Question

You have S12 to spend on fruit for a meeting. Grapes cost \(\$ 1\) per pound and peaches cost \(\$ 1.50\) per pound. Let \(x\) represent the number of pounds of grapes you can buy. Let \(y\) represent the number of pounds of peaches you can buy. Write and graph an inequality to model the amounts of grapes and peaches you can buy.

Short answer

You formulate the inequality as \(x + 1.5y \leq 12\). When graphed, this will be a downward sloping line with shading below, indicating all the possible combinations of grapes and peaches that can be bought with $12.

Step-by-step solution

Formulate the inequality

The money spent on grapes and peaches can not exceed $12. Therefore, we formulate the inequality as follows: \(x \cdot 1 + y \cdot 1.5 \leq 12\) where \(x\) is the number of pounds of grapes and \(y\) is the number of pounds of peaches.

Simplify the inequality

We simplify the inequality to its simplest form. As multiplying by 1 does not change a number, we can write the inequality as: \(x + 1.5y \leq 12\)

Graph the inequality

To graph this inequality, first plot it as if it were an equation (using solid line, since the inequality allows equal): \(x + 1.5y = 12\). Then, shade the side of the line where the inequality is true. Since this is a less than or equal to inequality, the shading should fall on the area below the line. This represents all possible combinations of grapes (x) and peaches (y) that could be bought with $12.