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

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Solving a System of Linear Inequalities by Graphing

Example

Solving a Calorie and Budget Problem Using a System of Inequalities

Problem: Omar needs at least 800 calories before team practice. He wants only hamburgers and cookies, and he cannot spend more than $5. Each hamburger has 240 calories and costs $1.40; each cookie has 160 calories and costs $0.50.

ⓐ Set up the system. Let $h$ = the number of hamburgers and $c$ = the number of cookies. Translating the two constraints:

"At least 800 calories" → $240h + 160c \geq 800$
"No more than $5" → $1.40h + 0.50c \leq 5$

The system is: $\left\{\begin{array}{l} 240h + 160c \geq 800 \\ 1.40h + 0.50c \leq 5 \end{array}\right.$

ⓑ Graph the system. Graph $240h + 160c = 800$ as a solid boundary line. Testing $(0, 0)$ : $0 \geq 800$ is false, so shade the side away from the origin. Graph $1.40h + 0.50c = 5$ as a solid boundary line. Testing $(0, 0)$ : $0 \leq 5$ is true, so shade the side containing the origin. The solution is the doubly-shaded overlap region.