1Cademy - Solving a Diet Calorie and Budget Problem Using a System of Inequalities

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Solving Applications of Systems of Linear Inequalities

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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 $140$ calories each) and juice (costing $1.25 per bottle and having $125$ calories each). He does not want to spend more than $12.

ⓐ Set up the system. Let $p$ = the number of protein bars and $j$ = the number of bottles of juice. Translating the two constraints:

"At least $1{,}000$ calories" → $140p + 125j \geq 1{,}000$
"No more than $12" → $1.80p + 1.25j \leq 12$

Because quantities cannot be negative, we also have $p \geq 0$ and $j \geq 0$ . The system is: $\left\{\begin{array}{l} 140p + 125j \geq 1{,}000 \\\\ 1.80p + 1.25j \leq 12 \\\\ p \geq 0 \\\\ j \geq 0 \end{array}\right.$

ⓑ Graph the system. Graph $140p + 125j = 1{,}000$ as a solid boundary line. Testing $(0, 0)$ : $0 \geq 1{,}000$ is false, so shade the side away from the origin. Graph $1.80p + 1.25j = 12$ as a solid boundary line. Testing $(0, 0)$ : $0 \leq 12$ is true, so shade the side containing the origin. The solution is the doubly-shaded region in the first quadrant.

Calories: $140(3) + 125(5) = 420 + 625 = 1{,}045 \geq 1{,}000$ (True)
Budget: $1.80(3) + 1.25(5) = 5.40 + 6.25 = 11.65 \leq 12$ (True) Since both constraints are met, he can buy $3$ protein bars and $5$ bottles of juice.

ⓓ To determine if $5$ protein bars and $3$ bottles of juice satisfy the needs, we test the point $(5, 3)$ :

Calories: $140(5) + 125(3) = 700 + 375 = 1{,}075 \geq 1{,}000$ (True)
Budget: $1.80(5) + 1.25(3) = 9.00 + 3.75 = 12.75 \leq 12$ (False) Since the budget constraint is not met, he cannot buy $5$ protein bars and $3$ bottles of juice.

Updated 2026-04-29

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OpenStax Intermediate Algebra

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