Graphing Systems of Linear Inequalities

Example 5.57

Christy sells her photographs at a booth at a street fair. At the start of the day, she wants to have at least 25 photos to display at her booth. Each small photo she displays costs her $4 and each large photo costs her $10. She doesn’t want to spend more than $200 on photos to display.

1. Write a system of inequalities to model this situation.
2. Graph the system.
3. Could she display 15 small and 5 large photos?
4. Could she display 3 large and 22 small photos?

Solution

1. Let \(x=\) the number of small photos.

   \(y=\) the number of large photos

   To find the system of inequalities, translate the information.

   \(\text{She wants to have at least 25 photos. The number of small plus the number of large should be at least 25.}\)

   \( x+y \geq 25 \)

   \(\text{\$4 for each small and \$10 for each large must be no more than \$200}\)

   \(4 x+10 y \leq 200 \)

   We have our system of inequalities. \(\left\{\begin{array}{l}x+y \geq 25 \\ 4 x+10 y \leq 200\end{array}\right.\)

2. To graph \(x+y \geq 25\), graph \(x+y=25\) as a solid line. Choose \((0,0)\) as a test point. Since it does not make the inequality true, shade the side that does not include the point \((0,0)\) red. To graph \(4 x+10 y \leq 200\), graph \(4 x+10 y=200\) as a solid line. Choose \((0,0)\) as a test point. Since it does not make the inequality true, shade the side that includes the point \((0,0)\) blue.

   
   The solution of the system is the region of the graph that is double shaded and so is shaded darker.

3. To determine if 10 small and 20 large photos would work, we see if the point (10, 20) is in the solution region. It is not. Christy would not display 10 small and 20 large photos.
4. To determine if 20 small and 10 large photos would work, we see if the point (20, 10) is in the solution region. It is. Christy could choose to display 20 small and 10 large photos.

Notice that we could also test the possible solutions by substituting the values into each inequality.

Try It 5.113

A trailer can carry a maximum weight of 160 pounds and a maximum volume of 15 cubic feet. A microwave oven weighs 30 pounds and has 2 cubic feet of volume, while a printer weighs 20 pounds and has 3 cubic feet of space.

1. Write a system of inequalities to model this situation.
2. Graph the system.
3. Could 4 microwaves and 2 printers be carried on this trailer?
4. Could 7 microwaves and 3 printers be carried on this trailer?

Try It 5.114

Mary needs to purchase supplies of answer sheets and pencils for a standardized test to be given to the juniors at her high school. The number of the answer sheets needed is at least 5 more than the number of pencils. The pencils cost $2 and the answer sheets cost $1. Mary’s budget for these supplies allows for a maximum cost of $400.

1. Write a system of inequalities to model this situation.
2. Graph the system.
3. Could Mary purchase 100 pencils and 100 answer sheets?
4. Could Mary purchase 150 pencils and 150 answer sheets?

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.

1. Write a system of inequalities to model this situation.
2. Graph the system.
3. Could he eat 3 hamburgers and 1 cookie?
4. Could he eat 2 hamburgers and 4 cookies?

Solution

1. 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.

   \(240 h+160 c \geq 800\)

   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.

   \(1.40 h+0.50 c \leq 5\)

   We have our system of inequalities. \(\left\{\begin{array}{l}240 h+160 c \geq 800 \\ 1.40 h+0.50 c \leq 5\end{array}\right.\)

2. To graph \(240 h+160 c \geq 800\) 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 \(1.40 h+0.50 c \leq 5\), 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.

3. 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.
4. 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.

Try It 5.115

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.

1. Write a system of inequalities that models this situation.
2. Graph the system.
3. Can he buy 8 donuts and 4 energy drinks?
4. Can he buy 1 donut and 3 energy drinks?

Try It 5.116

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.

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