Solving a Diet Calorie and Budget Problem Using a System of Inequalities
Problem: Philip's doctor tells him he should add at least 1{,}000 more calories per day to his usual diet. Philip wants to buy protein bars (costing $1.80 and having calories each) and juice (costing $1.25 per bottle and having calories each). He does not want to spend more than $12.
ⓐ Set up the system. Let = the number of protein bars and = the number of bottles of juice. Translating the two constraints:
- "At least 1{,}000 calories" →
- "No more than $12" →
Because quantities cannot be negative, we also have and . The system is:
ⓑ Graph the system. Graph as a solid boundary line. Testing (0, 0): is false, so shade the side away from the origin. Graph as a solid boundary line. Testing (0, 0): is true, so shade the side containing the origin. The solution is the doubly-shaded region in the first quadrant.
ⓒ To determine if protein bars and bottles of juice satisfy the needs, we test the point (3, 5) in the inequalities:
- Calories: (True)
- Budget: (True) Since both constraints are met, he can buy protein bars and bottles of juice.
ⓓ To determine if protein bars and bottles of juice satisfy the needs, we test the point (5, 3):
- Calories: (True)
- Budget: (False) Since the budget constraint is not met, he cannot buy protein bars and bottles of juice.
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Intermediate Algebra @ OpenStax
Ch.4 Systems of Linear Equations - Intermediate Algebra @ OpenStax
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