Question

David and Samantha decide to be flexible again and decide to go to the brand-new convenience store down the street. The chocolate bars are slightly more expensive at $1.25 each, but the lollipops are cheaper at $0.30 each.

Question 1

Write a linear inequality that represents how many chocolate bars and lollipops David and Samantha could possibly buy at the new convenience store if they put their money together. Use a for the number of chocolate bars and b for the number of lollipops.

Question 2

Write a system of linear inequalities that represents the combinations of candy that they could purchase at both stores.

Question 3

Graph the solution set to the system of linear inequalities on a coordinate grid.

A blank graph paper with x and y axes.

Answer 1

I’m missing two pieces of information needed to give a fully numeric answer: (1) the total amount of money David and Samantha have together, and (2) the prices at the other (first) store. I can give the general inequalities and explain how to graph the solution set; if you tell me the total money and the other store’s prices I will produce the exact numeric system and a plotted graph.

General answers:

Question 1
Let a = number of chocolate bars and b = number of lollipops. If their total money is T dollars, the cost at the new convenience store gives the linear inequality
1.25a + 0.30b ≤ T,
with a ≥ 0 and b ≥ 0 (and if you require whole items, a and b should be integers).

Question 2
Let the prices at the other store be Pc dollars per chocolate and Pl dollars per lollipop (those values come from the first store). The system describing what they can buy at both stores is
Pc a + Pl b ≤ T
1.25 a + 0.30 b ≤ T
together with a ≥ 0, b ≥ 0 (and optionally integer constraints).

Question 3 — how to graph the solution set
- Treat a as the x-axis and b as the y-axis.
- For each inequality, graph the corresponding boundary line by converting to equality.
Example for the new store: 1.25a + 0.30b = T.
Find intercepts: when b = 0, a = T/1.25; when a = 0, b = T/0.30.
Plot those two intercept points and draw the straight line between them.
- Since each inequality is “≤ T,” shade the region below and/or to the left of each line (the side that includes the origin because a = 0, b = 0 always satisfies the ≤ inequality).
- The feasible combinations purchasable at both stores are the intersection of the two shaded regions (and also restricted to a ≥ 0, b ≥ 0). If you require integer numbers of items, restrict the solution set to lattice points in that intersection.
- If you want, give me the numeric value of T and the other store’s prices and I’ll (a) write the numeric system and (b) draw or produce the plotted region for you.