Graphing Systems of Linear Inequalities: example 5.57

1. ⓐ Let $x=$ the number of small photos.
   $y=$ the number of large photos
   To find the system of inequalities, translate the information.
   $\begin{array}{ccccc}& & & & \text{She wants to have at least 25 photos.}\hfill \\ & & & & \text{The number of small plus the number of large should be at least 25.}\hfill \\ & & & & \hfill x+y\ge 25\hfill \\ & & & & \text{\$4 for each small and \$10 for each large must be no more than \$200}\hfill \\ & & & & \hfill 4x+10y\le 200\hfill \end{array}$
   We have our system of inequalities. $\{\begin{array}{c}x+y\ge 25\hfill \\ 4x+10y\le 200\hfill \end{array}$
2. ⓑ
   

   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 the side that does not include the point (0, 0) red. To graph $4x+10y\le 200$, graph 4x + 10y = 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.