Question

Martha has a $75 gift certificate to spend at an online comic book store. Including tax, the premium comics cost $8 each and the regular comics cost $5 each. Martha wants to buy at least 3 premium comics.
Graph a system of inequalities that represents this story. One inequality should represent Martha's budget constraint, and the other inequality should represent the number of premium comics she buys.

Answer 1

To represent Martha's situation with inequalities, let's define the variables:

- Let \( x \) be the number of premium comics Martha buys.
- Let \( y \) be the number of regular comics Martha buys.

### Step 1: Establish the inequalities

1. **Budget Constraint**:
Martha has a $75 gift certificate, and the cost of premium comics is $8 each while regular comics cost $5 each. The budget constraint can be written as:
\[
8x + 5y \leq 75
\]

2. **Premium Comics Requirement**:
Martha wants to buy at least 3 premium comics, which can be expressed as:
\[
x \geq 3
\]

### Step 2: Non-negativity constraints
Since Martha cannot buy a negative number of comics, we also need to state that:
\[
x \geq 0 \quad \text{and} \quad y \geq 0
\]

### Step 3: Summary of inequalities
Putting it all together, our system of inequalities is:
1. \( 8x + 5y \leq 75 \) (Budget constraint)
2. \( x \geq 3 \) (Requirement for premium comics)
3. \( x \geq 0 \) (Non-negativity for premium comics)
4. \( y \geq 0 \) (Non-negativity for regular comics)

### Step 4: Graphing the inequalities

To graph these inequalities:

1. **Graph the budget constraint** \( 8x + 5y = 75 \):
- Find the intercepts:
- When \( x = 0 \): \( 5y = 75 \Rightarrow y = 15 \) (y-intercept)
- When \( y = 0 \): \( 8x = 75 \Rightarrow x = 9.375 \) (x-intercept)

These points, \( (0, 15) \) and \( (9.375, 0) \), can be plotted on the graph, and the area below this line represents the inequality \( 8x + 5y \leq 75 \).

2. **Graph the premium comics requirement** \( x \geq 3 \):
- This is a vertical line at \( x = 3 \) and includes the area to the right of this line.

3. **Non-negativity constraints**:
- These constraints ensure that both \( x \) and \( y \) are plotted in the first quadrant, where \( x \geq 0 \) and \( y \geq 0 \).

### Final Graph
In summary, the solution region for this system will be where the area below the budget line intersects with the area to the right of the line \( x = 3 \) within the first quadrant.