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Example 5.58 from Elementary Algebra, 5.6 Graphing Systems of Linear Inequalities
Graphing Systems of Linear Inequalities: example 5.58
Omar needs to eat at least 800 calories before going to his team practice. All he wants is hamburgers and cookies, and he doesn’t want to spend more than $5. At the hamburger restaurant near his college, each hamburger has 240 calories and costs $1.40. Each cookie has 160 calories and costs $0.50.
- ⓐ Write a system of inequalities to model this situation.
- ⓑ Graph the system.
- ⓒ Could he eat 3 hamburgers and 1 cookie?
- ⓓ Could he eat 2 hamburgers and 4 cookies?
Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.
Solution
ⓐ Let the number of hamburgers.
the number of cookies
To find the system of inequalities, translate the information.
The calories from hamburgers at 240 calories each, plus the calories from cookies at 160 calories each must be more that 800.The amount spent on hamburgers at $1.40 each, plus the amount spent on cookies at $0.50 each must be no more than $5.00.
We have our system of inequalities.
ⓑ
To graph graph as a solid line.
Choose (0, 0) as a test point. it does not make the inequality true.
So, shade (red) the side that does not include the point (0, 0).
To graph , graph as a solid line.
Choose (0,0) as a test point. It makes the inequality true. So, shade
(blue) the side that includes the point.
The solution of the system is the region of the graph that is double shaded and so is shaded darker.ⓒ To determine if 3 hamburgers and 2 cookies would meet Omar’s criteria, we see if the point (3, 1) is in the solution region. It is. He might choose to eat 3 hamburgers and 2 cookies.
ⓓ To determine if 2 hamburgers and 4 cookies would meet Omar’s criteria, we see if the point (2, 4) is in the solution region. It is. He might choose to eat 2 hamburgers and 4 cookies.We could also test the possible solutions by substituting the values into each inequality.
Your turn
The book's Try It right after this example: same idea, new numbers. Only the answer is given.
Tension needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only $25 to spend on the extra food he needs and will spend it on $0.75 donuts which have 360 calories each and $2 energy drinks which have 110 calories.
ⓐ Write a system of inequalities that models this situation.
ⓑ Graph the system.
ⓒ Can he buy 8 donuts and 4 energy drinks?
ⓓ Can he buy 1 donut and 3 energy drinks?
How did it go?