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
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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: \[ 8x + 5y \leq 75 \]
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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:
- \( 8x + 5y \leq 75 \) (Budget constraint)
- \( x \geq 3 \) (Requirement for premium comics)
- \( x \geq 0 \) (Non-negativity for premium comics)
- \( y \geq 0 \) (Non-negativity for regular comics)
Step 4: Graphing the inequalities
To graph these inequalities:
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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 \( 8x + 5y \leq 75 \).
- Find the intercepts:
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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.
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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.