2.15.3 Solving Problems to Satisfy Multiple Constraints Simultaneously

Solve Applications of Systems of Inequalities

When we use variables other than $x$ and $y$ to define an unknown quantity, we must change the names of the axes of the graph as well.

Example

Use the following information to answer 1 - 4.

Omar needs to eat at least 800 calories before going to his team practice. All he wants is hamburgers, $h$, and cookies, $c$, 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.

Solution

Let $h$ equal the number of hamburgers.

Let $c$ equal the number of cookies.

To find the system of equations, translate the information.

• The calories from hamburgers at 240 calories each, plus the calories from cookies at 160 calories each must be 800 or more. $240h+160c\ge 800$
• 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. $1.40h+0.50c\le 5$
• The number of hamburgers must be greater than or equal to 0. $h\ge 0$
• The number of cookies must be greater than or equal to 0. $c\ge 0$
• We have our system of equations. $\{\begin{array}{c}240h+160c\ge 800\hfill \\ 1.40h+0.50c\le 5\hfill \\ h\ge 0\hfill \\ c\ge 0\hfill \end{array}$

2. Graph the system.

Solution

Since $h\ge 0$ and $c\ge 0$ all solutions will be in the first quadrant. As a result, our graph shows only quadrant one.

• To graph $240h+160c\ge 80$, graph $240h+160c=800$ as a solid line. Choose $(0,0)$ as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point $(0,0)$.
• Graph $1.40h+0.50c\le 5$. The boundary line is $1.40h+0.50c=5$. We test $(0,0)$ and it makes the inequality true. We shade the side of the line that includes $(0,0)$.
• The solution of the system is the region of the graph that is shaded the darkest. The boundary line sections that border the darkly shaded section are included in the solution as are the points on the $x$-axis from $(5,0)$ to $(10,0)$.

3. Could he eat 3 hamburgers and 1 cookie?

Solution

To determine if 3 hamburgers and 1 cookie would meet Omar’s criteria, we see if the point $(3,2)$ is in the solution region. It is, so Omar might choose to eat 3 hamburgers and 1 cookie.

4. Could he eat 2 hamburgers and 4 cookies?

Solution

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, so Omar 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

Solve Applications of Systems of Inequalities

Use the following information for 1 - 4.

Philip’s doctor tells him he should add at least 1,000 more calories per day to his usual diet. Philip wants to buy protein bars, $p$, that cost $1.80 each and have 140 calories and juice, $j$, that costs $1.25 per bottle and has 125 calories. He doesn’t want to spend more than $12.

1.

Write a system of inequalities that models this situation.

Compare your answer:

$\{\begin{array}{c}140p+125j\ge 1000\hfill \\ 1.80p+1.25j\le 12\hfill \end{array}$

2.

Use the graphing tool or technology outside the course. Graph the system of inequalities from question 1 using the Desmos tool below.

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Compare your answer:

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3.

Can he buy 3 protein bars and 5 bottles of juice?

Yes.

4.

Can he buy 5 protein bars and 3 bottles of juice?

No.