packages sold.
- \( y \): the number of special blend packages sold.

### Deluxe Blend:
- Each deluxe blend package contains 4 ounces of A grade coffee and 8 ounces of B grade coffee.

### Special Blend:
- Each special blend package contains 7 ounces of A grade coffee and 3 ounces of B grade coffee.

### Available Coffee:
- There are 700 ounces of A grade coffee available.
- There are 600 ounces of B grade coffee available.

### Constraints

**A grade coffee constraint**:  
For \( x \) deluxe packages, \( 4x \) ounces of A grade coffee are used.  
For \( y \) special packages, \( 7y \) ounces of A grade coffee are used.  
The total amount of A grade coffee used must be less than or equal to 700 ounces:
\[
4x + 7y \leq 700
\]

**B grade coffee constraint**:  
For \( x \) deluxe packages, \( 8x \) ounces of B grade coffee are used.  
For \( y \) special packages, \( 3y \) ounces of B grade coffee are used.  
The total amount of B grade coffee used must be less than or equal to 600 ounces:
\[
8x + 3y \leq 600
\]

### Non-negativity constraints:
Both \( x \) and \( y \) must be non-negative because they represent the number of packages:
\[
x \geq 0, \quad y \geq 0
\]

### System of inequalities:
The complete system of inequalities is:
\[
\begin{aligned}
    4x + 7y &\leq 700 \\
    8x + 3y &\leq 600 \\
    x &\geq 0 \\
    y &\geq 0
\end{aligned}
\]

To solve this, we need to graph the inequalities and shade the region that satisfies all the constraints. 

Would you like me to graph these inequalities or provide further explanations about the solution? Let me know if you'd like details on the graphical solution! 

### Follow-up questions:
1. How can we interpret the shaded region in terms of package combinations?
2. What happens if the availability of A or B grade coffee changes?
3. Can you derive the corner points of the solution region algebraically?
4. What is the maximum number of deluxe blend packages that can be sold?
5. What is the maximum number of special blend packages that can be sold?

**Tip**: When solving inequalities, remember to check for boundary points to identify the feasible region.

For a given week, Shen's Coffee House has available 700 ounces of A grade coffee and 600 ounces of B grade coffee. These are blended into packages as follows: a deluxe blend that contains 4 ounces of A grade coffee and 8 ounces of B grade coffee, and a special blend that contains 7 ounces of A grade coffee and 3 ounces of B grade coffee. Let x be the number of deluxe blend packages sold. Let y be the number of special blend packages sold. Shade the region corresponding to all values of x and y that satisfy these requirements.

Solve the problem of determining how many deluxe and special blend packages Shen's Coffee House can sell given their coffee stock. This example involves graphing inequalities based on coffee constraints and solving for the feasible region. Suitable for students in Grades 10-12 studying systems of inequalities.

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