The bookstore hopes to sell at least 30 binders and calculators each week. The store also hopes to
have sales revenue of at least $200 in binders and calculators.
a) How many binders and calculators could be sold to meet both of these sales goals if binders
cost $3.65 a piece and calculators cost $14.80 a piece?
b) Is it a unique solution? If not, use graph to shade the region of all possible solutions

Let's tackle the problem in two parts.

### Part (a) – Defining the constraints and solving the system of inequalities

We are given the following information:
- The bookstore wants to sell at least 30 binders and calculators each week.
- The bookstore wants to have sales revenue of at least $200 from binders and calculators.
- The cost of each binder is $3.65, and the cost of each calculator is $14.80.

Let:
- $ x $ = number of binders sold.
- $ y $ = number of calculators sold.

We now form the two constraints:

1. The store wants to sell at least 30 items (binders + calculators):
\[
x + y \geq 30
\]

2. The store wants to make at least $200 in revenue:
\[
3.65x + 14.80y \geq 200
\]

These two inequalities represent the problem mathematically.

### Part (b) – Graphing the inequalities

We need to graph the system of inequalities to find all possible solutions.

The inequalities are:
1. $ x + y \geq 30 $
2. $ 3.65x + 14.80y \geq 200 $

#### To graph these inequalities:
- First, solve each inequality for $ y $ in terms of $ x $.
  
**For the first inequality $ x + y \geq 30 $:**
\[
y \geq 30 - x
\]

This is a straight line with slope -1 and y-intercept 30. We will shade above this line because the inequality is $ y \geq 30 - x $.

**For the second inequality $ 3.65x + 14.80y \geq 200 $:**
\[
14.80y \geq 200 - 3.65x
\]
\[
y \geq \frac{200 - 3.65x}{14.80}
\]

This is another line with slope $ -\frac{3.65}{14.80} $ and y-intercept $ \frac{200}{14.80} \approx 13.51 $. We will shade above this line as well.

### Graphing Approach
We would graph both inequalities on the coordinate plane. The feasible region is the area where the shading overlaps, which gives all the possible combinations of binders and calculators that meet both sales goals.

### Is it a unique solution?
No, it is not a unique solution because we have two inequalities that describe a region of possible values rather than a single point. Any combination of $ x $ and $ y $ within the overlapping shaded region on the graph will satisfy both constraints.

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Would you like me to plot this graph for better visualization or dive deeper into any specific part of the problem?

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Here are five questions to expand on the problem:
1. What are the corner points of the feasible region in this problem?
2. What happens if the prices of binders and calculators change?
3. How would the solution change if the total revenue goal increases to $300?
4. How can we find the optimal number of binders and calculators to sell for maximum revenue?
5. What if the bookstore has a limit of 40 total items they can sell each week?

**Tip:** When solving systems of inequalities, always find where the constraints overlap to identify all possible solutions!

The bookstore hopes to sell at least 30 binders and calculators each week. The store also hopes to have sales revenue of at least $200 in binders and calculators. a) How many binders and calculators could be sold to meet both of these sales goals if binders cost $3.65 a piece and calculators cost $14.80 a piece? b) Is it a unique solution? If not, use graph to shade the region of all possible solutions.

Solve a system of inequalities to determine how many binders and calculators can be sold to meet the bookstore's sales and revenue goals. The problem involves key concepts such as linear inequalities and revenue calculations. Suitable for students in Grades 9-11.

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