Exercise 3 Page 402

There are two constraints in this problem. The first being the budget and the second being the amount of fruit needed.

Fruit Constraint

Let $c$ represent the $\text{cherries}$ and $g$ the $\text{grapes}.$ We need to buy at least $4\text{ lb}$ of fruit. We can express the total amount of fruit we buy as an inequality.

Budget Constraint

The $\text{cherries}$ cost $\$4/\text{lb}$ and the $\text{grapes}$ set us back $\$2.50/\text{lb}.$ Let's write an expression for the total amount of money we spend on fruit. We also know that our total spending cannot exceed $\$15,$ which can be described as the following inequality.

System of Inequalities

Combining our inequalities, we get a system of inequalities. To graph this we need to identify the boundary lines. To do that we have to solve each of them for $g.$ Let's start with the first inequality. We have a slope of $\text{-}1.6$ and a $y\text{-}$intercept of $6.$ Let's draw the graph of a linear function with these characteristics.

The inequality sign tells us that $g$ is less than or equal to the line. Therefore, we know that our boundary is a solid line and that we should shade the area below the line. When graphing we must keep in mind that $c$ and $g$ cannot take negative values.

Let's continue with the second inequality. The boundary line has a slope of $\text{-}1$ and a $y\text{-}$intercept of $4.$ We know that $g$ is greater than or equal to the line. The boundary line must then be a solid line and we need to shade the area above the line.

The common solutions for the given system lie where the shaded regions overlap. They are bounded by the boundary lines and the vertical axis.