Exercise 38 Page 405

First we will identify the constraints posed in the problem so we can write them as a system of inequalities.

System Constraints

Let $x$ be the number of $\text{smaller}$ pieces of art, which are $8.5$ inches wide, and $y$ be the number of $\text{larger}$ pieces, which are $11$ inches wide. There are two constraints posed in this problem.

1. The number of pieces to be posted.
2. The width of the wall.

The number of pieces to be posted on the wall should be at least $16.$ We can write this as an inequality. The second constraint concerns the width of the wall and total sum of the widths of the pieces that can fit on it. Since we must leave $3$ inches of space to the left of each piece, we will need a total of $11.5$ inches for the $\text{smaller}$ pieces and $14$ inches for the $\text{larger}$ pieces. Note that we only need to concern ourselves with the space on one side of each piece, since factoring in the space on both sides would cause us to double count each gap. Additionally, the wall itself is only $20$ feet long, or $240$ inches, and we cannot use more space than that. This introduces the second inequality. Combining these two constraints, we have the following system of inequalities.

Graphing the System

To graph the system of inequalities we will first isolate $y$ in both of the equations which will allow us to graph our boundary lines. We will start by drawing the first inequality. By drawing $y=\text{-}x+16,$ we get the boundary line. Also, since the inequality symbol is $\ge$ we include the boundary line in the solutions and shade everything above it. When graphing we must keep in mind that $x$ and $y$ only can take positive values.

Continuing with the second inequality, by drawing the equation $y=\text{-}\frac{23}{28}x+\frac{120}{7}$ we get the boundary line. Also, since the inequality symbol is $\le ,$ we include the boundary line in the solutions and shade everything below it.

The common solutions for the given system lie in the region where the shaded regions overlap, including the boundary lines. However, because $x$ and $y$ stand for the number of pieces of art, they can only be whole numbers. We need to only consider integer pairs contained within the overlapping shaded region. A few examples are presented below.