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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?

Solution

Let h = the number of hamburgers.
c = 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.

240 h + 160 c 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.

1.40 h + 0.50 c 5

We have our system of inequalities. { 240 h + 160 c 800 1.40 h + 0.50 c 5


To graph 240 h + 160 c 800 graph 240 h + 160 c = 800 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 1.40 h + 0.50 c 5 , graph 1.40 h + 0.50 c = 5 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, 2) 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 not. Omar would not choose to eat 2 hamburgers and 4 cookies.

We could also test the possible solutions by substituting the values into each inequality.

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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?

  1. { 0.75 d + 2 e 25 360 d + 110 e 1000

  2. This figure shows a graph on an x y-coordinate plane of 0.75d + 2e is less than or equal to 25 and 360d + 110e is greater than or equal to 1000. The area to the left or right of each line is shaded slightly different colors with the overlapping area also shaded a slightly different color.
  3. yes
  4. no
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Philip’s doctor tells him he should add at least 1000 more calories per day to his usual diet. Philip wants to buy protein bars that cost $1.80 each and have 140 calories and juice that costs $1.25 per bottle and have 125 calories. He doesn’t want to spend more than $12.

Write a system of inequalities that models this situation.
Graph the system.
Can he buy 3 protein bars and 5 bottles of juice?
Can he buy 5 protein bars and 3 bottles of juice?

  1. { 140 p + 125 j 1000 1.80 p + 1.25 j 12

  2. This figure shows a graph on an x y-coordinate plane of 140p + 125j is greater than or equal to 1000 and 1.80p + 1.25j is less than or equal to 12. The area to the left or right of each line is shaded slightly different colors with the overlapping area also shaded a slightly different color.
  3. yes
  4. no
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Access these online resources for additional instruction and practice with graphing systems of linear inequalities.

Key concepts

  • To Solve a System of Linear Inequalities by Graphing
    1. Graph the first inequality.
      • Graph the boundary line.
      • Shade in the side of the boundary line where the inequality is true.
    2. On the same grid, graph the second inequality.
      • Graph the boundary line.
      • Shade in the side of that boundary line where the inequality is true.
    3. The solution is the region where the shading overlaps.
    4. Check by choosing a test point.

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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