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

Problem: Tenison needs to eat at least an extra $1{,}000$ calories a day to prepare for a marathon. He has only $25 to spend on the extra food and will spend it on donuts (costing $0.75 and having $360$ calories each) and energy drinks (costing $2 and having $110$ calories each).

ⓐ Set up the system. Let $d$ = the number of donuts and $e$ = the number of energy drinks. Translating the two constraints:

"At least $1{,}000$ calories" → $360d + 110e \geq 1{,}000$
"No more than $25" → $0.75d + 2e \leq 25$

Because quantities cannot be negative, we also have $d \geq 0$ and $e \geq 0$ . The system is: $\left\{\begin{array}{l} 360d + 110e \geq 1{,}000 \\\\ 0.75d + 2e \leq 25 \\\\ d \geq 0 \\\\ e \geq 0 \end{array}\right.$

ⓑ Graph the system. Graph $360d + 110e = 1{,}000$ as a solid boundary line. Testing $(0, 0)$ : $0 \geq 1{,}000$ is false, so shade the side away from the origin. Graph $0.75d + 2e = 25$ as a solid boundary line. Testing $(0, 0)$ : $0 \leq 25$ is true, so shade the side containing the origin. The solution is the doubly-shaded region in the first quadrant.

Calories: $360(8) + 110(4) = 2{,}880 + 440 = 3{,}320 \geq 1{,}000$ (True)
Budget: $0.75(8) + 2(4) = 6 + 8 = 14 \leq 25$ (True) Since both constraints are met, he can buy $8$ donuts and $4$ energy drinks.

ⓓ To determine if $1$ donut and $3$ energy drinks satisfy the needs, we test the point $(1, 3)$ :

Calories: $360(1) + 110(3) = 360 + 330 = 690 \geq 1{,}000$ (False) Since the calorie constraint is not met, he cannot buy $1$ donut and $3$ energy drinks.

Updated 2026-04-29

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OpenStax Intermediate Algebra

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