2.14.3 Writing Systems of Inequalities that Represent Situations

Write a System of Inequalities for a Situation

For questions 1 – 4, use the following scenario to write, graph, and find solutions to a system of inequalities.

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.

Step 1 - Identify the Variables.

Let $x=$ the number of small photos.

Let $y=$ the number of large photos.

Step 2 - To find the system of equations, translate the information. She wants to have at least 25 photos. The number of small plus the number of large should be at least 25.

$x+y\ge 25$

$4 for each small and $10 for each large must be no more than $200.

$4x+10y\le 200$

To find the system of equations, translate the information.

She wants to have at least 25 photos.

The number of small plus the number of large should be at least 25.

$x+y\ge 25$

$4 for each small and $10 for each large must be no more than $200.

$4x+10y\le 200$

Step 3 - Identify constraints based on the context.

The number of small photos must be greater than or equal to 0.

$x\ge 0$

The number of large photos must be greater than or equal to 0.

$y\ge 0$

Step 4 - Write the system of inequalities.

We have our system of equations.

$\{\begin{array}{c}x+y\ge 25\hfill \\ 4x+10y\le 200\hfill \\ x\ge 0\hfill \\ y\ge 0\hfill \end{array}$

2. Graph the system

Since $x\ge 0$ and $y\ge 0$ (both are greater than or equal to zero), all solutions will be in the first quadrant. As a result, our graph shows only quadrant one.

To graph $x+y\ge 25$, graph $x+y=25$ as a solid line. Choose $(0,0)$ as a test point. Since it does not make the inequality true, shade (red) the side that does not include the point $(0,0)$.

To graph $4x+10y\le 200$, graph $4x+10y=200$ as a solid line. Choose $(0,0)$ as a test point. Since it does make the inequality true, shade (blue) the side that includes the point $(0,0)$.

3. Could she display 10 small and 20 large photos?

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 $(25,0)$ to $(50,0)$.

To determine if 10 small and 20 large photos would work, we look at the graph to see if the point $(10,20)$ is in the solution region. We could also test the point to see if it is a solution of both equations.

It is not, so Christy would not display 10 small and 20 large photos.

4. Could she display 20 large and 10 small photos?

To determine if 20 small and 10 large photos would work, we look at the graph to see if the point $(20,10)$ is in the solution region. We could also test the point to see if it is a solution of both equations.

It is, so 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.